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f:C 13:["$","$L1a",null,{"lessons":[{"id":"solve-an-equation","title":"Solve an equation","skill":"equation","problem":"What value of $x$ is the solution to the equation below?\n$3\\left(x-4\\right)+7=\\frac{x}{2}+5$","introduction":"Give each side of the equation its own graph. Where the two graphs cross, the sides are equal, and that point's x-coordinate solves the equation.","previewExpressions":["y=3(x-4)+7","y=x/2+5"],"window":{"left":-2,"right":10,"bottom":-6,"top":14},"steps":[{"kind":"think","title":"Where will the graph show the solution?","instruction":"You’ll graph the left side as one line and the right side as another.","options":[{"text":"Where the two graphs cross","correct":true,"feedback":"At the crossing, both sides have the same height, so that point's $x$ makes the equation true."},{"text":"Where the left side's graph crosses the x-axis","feedback":"There the left side equals $0$. The solution is where the left side equals the right side, which is where the two graphs meet."},{"text":"Where the left side's graph crosses the y-axis","feedback":"That point only shows $x=0$. The solution is where the left side equals the right side, so look where the two graphs meet."}],"hint":"You’ll graph the left side as one line and the right side as another.","success":"At the crossing, both sides have the same height, so that point's $x$ makes the equation true."},{"kind":"expression","row":0,"expression":"y=3(x-4)+7","title":"Graph the left side","instruction":"In input box 1, type $y=$ followed by the left side only. The line's height at each $x$ is what the left side equals there.","hint":"Type $y=3(x-4)+7$. Leave out the equals sign and everything after it.","success":"That's the left side. Next, the right side gets a graph of its own."},{"kind":"expression","row":1,"expression":"y=x/2+5","title":"Graph the right side","instruction":"In input box 2, type $y=$ followed by the right side. The slash starts the fraction, and the right arrow leaves it before you type $+5$.","hint":"Type $y=x/2$, press the right arrow, then type $+5$.","success":"Both sides are graphed. Where they cross, $3\\left(x-4\\right)+7$ and $\\frac{x}{2}+5$ are equal."},{"kind":"intersection","points":[{"x":4,"y":7}],"title":"Select the crossing","instruction":"Click the point where the two lines cross. Desmos shows its coordinates as $\\left(x,y\\right)$.","hint":"Look for the small point where the steeper line crosses the flatter one.","success":"The lines cross at $\\left(4,7\\right)$. The first coordinate is $x$; the second is $y$."},{"kind":"think","title":"What does the crossing at $\\left(4,7\\right)$ tell you?","instruction":"One line is the left side, $3\\left(x-4\\right)+7$. The other is the right side, $\\frac{x}{2}+5$.","options":[{"text":"When $x=4$, both sides equal $7$","correct":true,"feedback":"Both lines are at height $7$ when $x=4$, so the two sides are equal there. That makes $x=4$ the solution."},{"text":"The solution is $x=7$","feedback":"$$7$ is the height where the lines meet, the value both sides share. The crossing's x-coordinate is $4$."},{"text":"The equation has two solutions, $4$ and $7$","feedback":"A point has one x-value. $4$ is the solution; $7$ is what both sides equal when $x$ is $4$."}],"hint":"One line is the left side, $3\\left(x-4\\right)+7$. The other is the right side, $\\frac{x}{2}+5$.","success":"Both lines are at height $7$ when $x=4$, so the two sides are equal there. That makes $x=4$ the solution."},{"kind":"answer","values":[4],"coordinate":"x","answerLabel":"$$x=$","title":"Give the x-coordinate as the solution","instruction":"The question asks for $x$.\n\nUse the first coordinate of the crossing.","hint":"In $\\left(4,7\\right)$, $x$ is $4$. The $7$ is the height both lines share.","success":"With $x=4$, both sides equal $7$:\n$$\\begin{aligned}\\text{Left: }&\\quad3(4-4)+7=7\\\\\\text{Right: }&\\quad\\frac{4}{2}+5=7\\end{aligned}$$"}],"takeaway":"Graph each side of an equation as its own line. The x-coordinate of each crossing is a solution.","result":"$$x=4$","guideHref":"/reference/sat_math/desmos-mastery/graphing-linear-equations"},{"id":"solve-a-system","title":"Solve a system","skill":"system","problem":"What is the solution $\\left(x,y\\right)$ to the system of equations below?\n$y=2x+1$\n$y=-x+7$","introduction":"A system asks for values that satisfy both equations. Graph the two lines to find the point they share.","previewExpressions":["y=2x+1","y=-x+7"],"window":{"left":-5,"right":9,"bottom":-3,"top":11},"steps":[{"kind":"expression","row":0,"expression":"y=2x+1","title":"Graph the first equation","instruction":"Enter $y=2x+1$ in input box 1. Every point on this line satisfies the first equation.","hint":"Type $y=2x+1$ into the highlighted expression row.","success":"The first line is visible. Add the second to see where both equations hold."},{"kind":"expression","row":1,"expression":"y=-x+7","title":"Graph the second equation","instruction":"Enter $y=-x+7$ in input box 2. Keep the minus sign before x.","hint":"Type $y=-x+7$. This line slopes downward because its x-coefficient is negative.","success":"The lines meet at one point. That point satisfies both equations."},{"kind":"intersection","points":[{"x":2,"y":5}],"title":"Select the point both lines share","instruction":"Select the point where the lines cross. This question asks for both coordinates of the solution.","hint":"Select the small point where the rising and falling lines meet.","success":"The point $\\left(2,5\\right)$ lies on both lines: putting in $x=2$ gives $y=5$ in each equation."},{"kind":"answer","values":[2,5],"coordinate":"pair","answerLabel":"$$\\left(x,y\\right)=$","title":"Give both coordinates","instruction":"Enter the solution as an ordered pair, with x first and y second.","hint":"Type $\\left(2,5\\right)$. This question requests the ordered pair.","success":"The solution is $\\left(2,5\\right)$: $x=2$ and $y=5$ make both equations true."}],"takeaway":"A system's solution is the point on both lines. When the question asks for $\\left(x,y\\right)$, give both of its coordinates.","result":"$$\\left(x,y\\right)=\\left(2,5\\right)$","guideHref":"/reference/sat_math/algebra-foundations/systems-of-equations"},{"id":"find-both-solutions","title":"Solve an equation with two solutions","skill":"intersections","problem":"What are the two solutions to the equation below?\n$x^{2}=2x+3$","introduction":"A curved graph can meet a line more than once. Each intersection gives an x-value that makes the two sides equal.","previewExpressions":["y=x^2","y=2x+3"],"window":{"left":-5,"right":7,"bottom":-3,"top":13},"steps":[{"kind":"expression","row":0,"expression":"y=x^2","keys":["y=x","^","2","→"],"title":"Graph the squared term","instruction":"Enter $y=x^2$ in input box 1. The ^ key opens an exponent; press the right arrow to leave it after typing 2.","hint":"Type y=x^2. The ^ key starts the exponent.","success":"The curved graph is a parabola. Next, compare it with the other side."},{"kind":"expression","row":1,"expression":"y=2x+3","title":"Graph the other side","instruction":"Enter $y=2x+3$ in input box 2. Look for every place the line and curve meet.","hint":"Type $y=2x+3$ in the second expression row.","success":"The line crosses the parabola twice. Both crossings matter."},{"kind":"intersection","points":[{"x":-1,"y":1},{"x":3,"y":9}],"title":"Select both intersections","instruction":"Select both crossing points to reveal their coordinates. Keep track of the first coordinate in each pair.","hint":"One crossing is to the left of the y-axis. The other is above and to the right.","success":"The points are $\\left(-1,1\\right)$ and $\\left(3,9\\right)$. Their $x$-coordinates are $-1$ and $3$."},{"kind":"answer","values":[-1,3],"coordinate":"x","answerLabel":"$$x=$","title":"Give both x-values","instruction":"Each crossing gives one solution.\n\nEnter one x-value in each box. Either order is fine.","hint":"Type -1 in one box and 3 in the other. The y-coordinates, 1 and 9, are not the values the question asks for.","success":"Both work: at $x=-1$ each side equals 1, and at $x=3$ each side equals 9."}],"takeaway":"Check the whole graph for intersections. A single equation can have more than one solution.","result":"$$x=-1$ or $x=3$","guideHref":"/reference/sat_math/desmos-mastery/finding-intersections"},{"id":"table-linear-model","title":"Match a line to its table values","skill":"table","category":"tables","problem":"The table shows three points on a line. What is the slope of the line?","problemTable":{"type":"table","altText":"x-values 1, 2, 3; y-values 5, 7, 9","headers":[{"label":"x","align":"center"},{"label":"y","align":"center"}],"rows":[[1,5],[2,7],[3,9]],"size":"compact"},"introduction":"Enter the pairs in a table, then choose the linear model. A regression finds the coefficients of a model from the data.","previewExpressions":[""],"previewTable":[{"x":"1","y":"5"},{"x":"2","y":"7"},{"x":"3","y":"9"}],"previewModel":"linear","window":{"left":-1,"right":6,"bottom":-2,"top":20},"steps":[{"kind":"table","points":[{"x":1,"y":5},{"x":2,"y":7},{"x":3,"y":9}],"marks":["The table shows"],"title":"Copy the table into Desmos","instruction":"Open the + menu and choose table, or type table in an empty input box. Copy the question's table: its x-values in the first column and its y-values in the second.","hint":"The x-values go in the first column; each matching y-value goes in the second. Press Enter to continue down a column; the table adds rows as you type.","success":"The table now plots the given pairs. The model must describe their relationship."},{"kind":"think","title":"Desmos will fit $y=mx+b$ to the table.\n\nWhich part will be the slope?","instruction":"The regression gives two numbers, $m$ and $b$.\n\nDecide which one answers the question.","options":[{"text":"$$m$, the number multiplied by $x$","correct":true,"feedback":"The slope is how much $y$ changes when $x$ goes up by $1$, and in $y=mx+b$ that is $m$."},{"text":"$$b$, the number added at the end","feedback":"$$b$ is where the line crosses the y-axis, the value of $y$ when $x=0$. The slope is $m$, the number multiplied by $x$."},{"text":"$$5$, the first y-value in the table","feedback":"$$5$ is only one point's height. The slope is how much $y$ changes each time $x$ goes up by $1$, which the regression gives as $m$."}],"hint":"The regression gives two numbers, $m$ and $b$.\n\nDecide which one answers the question.","success":"The slope is how much $y$ changes when $x$ goes up by $1$, and in $y=mx+b$ that is $m$."},{"kind":"regression","model":"linear","title":"Fit a line to the table","instruction":"Press and hold the colored icon beside the second column heading (or right-click it), then choose Add Regression. A new row starts with Linear Regression; read $m$ and $b$ under Parameters.","hint":"The slope is the coefficient of x. The constant is the value when x is zero.","success":"The regression works out the model's coefficients from your rows: $y=2x+3$."},{"kind":"answer","values":[2],"coordinate":"value","answerLabel":"Slope =","title":"Give the slope","instruction":"The slope is $m$, the regression's coefficient of $x$.\n\nEnter it.","hint":"The equation is $y=2x+3$. Each increase of $1$ in $x$ adds $2$ to $y$, so the slope is $2$.","success":"The equation is $y=2x+3$. Each increase of $1$ in $x$ adds $2$ to $y$, so the slope is $2$."}],"takeaway":"For points on a line, add a linear regression to the table. The slope is $m$, the coefficient of $x$.","result":"Slope $=2$","guideHref":"/reference/sat_math/desmos-mastery/regression-tools"},{"id":"table-quadratic-model","title":"Find the quadratic model that matches a table","skill":"table","category":"tables","problem":"The table shows four points on the graph of $y=ax^{2}+bx+c$. What is the value of $b$?","problemTable":{"type":"table","altText":"x-values 0, 1, 2, 3; y-values 3, 6, 11, 18","headers":[{"label":"x","align":"center"},{"label":"y","align":"center"}],"rows":[[0,3],[1,6],[2,11],[3,18]],"size":"compact"},"introduction":"Enter the pairs in a table, then choose the quadratic model. A regression finds the coefficients of a model from the data.","previewExpressions":[""],"previewTable":[{"x":"0","y":"3"},{"x":"1","y":"6"},{"x":"2","y":"11"},{"x":"3","y":"18"}],"previewModel":"quadratic","window":{"left":-1,"right":6,"bottom":-2,"top":20},"steps":[{"kind":"table","points":[{"x":0,"y":3},{"x":1,"y":6},{"x":2,"y":11},{"x":3,"y":18}],"marks":["The table shows"],"title":"Copy the table into Desmos","instruction":"Open the + menu and choose table, or type table in an empty input box. Copy the question's table: its x-values in the first column and its y-values in the second.","hint":"The x-values go in the first column; each matching y-value goes in the second. Press Enter to continue down a column; the table adds rows as you type.","success":"The table now plots the given pairs. The model must describe their relationship."},{"kind":"regression","model":"quadratic","title":"Fit a quadratic model to the table","instruction":"Press and hold the colored icon beside the second column heading (or right-click it), then choose Add Regression. In the new row, change Linear Regression to Quadratic Regression and read the parameters.","hint":"Choose Quadratic Regression for a model with an $x^{2}$ term.","success":"The regression works out the model's coefficients from your rows: $y=x^{2}+2x+3$."},{"kind":"answer","values":[2],"coordinate":"value","answerLabel":"$$b$ =","title":"Give the value of $b$","instruction":"Enter $b$, the coefficient of $x$ in the regression's parameters.","hint":"The equation is $y=x^{2}+2x+3$. The coefficient $b$ multiplies $x$, so $b=2$. It is not the $x^{2}$ coefficient or the constant.","success":"The equation is $y=x^{2}+2x+3$. The coefficient $b$ multiplies $x$, so $b=2$. It is not the $x^{2}$ coefficient or the constant."}],"takeaway":"Choose Quadratic Regression, then match each parameter to its place in $y=ax^{2}+bx+c$ before you answer.","result":"$$b=2$","guideHref":"/reference/sat_math/desmos-mastery/regression-tools"},{"id":"table-exponential-model","title":"Find an exponential model's percent increase","skill":"table","category":"tables","problem":"The table shows four points on the graph of $y=a\\cdot b^{x}$. By what percent does $y$ increase each time $x$ increases by $1$?","problemTable":{"type":"table","altText":"x-values 0, 1, 2, 3; y-values 4, 8, 16, 32","headers":[{"label":"x","align":"center"},{"label":"y","align":"center"}],"rows":[[0,4],[1,8],[2,16],[3,32]],"size":"compact"},"introduction":"Enter the pairs in a table, then choose the exponential model. A regression finds the coefficients of a model from the data.","previewExpressions":[""],"previewTable":[{"x":"0","y":"4"},{"x":"1","y":"8"},{"x":"2","y":"16"},{"x":"3","y":"32"}],"previewModel":"exponential","window":{"left":-1,"right":6,"bottom":-2,"top":40},"steps":[{"kind":"table","points":[{"x":0,"y":4},{"x":1,"y":8},{"x":2,"y":16},{"x":3,"y":32}],"marks":["The table shows"],"title":"Copy the table into Desmos","instruction":"Open the + menu and choose table, or type table in an empty input box. Copy the question's table: its x-values in the first column and its y-values in the second.","hint":"The x-values go in the first column; each matching y-value goes in the second. Press Enter to continue down a column; the table adds rows as you type.","success":"The table now plots the given pairs. The model must describe their relationship."},{"kind":"regression","model":"exponential","title":"Fit an exponential model to the table","instruction":"Press and hold the colored icon beside the second column heading (or right-click it), then choose Add Regression. In the new row, change Linear Regression to Exponential Regression and read the parameters.","hint":"In $y=a\\cdot b^{x}$, $b$ is the multiplier for each increase of $1$ in $x$.","success":"The regression works out the model's coefficients from your rows: $y=4\\cdot 2^{x}$."},{"kind":"answer","values":[100],"coordinate":"value","answerLabel":"Percent increase =","title":"Give the percent increase","instruction":"The factor $b$ multiplies $y$ each time $x$ goes up by $1$.\n\nTurn that factor into a percent increase and enter the number.","hint":"The model is $y=4\\cdot 2^{x}$. A factor of $2$ makes each new value $200\\%$ of the old one, which is a $100\\%$ increase.","success":"The model is $y=4\\cdot 2^{x}$. A factor of $2$ makes each new value $200\\%$ of the old one, which is a $100\\%$ increase."}],"takeaway":"In $y=a\\cdot b^{x}$, the factor $b$ sets the percent change. Subtract $1$ from $b$ and write the result as a percent.","result":"Percent increase $=100\\%$","guideHref":"/reference/sat_math/desmos-mastery/regression-tools"},{"id":"fraction-decimal","title":"Switch fractions and decimals","skill":"typing","category":"typing","problem":"What is $\\frac{3}{8}$ written as a decimal?","introduction":"A fraction and its decimal describe the same value. The conversion button changes the result's display; it leaves your expression intact.","previewExpressions":["3/8"],"window":{"left":-5,"right":10,"bottom":-4,"top":12},"steps":[{"kind":"expression","keys":"$undefined","row":0,"expression":"3/8","title":"Use a slash for division","instruction":"In input box 1, type 3/8. The slash makes the fraction $\\frac{3}{8}$, and the result appears below the expression.","hint":"Compare each sign, exponent, and pair of parentheses with the expression below.","success":"The result is 0.375. You can also display the exact fraction."},{"kind":"format","row":0,"format":"fraction","title":"Show the fraction","instruction":"Click the $\\frac{a}{b}$ button at the left end of input box 1. It switches the result below the expression to a fraction.","hint":"The button sits in the gray strip at the left end of the row.","success":"The result now shows as $\\frac{3}{8}$. Only its display changed; the value is still $0.375$."},{"kind":"format","row":0,"format":"decimal","title":"Switch back to a decimal","instruction":"Click the same conversion button again to restore the decimal result.","hint":"Use the same button for both directions.","success":"Both forms represent 0.375."},{"kind":"answer","values":[0.375],"coordinate":"value","answerLabel":"Decimal =","title":"Give the decimal","instruction":"Enter the decimal shown under input box 1.","hint":"The decimal stops after three places, so $0.375$ is exact, not rounded.","success":"The decimal stops after three places, so $0.375$ is exact, not rounded."}],"takeaway":"Choose the form that fits the question. For a repeating decimal, keep the fraction until you need to round.","result":"$$\\frac{3}{8}=0.375$","guideHref":"/reference/sat_math/desmos-mastery/fractions-and-decimals"},{"id":"roots-and-powers","title":"Type square and cube roots","skill":"typing","category":"typing","problem":"What is the value of $\\sqrt{81}+\\sqrt[3]{64}$?","introduction":"The square-root symbol gives the nonnegative number whose square equals the number inside. A cube root gives the number whose cube equals the number inside.","previewExpressions":["sqrt(81)","cbrt(64)"],"window":"$13:props:lessons:6:window","steps":[{"kind":"expression","keys":["sqrt","81","→"],"row":0,"expression":"sqrt(81)","title":"Enter the square root","instruction":"In input box 1, type sqrt, then 81. The letters immediately become a square-root sign. Press the right arrow to leave the root.","hint":"Compare each sign, exponent, and pair of parentheses with the expression below.","success":"The square root is $9$ because $9^{2}=81$. The root sign means the nonnegative root, so it is not $-9$, even though $(-9)^{2}=81$ as well."},{"kind":"expression","keys":["cbrt","64"],"row":1,"expression":"cbrt(64)","title":"Enter the cube root","instruction":"In input box 2, type cbrt, then 64. The letters become a cube-root sign with a small 3 already in place. The display should show $\\sqrt[3]{64}$.","hint":"Compare each sign, exponent, and pair of parentheses with the expression below.","success":"The cube root is 4 because $4^{3}=64$."},{"kind":"answer","values":[13],"coordinate":"value","answerLabel":"Sum =","title":"Give the sum of the two roots","instruction":"Add the square root, $9$, and the cube root, $4$.","hint":"$$9+4=13$. Keep the cube-root index separate from the number inside the root.","success":"$$9+4=13$. Keep the cube-root index separate from the number inside the root."}],"takeaway":"Type sqrt for a square root and cbrt for a cube root. Any other root uses nthroot, with the degree in its small index.","result":"$$\\sqrt{81}+\\sqrt[3]{64}=13$","guideHref":"/reference/sat_math/desmos-mastery/typing-shortcuts"},{"id":"nested-fractions","title":"Enter a complex fraction","skill":"typing","category":"typing","problem":"What is the value of the expression below?\n$\\dfrac{\\;\\dfrac{1}{2}\\;}{\\;\\dfrac{3}{4}\\;}$","introduction":"A complex fraction contains a fraction in its numerator or denominator. Parentheses make the intended grouping explicit.","previewExpressions":["(1/2)/(3/4)","1/2*4/3"],"window":"$13:props:lessons:6:window","steps":[{"kind":"expression","keys":["1/2","→","/","3/4"],"row":0,"expression":"(1/2)/(3/4)","title":"Group both fractions","instruction":"In input box 1, type $\\frac{1}{2}$, press the right arrow to leave that fraction, then type / to divide the whole fraction. In the new denominator, type $\\frac{3}{4}$. Check that the main bar separates the two fractions.","hint":"Compare each sign, exponent, and pair of parentheses with the expression below.","success":"The result is about $0.6666666667$. Check that the numerator is $\\frac{1}{2}$ and the denominator is $\\frac{3}{4}$: a fraction inside a fraction is what makes this a complex fraction."},{"kind":"think","title":"Will $\\frac{1}{2}\\cdot\\frac{4}{3}$ have the same value as input box 1?","instruction":"Input box 1 divides $\\frac{1}{2}$ by $\\frac{3}{4}$. The new expression multiplies $\\frac{1}{2}$ by $\\frac{4}{3}$, which is $\\frac{3}{4}$ flipped.","options":[{"text":"Yes, the same value","correct":true,"feedback":"Dividing by $\\frac{3}{4}$ and multiplying by $\\frac{4}{3}$ do the same thing, because $\\frac{3}{4}\\cdot\\frac{4}{3}=1$. Type the second one in input box 2 to see the match."},{"text":"No, flipping $\\frac{3}{4}$ changes the value","feedback":"Flipping alone would change it, but the division also turned into multiplication. Together the two changes keep the value, because $\\frac{3}{4}\\cdot\\frac{4}{3}=1$."},{"text":"Only if $\\frac{1}{2}$ is flipped too","feedback":"Only the number you divide by flips. Flipping $\\frac{1}{2}$ as well would give $2\\cdot\\frac{4}{3}$, which is a different number."}],"hint":"Input box 1 divides $\\frac{1}{2}$ by $\\frac{3}{4}$. The new expression multiplies $\\frac{1}{2}$ by $\\frac{4}{3}$, which is $\\frac{3}{4}$ flipped.","success":"Dividing by $\\frac{3}{4}$ and multiplying by $\\frac{4}{3}$ do the same thing, because $\\frac{3}{4}\\cdot\\frac{4}{3}=1$. Type the second one in input box 2 to see the match."},{"kind":"expression","row":1,"expression":"1/2*4/3","marks":["$$\\dfrac{\\;\\dfrac{1}{2}\\;}{\\;\\dfrac{3}{4}\\;}$"],"uses":["*"],"usesLabel":"a multiplication sign","keys":["1/2","→","*","4/3"],"title":"Check it by multiplying by the flip","instruction":"Test your prediction. In input box 2, type $\\frac{1}{2}$, press the right arrow, type *, then type $\\frac{4}{3}$.","hint":"The flip of $\\frac{3}{4}$ is $\\frac{4}{3}$. Desmos shows * as a multiplication dot.","success":"Both rows show the same value. Dividing by $\\frac{3}{4}$ gives the same result as multiplying by $\\frac{4}{3}$."},{"kind":"format","row":0,"format":"fraction","title":"Keep the exact value","instruction":"Click the $\\frac{a}{b}$ button at the left end of input box 1 to show the exact fraction instead of a rounded repeating decimal.","hint":"Click $\\frac{a}{b}$ to show the reduced fraction.","success":"The exact result is $\\frac{2}{3}$."},{"kind":"answer","values":[0.6666666666666666],"coordinate":"value","answerLabel":"Exact value =","title":"Give the exact fraction","instruction":"Enter the exact result as a fraction using a slash.","hint":"The exact value is $\\frac{2}{3}$, and both of your rows agree on it.","success":"The exact value is $\\frac{2}{3}$, and both of your rows agree on it."}],"takeaway":"Group the complete numerator and denominator. In Desmos's visual editor, check where the cursor is before adding another slash.","result":"$$\\dfrac{\\;\\dfrac{1}{2}\\;}{\\;\\dfrac{3}{4}\\;}=\\dfrac{2}{3}$","guideHref":"/reference/sat_math/desmos-mastery/typing-shortcuts"},{"id":"evaluate-a-function","workspace":"calculation","title":"Evaluate a function","skill":"function","category":"functions","problem":"The function $f$ is defined below.\n$f\\left(x\\right)=2x^{2}-3x+1$\nWhat is the value of $f\\left(4\\right)$?","introduction":"$$f\\left(4\\right)$ asks for the output when the input is 4. Define the function, then ask Desmos to evaluate it at that input.","previewExpressions":["f(x)=2x^2-3x+1","f(4)"],"window":{"left":-2,"right":6,"bottom":-3,"top":25},"steps":[{"kind":"think","marks":["$$f\\left(4\\right)$"],"title":"What does $f\\left(4\\right)$ ask for?","instruction":"Before you type the function, decide what the $4$ in $f\\left(4\\right)$ is.","options":[{"text":"The output of $f$ when the input is $4$","correct":true,"feedback":"The number in parentheses is the input, so $f\\left(4\\right)$ is the output $f$ gives for $x=4$."},{"text":"$$f$ times $4$","feedback":"Parentheses after a function name hold its input. They don't mean multiply."},{"text":"The input that makes the output $4$","feedback":"That question would be written $f\\left(x\\right)=4$. In $f\\left(4\\right)$, the $4$ is the input."}],"hint":"Before you type the function, decide what the $4$ in $f\\left(4\\right)$ is.","success":"The number in parentheses is the input, so $f\\left(4\\right)$ is the output $f$ gives for $x=4$."},{"kind":"expression","keys":"$undefined","row":0,"expression":"f(x)=2x^2-3x+1","title":"Define $f(x)=2x^2-3x+1$","instruction":"Desmos knows nothing about f until a line gives it the whole rule, everything after the equals sign.\n\nIn input box 1, type f(x)=2x^2, press the right arrow to leave the exponent, then type -3x+1.","hint":"Compare each sign, exponent, and pair of parentheses with the expression below.","success":"Desmos now knows the rule for f, so f with any input has an output."},{"kind":"expression","keys":"$undefined","row":1,"expression":"f(4)","title":"Evaluate $f(4)$","instruction":"Enter $f\\left(4\\right)$ in input box 2. The number in parentheses is the input, not a multiplier.","hint":"Compare each sign, exponent, and pair of parentheses with the expression below.","success":"The result is 21: $2\\left(4^{2}\\right)-3\\left(4\\right)+1=21$."},{"kind":"think","title":"Which point on the graph of $f$ shows $f\\left(4\\right)=21$?","instruction":"On a graph, the input is how far across a point is, and the output is how high it is.","options":[{"text":"$$\\left(4,21\\right)$","correct":true,"feedback":"Go across to the input $4$, and the graph is at height $21$, the output."},{"text":"$$\\left(21,4\\right)$","feedback":"The input comes first. $\\left(21,4\\right)$ would mean an input of $21$ gives an output of $4$."},{"text":"The point where the graph reaches height $4$","feedback":"Height is the output. Here $4$ is the input, so go across to $x=4$ and read the height there."}],"hint":"On a graph, the input is how far across a point is, and the output is how high it is.","success":"Go across to the input $4$, and the graph is at height $21$, the output."},{"kind":"answer","values":[21],"coordinate":"value","answerLabel":"$$f(4)=$","title":"Give the value of $f(4)$","instruction":"Enter the output shown under $f\\left(4\\right)$.","hint":"The point $\\left(4,21\\right)$ lies on the function's graph.","success":"The point $\\left(4,21\\right)$ lies on the function's graph."}],"takeaway":"The number inside $f(\\;)$ is an input. Define $f\\left(x\\right)$ once, then type $f$ with any given input to get its output.","result":"$$f\\left(4\\right)=21$","guideHref":"/reference/sat_math/desmos-mastery/function-notation"},{"id":"find-function-input","title":"Solve for the input that gives an output","skill":"function","category":"functions","problem":"The function $f$ is defined below.\n$f\\left(x\\right)=3x-2$\nIf $f\\left(x\\right)=10$, what is the value of $x$?","introduction":"Here the output is known and the input is missing. Graph the function and the target height to find that input.","previewExpressions":["f(x)=3x-2","y=10"],"window":{"left":-2,"right":8,"bottom":-5,"top":16},"steps":[{"kind":"think","marks":["If $f\\left(x\\right)=10$"],"title":"In $f\\left(x\\right)=10$, what is the $10$?","instruction":"Decide whether the question gives you an input of $f$ or an output of $f$.","options":[{"text":"An output of $f$","correct":true,"feedback":"$$f\\left(x\\right)$ is the output, so $10$ is an output. You need the input $x$ that produces it."},{"text":"An input of $f$","feedback":"An input goes inside the parentheses, as in $f\\left(10\\right)$. Here $10$ is what $f\\left(x\\right)$ equals, so it is an output."}],"hint":"Decide whether the question gives you an input of $f$ or an output of $f$.","success":"$$f\\left(x\\right)$ is the output, so $10$ is an output. You need the input $x$ that produces it."},{"kind":"expression","keys":"$undefined","row":0,"expression":"f(x)=3x-2","title":"Define $f(x)=3x-2$","instruction":"In input box 1, type f(x)=3x-2.","hint":"Compare each sign, exponent, and pair of parentheses with the expression below.","success":"The line shows every input-output pair for f."},{"kind":"think","title":"Which points make up the line $y=10$?","instruction":"A graph's height is its output. $y=10$ has no $x$ in it, so $x$ can be anything.","options":[{"text":"Every point at height $10$","correct":true,"feedback":"Every point on $y=10$ has height $10$. Where the graph of $f$ meets it, $f$'s output is $10$."},{"text":"Every point with an input of $10$","feedback":"Inputs run across the x-axis, and the points with input $10$ form the vertical line $x=10$. $y=10$ fixes the height."},{"text":"Only the point $\\left(10,10\\right)$","feedback":"$$y=10$ says nothing about $x$, so it holds every point at height $10$, from far left to far right."}],"hint":"A graph's height is its output. $y=10$ has no $x$ in it, so $x$ can be anything.","success":"Every point on $y=10$ has height $10$. Where the graph of $f$ meets it, $f$'s output is $10$."},{"kind":"expression","keys":"$undefined","row":1,"expression":"y=10","title":"Graph the given output","instruction":"Enter $y=10$ in input box 2: a horizontal line at height 10, the output you know.","hint":"Compare each sign, exponent, and pair of parentheses with the expression below.","success":"At the crossing, $f$ is at height 10, so its output is 10."},{"kind":"intersection","points":[{"x":4,"y":10}],"title":"Select the crossing to get the input","instruction":"The crossing is on both graphs, so it's the point where $f$'s output is $10$.\n\nSelect it. Its x-coordinate is the input that gives that output.","hint":"The point is $\\left(4,10\\right)$. The input comes first.","success":"The input 4 gives the output 10."},{"kind":"answer","values":[4],"coordinate":"value","answerLabel":"$$x=$","title":"Give the input $x$","instruction":"Enter the x-coordinate of the crossing, the input that gives an output of $10$.","hint":"$$3\\left(4\\right)-2=10$, so $x=4$.","success":"$$3\\left(4\\right)-2=10$, so $x=4$."}],"takeaway":"For $f\\left(a\\right)$, calculate an output. For $f\\left(x\\right)=b$, find the input where the graph reaches height $b$.","result":"$$x=4$","guideHref":"/reference/sat_math/desmos-mastery/function-notation"},{"id":"compose-functions","workspace":"calculation","title":"Evaluate a composed function from the inside out","skill":"function","category":"functions","problem":"The functions $f$ and $g$ are defined below.\n$f\\left(x\\right)=x^{2}+1$\n$g\\left(x\\right)=2x-1$\nWhat is the value of $f\\left(g\\left(3\\right)\\right)$?","introduction":"In $f\\left(g\\left(3\\right)\\right)$, g acts first. Its output becomes the input to f. Defining both rules keeps that order visible.","previewExpressions":["f(x)=x^2+1","g(x)=2x-1","f(g(3))"],"window":"$13:props:lessons:6:window","steps":[{"kind":"think","marks":["$$f\\left(g\\left(3\\right)\\right)$"],"title":"In $f\\left(g\\left(3\\right)\\right)$, which function acts on $3$ first?","instruction":"Read the notation from the inside out before you type anything.","options":[{"text":"$$g$ acts first","correct":true,"feedback":"$$g$ sits right next to the $3$, so it acts first. Its output then becomes the input of $f$."},{"text":"$$f$ acts first","feedback":"$$f$ is on the outside, so it acts last. The function next to the $3$ acts first, and that is $g$."}],"hint":"Read the notation from the inside out before you type anything.","success":"$$g$ sits right next to the $3$, so it acts first. Its output then becomes the input of $f$."},{"kind":"expression","keys":"$undefined","row":0,"expression":"f(x)=x^2+1","title":"Define the outside function","instruction":"In input box 1, type f(x)=x^2, press the right arrow to leave the exponent, then type +1.","hint":"Compare each sign, exponent, and pair of parentheses with the expression below.","success":"f squares its input, then adds 1."},{"kind":"expression","keys":"$undefined","row":1,"expression":"g(x)=2x-1","title":"Define the inside function","instruction":"Enter $g(x)=2x-1$ in input box 2.","hint":"Compare each sign, exponent, and pair of parentheses with the expression below.","success":"g doubles its input, then subtracts 1."},{"kind":"think","title":"If $g\\left(3\\right)=5$, what input does $f$ get?","instruction":"$$g$ has acted on $3$.\n\nDecide what $f$ works on next.","options":[{"text":"$$5$, the output of $g$","correct":true,"feedback":"$$g$'s output becomes $f$'s input, so $f\\left(g\\left(3\\right)\\right)$ is $f\\left(5\\right)$."},{"text":"$$3$, the number in the question","feedback":"$$3$ went into $g$. $f$ doesn't see the $3$; it gets what $g$ turned it into, which is $5$."},{"text":"$$x^{2}+1$, the rule for $f$","feedback":"That is what $f$ does to its input. The input itself is the number $g$ hands over: $5$."}],"hint":"$$g$ has acted on $3$.\n\nDecide what $f$ works on next.","success":"$$g$'s output becomes $f$'s input, so $f\\left(g\\left(3\\right)\\right)$ is $f\\left(5\\right)$."},{"kind":"expression","keys":"$undefined","row":2,"expression":"f(g(3))","title":"Evaluate $f(g(3))$","instruction":"Enter $f\\left(g\\left(3\\right)\\right)$ in input box 3. Desmos evaluates $g\\left(3\\right)$ first, then gives that result to f.","hint":"Compare each sign, exponent, and pair of parentheses with the expression below.","success":"$$g\\left(3\\right)=5$, then $f\\left(5\\right)=26$."},{"kind":"answer","values":[26],"coordinate":"value","answerLabel":"$$f(g(3))=$","title":"Give the value of $f(g(3))$","instruction":"Enter the final output, after both functions act.","hint":"$$f\\left(g\\left(3\\right)\\right)=f\\left(5\\right)=26$. Reversing the order would give a different result.","success":"$$f\\left(g\\left(3\\right)\\right)=f\\left(5\\right)=26$. Reversing the order would give a different result."}],"takeaway":"Read nested function notation from the inside out. The order is part of the question.","result":"$$f\\left(g\\left(3\\right)\\right)=26$","guideHref":"/reference/sat_math/desmos-mastery/function-notation"},{"id":"function-table","title":"Build a function table","skill":"function","category":"functions","problem":"The function $f$ is defined below.\n$f\\left(x\\right)=x^{2}+2$\nWhat is the greatest of the values $f\\left(-2\\right)$, $f\\left(0\\right)$, and $f\\left(3\\right)$?","introduction":"A function table calculates several outputs from one rule. Put the inputs in $x_{1}$ and the function in the second column heading.","previewExpressions":["f(x)=x^2+2"],"previewTable":[{"x":"-2","y":""},{"x":"0","y":""},{"x":"3","y":""}],"tableOutput":"f(x_1)","window":"$13:props:lessons:6:window","steps":[{"kind":"expression","keys":"$undefined","row":0,"expression":"f(x)=x^2+2","title":"Define $f$","instruction":"In input box 1, type f(x)=x^2, press the right arrow to leave the exponent, then type +2. The table uses this rule to calculate its outputs.","hint":"Compare each sign, exponent, and pair of parentheses with the expression below.","success":"The function is ready to calculate outputs."},{"kind":"table","points":[{"x":-2,"y":6},{"x":0,"y":2},{"x":3,"y":11}],"output":"f(x_1)","keys":["f(x1)"],"marks":["$$f\\left(-2\\right)$, $f\\left(0\\right)$, and $f\\left(3\\right)$"],"target":{"columns":["x_{1}","f\\left(x_{1}\\right)"],"inputs":["-2","0","3"],"typed":{"outputHeading":true,"inputs":true}},"title":"Calculate a column of outputs with $f\\left(x_{1}\\right)$","instruction":"Each row of a table's $x_{1}$ column holds one input, a single number. A second column headed $f\\left(x_{1}\\right)$ fills each row with that input's output.\n\nOpen the + menu and choose table, replace the heading $y_{1}$ with f(x1), then type $-2$, $0$, and $3$ down the $x_{1}$ column, one per row.","hint":"The heading's keys are f(x1). The 1 becomes a subscript by itself. Each $x_{1}$ cell takes one number, such as $-2$, never a point.","success":"Each output sits beside its input: $f\\left(-2\\right)=6$, $f\\left(0\\right)=2$, and $f\\left(3\\right)=11$."},{"kind":"think","title":"Which column holds the values the question compares?","instruction":"The question asks for the greatest of $f\\left(-2\\right)$, $f\\left(0\\right)$, and $f\\left(3\\right)$.\n\nDecide where those three values are in your table.","marks":["greatest of the values $f\\left(-2\\right)$, $f\\left(0\\right)$, and $f\\left(3\\right)$"],"options":[{"text":"The output column, $f\\left(x_{1}\\right)$","correct":true,"feedback":"$$f\\left(3\\right)$ is the output for the input $3$, so all three values are outputs, in the $f\\left(x_{1}\\right)$ column."},{"text":"The input column, $x_{1}$","feedback":"$$-2$, $0$, and $3$ are the inputs inside the parentheses. $f\\left(3\\right)$ is the output for the input $3$, and outputs fill the other column."}],"hint":"The question asks for the greatest of $f\\left(-2\\right)$, $f\\left(0\\right)$, and $f\\left(3\\right)$.\n\nDecide where those three values are in your table.","success":"$$f\\left(3\\right)$ is the output for the input $3$, so all three values are outputs, in the $f\\left(x_{1}\\right)$ column."},{"kind":"answer","values":[11],"coordinate":"value","answerLabel":"Greatest output =","title":"Give the greatest of the three outputs","instruction":"Compare the three numbers in the $f\\left(x_{1}\\right)$ column and enter the greatest one.","hint":"The greatest output is $11$, on the row where $x_{1}=3$, so $f\\left(3\\right)=11$ is the greatest value.","success":"The greatest output is $11$, on the row where $x_{1}=3$, so $f\\left(3\\right)=11$ is the greatest value.","retry":"That isn't the greatest output. The question compares $f\\left(-2\\right)$, $f\\left(0\\right)$, and $f\\left(3\\right)$, the three numbers in the $f\\left(x_{1}\\right)$ column."}],"takeaway":"A calculated column keeps one rule attached to every input. Match each output to its own row.","result":"$$f\\left(3\\right)=11$","guideHref":"/reference/sat_math/desmos-mastery/tables"},{"id":"read-a-maximum","title":"Find the maximum value of a function","skill":"intersections","category":"graphing","problem":"The function $f$ is defined below.\n$f\\left(x\\right)=-\\left(x-3\\right)^{2}+9$\nWhat is the maximum value of $f\\left(x\\right)$?","introduction":"This parabola opens downward, so its vertex is its highest point. The maximum value is the output at that point.","previewExpressions":["f(x)=-(x-3)^2+9"],"window":{"left":-2,"right":8,"bottom":-5,"top":12},"steps":[{"kind":"think","marks":["maximum value of $f\\left(x\\right)$"],"title":"Which coordinate of the highest point is the maximum value?","instruction":"The graph's highest point will have two coordinates.\n\nDecide which one the question asks for.","options":[{"text":"The y-coordinate, the greatest output","correct":true,"feedback":"$$f\\left(x\\right)$ is an output, so its maximum value is the y-coordinate of the highest point."},{"text":"The x-coordinate, where the maximum happens","feedback":"The x-coordinate tells you where the maximum occurs. The maximum value itself is the output there, the y-coordinate."}],"hint":"The graph's highest point will have two coordinates.\n\nDecide which one the question asks for.","success":"$$f\\left(x\\right)$ is an output, so its maximum value is the y-coordinate of the highest point."},{"kind":"expression","keys":"$undefined","row":0,"expression":"f(x)=-(x-3)^2+9","title":"Graph $f(x)=-(x-3)^2+9$","instruction":"In input box 1, type f(x)=-(x-3)^2, press the right arrow to leave the exponent, then type +9. Keep the minus sign outside the squared parentheses.","hint":"Compare each sign, exponent, and pair of parentheses with the expression below.","success":"The curve opens downward. Its turning point gives the greatest output."},{"kind":"intersection","points":[{"x":3,"y":9}],"title":"Select the highest point","instruction":"Select the curve to reveal its key points, then select the vertex at the top of the parabola.","hint":"Select the graph point $\\left(3,9\\right)$. The first coordinate is the input; the second is the output.","success":"At the vertex $\\left(3,9\\right)$, the output is 9. The input that gives it is 3."},{"kind":"answer","values":[9],"coordinate":"value","answerLabel":"Maximum value =","title":"Give the maximum value","instruction":"The question asks for the greatest output, so give the vertex's y-coordinate.","hint":"The maximum value is 9, attained when $x=3$.","success":"The maximum value is 9, attained when $x=3$.","retry":"The vertex is $\\left(3,9\\right)$. Its output, 9, is the maximum value; 3 tells you where it occurs."}],"takeaway":"A parabola that opens downward reaches its maximum at the vertex. The maximum value is the vertex's $y$-coordinate.","result":"Maximum value = 9","guideHref":"/reference/sat_math/desmos-mastery/graphing-strategies"},{"id":"named-points","title":"Plot and name a point","skill":"function","category":"graphing","problem":"Point $A$ is at $(3,2)$. What is $2x-y$ at $A$?","introduction":"An ordered pair gives a horizontal position first and a vertical position second. A name helps you identify the point without changing its coordinates.","previewExpressions":["A=(3,2)","2*3-2"],"window":{"left":-5,"right":10,"bottom":-5,"top":15},"steps":[{"kind":"expression","row":0,"expression":"A=(3,2)","title":"Name the point $A$","instruction":"In input box 1, type $A=(3,2)$. The capital letter names the point. Hold or right-click the colored icon beside the expression, then turn on Label if it is off to show the name.","success":"The point is three units right of the origin and two units up. Its name is $A$.","hint":"Compare each sign, exponent, and pair of parentheses with the expression below."},{"kind":"expression","row":1,"expression":"2(3)-2","marks":["$$(3,2)$","$$2x-y$"],"title":"Put $A$'s coordinates into $2x-y$","instruction":"In input box 2, type 2(3)-2: that is $2x-y$ with $A$'s coordinates, $x=3$ and $y=2$.","success":"The question asks for $2x-y$, so substitute both coordinates: $2(3)-2=4$.","hint":"Compare each sign, exponent, and pair of parentheses with the expression below."},{"kind":"answer","values":[4],"coordinate":"value","answerLabel":"$$2x-y=$","title":"Give the value of $2x-y$","instruction":"Enter the value of $2x-y$, rather than the ordered pair.","success":"The question asks for $2x-y$, not the point, so the answer is a single number.","hint":"The question asks for $2x-y$, not the point, so the answer is a single number."}],"takeaway":"The point gives two values. The question tells you how to use them.","result":"$$2x-y=4$","guideHref":"/reference/sat_math/desmos-mastery/graphing-strategies"},{"id":"system-requested-quantity","title":"Use a system's solution to evaluate an expression","skill":"system","category":"graphing","problem":"The system of equations below has one solution $(x,y)$.\n$3x+2y=13$\n$5x-y=13$\nWhat is the value of $2x-y$?","introduction":"The intersection supplies both coordinates. This question asks you to combine them, so finding the intersection is an intermediate step.","previewExpressions":["3x+2y=13","5x-y=13"],"window":{"left":0,"right":7,"bottom":-5,"top":10},"steps":[{"kind":"think","marks":["What is the value of $2x-y$?"],"title":"Which of these does the question ask for?","instruction":"No calculating yet. The graph will show the point $(x,y)$ where both equations are true.\n\nPick what the last line of the question wants from it.","options":[{"text":"The point $(x,y)$ itself","feedback":"The point is only a middle step. The question asks for $2x-y$, so you'll put the point's coordinates into it."},{"text":"The value of $2x-y$ at that point","correct":true,"feedback":"Find the point first, then put its coordinates into $2x-y$."}],"hint":"No calculating yet. The graph will show the point $(x,y)$ where both equations are true.\n\nPick what the last line of the question wants from it.","success":"Find the point first, then put its coordinates into $2x-y$."},{"kind":"expression","row":0,"expression":"3x+2y=13","title":"Type $3x+2y=13$ as printed","instruction":"Enter $3x+2y=13$ in input box 1, exactly as printed. Desmos graphs it without solving for $y$, so there is no rearranging to get wrong.","success":"This line holds every point that makes the first equation true.","hint":"Compare each sign, exponent, and pair of parentheses with the expression below."},{"kind":"expression","row":1,"expression":"5x-y=13","title":"Type $5x-y=13$ as printed","instruction":"Enter $5x-y=13$ in input box 2, exactly as printed.","success":"The two lines cross once, so the system has one solution.","hint":"Compare each sign, exponent, and pair of parentheses with the expression below."},{"kind":"intersection","points":[{"x":3,"y":2}],"title":"Select the point that makes both equations true","instruction":"Select the crossing point.","hint":"The lines meet above $x=3$.","success":"The intersection is $(3,2)$. Both original equations are true there."},{"kind":"answer","values":[4],"coordinate":"value","answerLabel":"$$2x-y=$","title":"Give the value of $2x-y$","instruction":"Calculate $2x-y$ from the intersection's coordinates.","success":"With $x=3$ and $y=2$, $2x-y=2(3)-2=4$.","hint":"With $x=3$ and $y=2$, $2x-y=2(3)-2=4$."}],"takeaway":"Read the requested expression before submitting a coordinate.","result":"$$2x-y=4$","guideHref":"/reference/sat_math/desmos-mastery/finding-intersections"},{"id":"percent-discount","title":"Calculate a percent decrease","problem":"An item priced at $72$ dollars is discounted by $15\\%$. What is the sale price in dollars?","skill":"typing","category":"typing","workspace":"calculation","introduction":"Subtract the discount from the price in one input box: type 72-15%72. The % key inserts “of” for you. Desmos finds $15\\%$ of $72$ before subtracting it.","previewExpressions":["72-15%of72"],"window":"$13:props:lessons:14:window","steps":[{"kind":"think","marks":["discounted by $15\\%$"],"title":"Will the sale price be more or less than $72$ dollars?","instruction":"Before you type, decide what the discount does to the price.","options":[{"text":"Less than $72$ dollars","correct":true,"feedback":"A discount takes $15\\%$ of the price away, so the sale price is $72$ minus $15\\%$ of $72$."},{"text":"More than $72$ dollars","feedback":"A discount lowers a price. Adding $15\\%$ would be a markup, not a sale."}],"hint":"Before you type, decide what the discount does to the price.","success":"A discount takes $15\\%$ of the price away, so the sale price is $72$ minus $15\\%$ of $72$."},{"kind":"expression","row":0,"expression":"72-15%of72","marks":["priced at $72$ dollars","discounted by $15\\%$"],"title":"Subtract $15\\%$ of $72$ from $72$","instruction":"Subtract the discount from the price in one input box: type 72-15%72. The % key inserts “of” for you. Desmos finds $15\\%$ of $72$ before subtracting it.","success":"The sale price is $61.20$ dollars. The $10.80$ dollar discount is a different quantity.","hint":"Compare each sign, exponent, and pair of parentheses with the expression below."},{"kind":"answer","values":[61.2],"coordinate":"value","answerLabel":"Sale price =","title":"Give the sale price","instruction":"Enter the sale price shown under input box 1.","success":"The sale price is $61.20$ dollars. The $10.80$ dollar discount is a different quantity.","hint":"The sale price is $61.20$ dollars. The $10.80$ dollar discount is a different quantity."}],"takeaway":"For a percent off, subtract the percent of the price from the price in one input box. Give the sale price, not the discount.","result":"Sale price $=61.20$ dollars","guideHref":"/reference/sat_math/desmos-mastery/percents-rates-and-units"},{"id":"convert-speed","title":"Convert units in one calculation","problem":"A cyclist rides at $54$ kilometers per hour. What is this speed in meters per second?","skill":"typing","category":"mixed","workspace":"calculation","introduction":"In input box 1, type 54*1000/3600. Multiplying by $1{,}000$ turns kilometers into meters, and dividing by $3{,}600$ turns hours into seconds.","previewExpressions":["54*1000/3600"],"window":"$13:props:lessons:14:window","steps":[{"kind":"think","marks":["What is this speed in meters per second?"],"title":"Which tool turns $54$ kilometers per hour into meters per second?","instruction":"Pick the Desmos tool before you work out any numbers.","options":[{"text":"Type the whole conversion as one calculation","correct":true,"feedback":"The speed is known and only its units change, so one line of arithmetic does it."},{"text":"Graph both sides of an equation and click the crossing","feedback":"A crossing solves for an unknown. Nothing is unknown here: the speed is given, and only its units change."},{"text":"Enter a table and fit a regression","feedback":"A regression finds a model from several data points. This question gives one speed, not a set of data."}],"hint":"Pick the Desmos tool before you work out any numbers.","success":"The speed is known and only its units change, so one line of arithmetic does it."},{"kind":"think","marks":["meters per second"],"title":"Will the speed in meters per second be more or less than $54$?","instruction":"Decide which way each unit change pushes the number before you calculate.","options":[{"text":"Less than $54$","correct":true,"feedback":"A kilometer has $1{,}000$ meters, but an hour has $3{,}600$ seconds. Dividing by $3{,}600$ outweighs multiplying by $1{,}000$, so the number shrinks."},{"text":"More than $54$","feedback":"Meters push the number up by a factor of $1{,}000$, but seconds pull it down by a factor of $3{,}600$. The pull down is bigger."}],"hint":"Decide which way each unit change pushes the number before you calculate.","success":"A kilometer has $1{,}000$ meters, but an hour has $3{,}600$ seconds. Dividing by $3{,}600$ outweighs multiplying by $1{,}000$, so the number shrinks."},{"kind":"expression","row":0,"expression":"54*1000/3600","marks":["$$54$ kilometers per hour","meters per second"],"title":"Multiply by $1{,}000$ and divide by $3{,}600$","instruction":"In input box 1, type 54*1000/3600. Multiplying by $1{,}000$ turns kilometers into meters, and dividing by $3{,}600$ turns hours into seconds.","success":"Both units changed: kilometers became meters, and hours became seconds.","hint":"Compare each sign, exponent, and pair of parentheses with the expression below."},{"kind":"answer","values":[15],"coordinate":"value","answerLabel":"Speed =","title":"Give the speed in meters per second","instruction":"Enter the speed in meters per second shown under input box 1.","success":"Both units changed: kilometers became meters, and hours became seconds.","hint":"Both units changed: kilometers became meters, and hours became seconds."}],"takeaway":"To convert a rate, multiply by the factor that changes its top unit and divide by the factor that changes its bottom unit, all in one line.","result":"Speed $=15$ meters per second","guideHref":"/reference/sat_math/desmos-mastery/percents-rates-and-units"},{"id":"exponential-interval","title":"Match the exponent to the time interval","problem":"The population of a town $t$ years from now is modeled by the function below.\n$P(t)=500(0.8)^{t/3}$\nWhat is the population $6$ years from now?","skill":"typing","category":"mixed","workspace":"calculation","introduction":"The factor $0.8$ applies every three years.\n\nIn input box 1, type 500*(0.8)^(6/3, press the right arrow to leave the fraction, then type ) to close the exponent. The screen shows $500(0.8)^{6/3}$.","previewExpressions":["500*(0.8)^(6/3)"],"window":"$13:props:lessons:14:window","steps":[{"kind":"think","marks":["What is the population $6$ years from now?"],"title":"Which tool gives the population $6$ years from now?","instruction":"Pick the Desmos tool before you work out any numbers.","options":[{"text":"Put $t=6$ into the model and calculate","correct":true,"feedback":"The model is given and $6$ is the input $t$, so calculating the model at $t=6$ answers the question."},{"text":"Graph the model and find where it reaches a height of $6$","feedback":"That finds the time when the population is $6$. Here $6$ is a number of years, an input, so it goes into the model."},{"text":"Enter a table and fit an exponential regression","feedback":"A regression finds a model from data. This question already gives the model."}],"hint":"Pick the Desmos tool before you work out any numbers.","success":"The model is given and $6$ is the input $t$, so calculating the model at $t=6$ answers the question."},{"kind":"think","marks":["$$P(t)=500(0.8)^{t/3}$"],"title":"How many times does the factor $0.8$ apply in $6$ years?","instruction":"The exponent $\\frac{t}{3}$ counts how many times the population is multiplied by $0.8$.","options":[{"text":"$$2$ times","correct":true,"feedback":"The factor applies once every $3$ years, and $6$ years hold two $3$-year intervals."},{"text":"$$6$ times","feedback":"That would be true for an exponent of $t$. The exponent is $\\frac{t}{3}$, so the factor applies once every $3$ years."}],"hint":"The exponent $\\frac{t}{3}$ counts how many times the population is multiplied by $0.8$.","success":"The factor applies once every $3$ years, and $6$ years hold two $3$-year intervals."},{"kind":"expression","row":0,"expression":"500*(0.8)^(6/3)","marks":["$$P(t)=500(0.8)^{t/3}$","$$6$ years from now"],"title":"Put $t=6$ into the model","instruction":"The factor $0.8$ applies every three years.\n\nIn input box 1, type 500*(0.8)^(6/3, press the right arrow to leave the fraction, then type ) to close the exponent. The screen shows $500(0.8)^{6/3}$.","success":"Six years is two three-year intervals, so $500$ is multiplied by $0.8$ twice. That is a $36\\%$ decrease, because $0.8^2=0.64$.","hint":"Compare each sign, exponent, and pair of parentheses with the expression below."},{"kind":"answer","values":[320],"coordinate":"value","answerLabel":"Population =","title":"Give the population after $6$ years","instruction":"Enter the population after $6$ years shown under input box 1.","success":"Six years is two three-year intervals, so $500$ is multiplied by $0.8$ twice. That is a $36\\%$ decrease, because $0.8^2=0.64$.","hint":"Six years is two three-year intervals, so $500$ is multiplied by $0.8$ twice. That is a $36\\%$ decrease, because $0.8^2=0.64$."}],"takeaway":"When the exponent is $t/3$, the factor applies once every $3$ years. Put the whole time in for $t$ and let the exponent count the intervals.","result":"Population $=320$","guideHref":"/reference/sat_math/desmos-mastery/exponential-models"},{"id":"trig-angle-units","title":"Check degrees before calculating","skill":"typing","category":"mixed","workspace":"calculation","problem":"A right triangle has hypotenuse $18$ and an acute angle of $30^\\circ$. What is the length of the side opposite that angle?","introduction":"Sine compares the opposite side to the hypotenuse. The angle is in degrees, so choose Degrees before evaluating the expression.","previewExpressions":["18sin(pi/6)"],"window":"$13:props:lessons:14:window","steps":[{"kind":"think","marks":["an acute angle of $30^\\circ$"],"title":"What must you check before typing $\\sin(30)$?","instruction":"Decide what could make the calculator read the $30$ wrong.","options":[{"text":"That the calculator is set to degrees","correct":true,"feedback":"The angle is $30^\\circ$. In radian mode, $\\sin(30)$ means the sine of $30$ radians, a different angle, so check Graph Settings first."},{"text":"Nothing, because the calculator always uses degrees","feedback":"The calculator can be set to radians, and then $\\sin(30)$ means the sine of $30$ radians. Check Graph Settings before any trig calculation."}],"hint":"Decide what could make the calculator read the $30$ wrong.","success":"The angle is $30^\\circ$. In radian mode, $\\sin(30)$ means the sine of $30$ radians, a different angle, so check Graph Settings first."},{"kind":"angle","mode":"degrees","title":"Match the calculator's angle unit","instruction":"Open Graph Settings with the wrench and choose Degrees.","hint":"Degrees should be selected in Graph Settings because the given angle is $30^\\circ$.","success":"The calculator now treats trig inputs as degrees."},{"kind":"expression","row":0,"expression":"18sin(30)","marks":["hypotenuse $18$","angle of $30^\\circ$","the side opposite"],"title":"Multiply the hypotenuse by $\\sin(30)$","instruction":"Sine is the opposite side divided by the hypotenuse, so the opposite side is the hypotenuse times $\\sin(30)$.\n\nIn input box 1, type $18\\sin(30)$. When sin opens parentheses, put $30$ inside them.","success":"Since $\\sin(30^\\circ)=1/2$, the opposite side is $18(1/2)=9$.","hint":"Compare each sign, exponent, and pair of parentheses with the expression below."},{"kind":"answer","values":[9],"coordinate":"value","answerLabel":"Opposite side =","title":"Give the opposite side's length","instruction":"Enter the length shown under input box 1.","success":"Degree mode was necessary because the angle is $30^\\circ$.","hint":"Degree mode was necessary because the angle is $30^\\circ$."}],"takeaway":"Match the angle units before using a trig result.","result":"Opposite side $=9$","guideHref":"/reference/sat_math/desmos-mastery/trigonometry-and-angle-units"},{"id":"circle-radius","title":"Find a circle's radius from its top and bottom points","skill":"intersections","category":"mixed","workspace":"graph","equalScale":true,"problem":"A circle in the xy-plane has the equation below.\n$x^2+y^2-4x+6y=12$\nWhat is the radius of the circle?","introduction":"A circle's top and bottom points sit straight above and below its center, so the distance between them is a diameter.","previewExpressions":["x^2+y^2-4x+6y=12"],"window":{"left":-6,"right":10,"bottom":-10,"top":4},"steps":[{"kind":"think","marks":["$$x^2+y^2-4x+6y=12$"],"title":"Which tool gets the radius from $x^2+y^2-4x+6y=12$?","instruction":"Pick the Desmos tool before you work out any numbers.","options":[{"text":"Graph the equation as printed and measure the circle","correct":true,"feedback":"Desmos graphs an equation in $x$ and $y$ as printed, and the circle's top and bottom points are a diameter apart."},{"text":"Solve the equation for $y$ before graphing it","feedback":"No rearranging is needed. Desmos graphs an equation in $x$ and $y$ exactly as printed."},{"text":"Add a slider for the radius","feedback":"A slider is for an unknown constant. This equation has none: its circle is already fixed, and you only need to measure it."}],"hint":"Pick the Desmos tool before you work out any numbers.","success":"Desmos graphs an equation in $x$ and $y$ as printed, and the circle's top and bottom points are a diameter apart."},{"kind":"expression","row":0,"expression":"x^2+y^2-4x+6y=12","title":"Type $x^2+y^2-4x+6y=12$ as printed","instruction":"Enter $x^2+y^2-4x+6y=12$ in input box 1, exactly as printed. Press the right arrow to leave each exponent.","success":"Desmos draws the circle, because it plots every point whose $x$ and $y$ make the equation true.","hint":"Compare each sign, exponent, and pair of parentheses with the expression below."},{"kind":"intersection","points":[{"x":2,"y":2},{"x":2,"y":-8}],"title":"Click the circle's top and bottom points","instruction":"Click the circle to show its gray points, then click the point at the top and the point at the bottom.","hint":"The top and bottom points share the same x-value.","success":"The top is $(2,2)$ and the bottom is $(2,-8)$, so they are $2-(-8)=10$ units apart. That distance runs straight through the center, so it is a diameter."},{"kind":"answer","values":[5],"coordinate":"value","answerLabel":"Radius =","title":"Give the radius","instruction":"The radius is half the distance between the top and bottom points.\n\nEnter it.","success":"Completing the squares agrees: the equation becomes $(x-2)^2+(y+3)^2=25$, and $\\sqrt{25}=5$.","hint":"Completing the squares agrees: the equation becomes $(x-2)^2+(y+3)^2=25$, and $\\sqrt{25}=5$.","retry":"The distance from the top to the bottom, $10$, is the diameter. The radius is half of it."}],"takeaway":"Graph a circle's equation as printed, click its top and bottom points, and halve the distance between them.","result":"Radius $=5$","guideHref":"/reference/sat_math/desmos-mastery/circles-and-geometry"},{"id":"quadratic-one-solution","title":"Find the constant that gives a quadratic exactly one solution","skill":"intersections","category":"constants","workspace":"graph","problem":"In the equation below, $k$ is a constant.\n$x^2-8x+k=0$\nFor what value of $k$ does the equation have exactly one real solution?","introduction":"Each real solution of $x^2-8x+k=0$ is a place where the graph of $y=x^2-8x+k$ meets the x-axis. A slider for $k$ moves the parabola up and down.","previewExpressions":["y=x^2-8x+k",""],"window":{"left":-3,"right":11,"bottom":-18,"top":10},"steps":[{"kind":"think","marks":["exactly one real solution"],"title":"How does the parabola meet the x-axis when there is exactly one real solution?","instruction":"Each real solution is a place where the graph of $y=x^2-8x+k$ meets the x-axis.\n\nDecide what one solution looks like.","options":[{"text":"It touches the x-axis at one point","correct":true,"feedback":"One meeting point is one solution, so the parabola just touches the axis, at its vertex."},{"text":"It crosses the x-axis twice","feedback":"Two crossings are two real solutions."},{"text":"It stays above the x-axis","feedback":"A parabola that never reaches the x-axis gives no real solutions."}],"hint":"Each real solution is a place where the graph of $y=x^2-8x+k$ meets the x-axis.\n\nDecide what one solution looks like.","success":"One meeting point is one solution, so the parabola just touches the axis, at its vertex."},{"kind":"expression","row":0,"expression":"y=x^2-8x+k","marks":["$$x^2-8x+k=0$"],"title":"Graph $y=x^2-8x+k$ with $k$ still in it","instruction":"In input box 1, type y=x^2, press the right arrow to leave the exponent, then type -8x+k. This graphs the equation's left side, $x^2-8x+k$.","success":"Desmos can’t draw the parabola until $k$ has a value, so it offers “add slider” under it.","hint":"Compare each sign, exponent, and pair of parentheses with the expression below."},{"kind":"slider","variable":"k","title":"Add a slider for $k$","instruction":"Click $k$ beside “add slider” under input box 1. A new line, $k=1$, appears in input box 2 with a slider under it.","hint":"You can also type $k=1$ in input box 2.","success":"At $k=1$ the parabola crosses the x-axis twice, so the equation has two real solutions."},{"kind":"intersection","points":[{"x":4,"y":-15}],"movingPoints":[{"x":"4","y":"k-16"}],"title":"Click the parabola's lowest point","instruction":"Click the parabola to show its gray points, then click its lowest point.","hint":"The lowest point sits halfway between the two x-axis crossings.","success":"The lowest point is $(4,{{k-16}})$. Raising $k$ by $1$ lifts the whole parabola by $1$, so adding ${{16-k}}$ to $k$ puts the lowest point on the x-axis."},{"kind":"slider","variable":"k","value":16,"title":"Raise $k$ by ${{16-k}}$ to lift the vertex onto the x-axis","instruction":"Change input box 2 to $k=16$, because ${{k}}{{+16-k}}=16$. Type it, since the slider only runs to $10$.","hint":"Click input box 2 and change the number after $k=$ to $16$.","success":"The vertex now sits on the x-axis at $(4,0)$, so the parabola touches the axis once."},{"kind":"answer","values":[16],"coordinate":"value","answerLabel":"$$k=$","title":"Give the value of $k$","instruction":"Enter the value of $k$ that put the vertex on the x-axis.","success":"The equation becomes $x^2-8x+16=0$, which is $(x-4)^2=0$, so its only solution is $x=4$.","hint":"The equation becomes $x^2-8x+16=0$, which is $(x-4)^2=0$, so its only solution is $x=4$.","retry":"At that value the parabola crosses the x-axis twice or misses it. Use the value that put the vertex on the axis. The solution, $4$, is not the constant."}],"takeaway":"Graph the equation's left side, add a slider for the constant, and lift or lower the vertex onto the x-axis for exactly one solution.","result":"$$k=16$","guideHref":"/reference/sat_math/desmos-mastery/parameters-and-equivalence"},{"id":"absolute-value-solutions","title":"Solve an absolute value equation with two solutions","skill":"intersections","category":"graphing","problem":"What are the solutions to the equation below?\n$|2x-5|=9$","introduction":"Graph the absolute value and the known output. Each crossing identifies an input with the requested output.","previewExpressions":["y=abs(2x-5)","y=9","abs(2x-5)=9"],"window":{"left":-5,"right":10,"bottom":-2,"top":15},"steps":[{"kind":"expression","row":0,"expression":"y=abs(2x-5)","title":"Graph the left side, $y=|2x-5|$","instruction":"In input box 1, type $y=$, then abs and $2x-5$ inside its parentheses. The function abs takes the absolute value: $y=|2x-5|$.","success":"The absolute value graph has two arms because either sign of the inside expression can give a positive output.","hint":"Compare each sign, exponent, and pair of parentheses with the expression below."},{"kind":"expression","row":1,"expression":"y=9","title":"Graph the right side, $y=9$","instruction":"Enter $y=9$ in input box 2.","success":"The two crossings show where the absolute value is $9$.","hint":"Compare each sign, exponent, and pair of parentheses with the expression below."},{"kind":"intersection","points":[{"x":-2,"y":9},{"x":7,"y":9}],"title":"Select both crossings","instruction":"Select the crossing on each arm of the graph.","hint":"One input is negative; the other is positive.","success":"The crossings are $(-2,9)$ and $(7,9)$."},{"kind":"expression","row":2,"expression":"abs(2x-5)=9","title":"Type the whole equation, $|2x-5|=9$","instruction":"Here is a second way to see the same answers.\n\nIn input box 3, type abs and $2x-5$ inside its parentheses, press the right arrow to leave them, then type $=9$.","hint":"Type abs, then 2x-5, press the right arrow, then type =9.","success":"Desmos draws two vertical lines, at $x=-2$ and $x=7$. Each one passes through a crossing you clicked."},{"kind":"think","title":"What does each vertical line show?","instruction":"The equation in input box 3 has $x$ in it but no $y$.\n\nDecide what Desmos drew.","options":[{"text":"An x-value that makes $|2x-5|=9$ true","correct":true,"feedback":"A solution such as $x=7$ works whatever $y$ is, so Desmos draws every point with $x=7$, a vertical line. Both ways give $-2$ and $7$. Graphing each side is easier to read and check, because you see both sides meet."},{"text":"The graph of $|2x-5|$","feedback":"That graph is the V from input box 1. The vertical lines come from the whole equation, and each one stands at an x-value that solves it."},{"text":"Where the V crosses the x-axis","feedback":"The V touches the x-axis only at $x=2.5$. The lines stand at $-2$ and $7$, where $|2x-5|$ equals $9$, so each is a solution."}],"hint":"The equation in input box 3 has $x$ in it but no $y$.\n\nDecide what Desmos drew.","success":"A solution such as $x=7$ works whatever $y$ is, so Desmos draws every point with $x=7$, a vertical line. Both ways give $-2$ and $7$. Graphing each side is easier to read and check, because you see both sides meet."},{"kind":"answer","values":[-2,7],"coordinate":"x","answerLabel":"$$x=$","title":"Give both solutions","instruction":"Each crossing gives one solution.\n\nEnter one x-value in each box. Either order is fine.","hint":"Use the first coordinate from each point, one in each box.","success":"Both work: putting in $-2$ or 7 makes $|2x-5|$ equal 9."}],"takeaway":"Check the whole graph. An absolute value graph has two arms, so the equation can have a solution on each one.","result":"$$x=-2$ or $x=7$","guideHref":"/reference/sat_math/desmos-mastery/absolute-radical-and-rational"},{"id":"shaded-inequality-graph","title":"Match an inequality to its shaded graph","skill":"equation","category":"choices","workspace":"graph","problem":"A solid line passes through $(0,4)$ and $(4,2)$, and the region below it is shaded. Which inequality does the graph show?","choices":["$$y\\ge -\\frac{1}{2}x+4$","$$y\\le \\frac{1}{2}x+4$","$$y<-\\frac{1}{2}x+4$","$$y\\le -\\frac{1}{2}x+4$"],"introduction":"The question describes its graph in words: a solid line through $(0,4)$ and $(4,2)$, shaded below. Type each choice and check its line, whether the line is solid or dashed, and which side is shaded.","previewExpressions":["(0,4)","(4,2)","",""],"window":{"left":-6,"right":6,"bottom":-6,"top":6},"steps":[{"kind":"expression","row":0,"expression":"y>=-1/2x+4","title":"Type choice A","instruction":"In input box 1, type y>=-1/2, press the right arrow to leave the fraction, then type x+4. The screen shows $y\\ge -\\frac{1}{2}x+4$.","hint":"Press the right arrow before x, or the x lands in the denominator.","success":"The solid line passes through $(0,4)$ and $(4,2)$, but Desmos shades above it. The question's graph is shaded below, so A is out.","eliminates":[0]},{"kind":"expression","row":1,"expression":"y<=1/2x+4","title":"Type choice B","instruction":"In input box 2, type y<=1/2, press the right arrow to leave the fraction, then type x+4. The screen shows $y\\le \\frac{1}{2}x+4$.","hint":"Press the right arrow before x, or the x lands in the denominator.","success":"This line rises from left to right, but the question's line falls from $(0,4)$ to $(4,2)$, so B is out.","eliminates":[1]},{"kind":"visibility","hide":["y>=-1/2x+4","y<=1/2x+4"],"show":[],"title":"Hide choices A and B","instruction":"Two shadings on top of each other are hard to read.\n\nClick the colored circle beside input box 1, then the one beside input box 2. Each circle turns empty and its graph disappears, while the expression stays in the list.","hint":"The circles are on the left edge of input boxes 1 and 2.","success":"A and B are still typed, just hidden, so the next choice's line and shading stand alone."},{"kind":"expression","row":2,"expression":"y<-1/2x+4","title":"Type choice C","instruction":"In input box 3, type y<-1/2, press the right arrow to leave the fraction, then type x+4. The screen shows $y<-\\frac{1}{2}x+4$.","hint":"Press the right arrow before x, or the x lands in the denominator.","success":"The shading is below the line, which matches, but the line is dashed, and the question's line is solid. C is out.","eliminates":[2]},{"kind":"expression","row":3,"expression":"y<=-1/2x+4","title":"Type choice D","instruction":"Hide choice C with its circle, the same way. In input box 4, type y<=-1/2, press the right arrow to leave the fraction, then type x+4. The screen shows $y\\le -\\frac{1}{2}x+4$.","hint":"Same keys as C, but with <= instead of <.","success":"A solid line through $(0,4)$ and $(4,2)$, shaded below."},{"kind":"choice","answer":3,"title":"Choose the inequality that matched every check","instruction":"Choose the inequality whose graph matched the line, the line style, and the shading.","hint":"A, B, and C each failed one of the checks.","success":"D is the only choice that passed all three checks.","retry":"That choice failed one of your checks. Look at which choice is still not crossed off."}],"takeaway":"Type each choice, and compare its line, whether the line is solid or dashed, and its shading with the graph the question describes.","result":"D) $y\\le -\\frac{1}{2}x+4$","guideHref":"/reference/sat_math/desmos-mastery/choosing-a-strategy"},{"id":"line-through-two-points","title":"Pick the equation of a line through two points","skill":"equation","category":"choices","workspace":"graph","problem":"Line $k$ passes through the points $(-1,1)$ and $(2,7)$. Which equation defines line $k$?","choices":["$$y=x+2$","$$y=2x+1$","$$y=2x+3$","$$y=3x+4$"],"introduction":"The two points give you something to match, just like a graph would. Plot them, then type the choices. The right line has to pass through both points, not just one.","previewExpressions":["(-1,1)","(2,7)","","",""],"window":{"left":-5,"right":5,"bottom":-3,"top":11},"steps":[{"kind":"expression","row":0,"expression":"(-1,1)","title":"Plot $(-1,1)$","instruction":"In input box 1, type (-1,1).","hint":"Type the parentheses and the comma, with x first.","success":"The point $(-1,1)$ is on the graph."},{"kind":"expression","row":1,"expression":"(2,7)","title":"Plot $(2,7)$","instruction":"In input box 2, type (2,7).","hint":"x first, then y.","success":"Both points are on the graph now."},{"kind":"think","marks":["passes through the points $(-1,1)$ and $(2,7)$"],"title":"What must the graph of the right equation do?","instruction":"Line $k$ passes through both points.\n\nDecide what to look for before you type a choice.","options":[{"text":"Pass through both $(-1,1)$ and $(2,7)$","correct":true,"feedback":"Line $k$ goes through both points, so its equation's graph has to hit both. A line that hits only one is a different line."},{"text":"Pass through at least one of the points","feedback":"Many lines pass through $(-1,1)$ alone, and most of them miss $(2,7)$. Line $k$ passes through both, so the right graph has to hit both."},{"text":"Rise from left to right","feedback":"Line $k$ does rise, since $(2,7)$ is higher and to the right, but every choice rises too. Look for the line that hits both points."}],"hint":"Line $k$ passes through both points.\n\nDecide what to look for before you type a choice.","success":"Line $k$ goes through both points, so its equation's graph has to hit both. A line that hits only one is a different line."},{"kind":"expression","row":2,"expression":"y=x+2","title":"Type choice A","instruction":"Enter $y=x+2$ in input box 3.","hint":"Type y=x+2.","success":"This line goes through $(-1,1)$ but misses $(2,7)$. Hitting one point isn't enough, so A is out.","eliminates":[0]},{"kind":"expression","row":3,"expression":"y=2x+1","title":"Type choice B","instruction":"Enter $y=2x+1$ in input box 4.","hint":"Type y=2x+1.","success":"This line misses both points, so B is out.","eliminates":[1]},{"kind":"expression","row":4,"expression":"y=2x+3","title":"Type choice C","instruction":"Enter $y=2x+3$ in input box 5.","hint":"Type y=2x+3.","success":"This line passes through both $(-1,1)$ and $(2,7)$."},{"kind":"choice","answer":2,"title":"Choose the equation of line $k$","instruction":"You haven't typed D.\n\nDecide whether you need to before you choose.","hint":"How many different lines can pass through the same two points?","success":"C is right. Two points fix exactly one line, so once C passed through both, D had to be a different line. When one choice passes every check, you can skip typing the rest.","retry":"That line misses at least one of the points. Only one line can pass through both $(-1,1)$ and $(2,7)$, and you've already graphed it."}],"takeaway":"When the question gives points and the choices are equations, plot the points and keep the line that passes through all of them.","result":"C) $y=2x+3$","guideHref":"/reference/sat_math/desmos-mastery/choosing-a-strategy"},{"id":"exponents","title":"Type an exponent, then step out of it","skill":"typing","category":"typing","workspace":"calculation","problem":"What is the value of $2^{5}+8^{\\frac{2}{3}}$?","introduction":"The ^ key opens an exponent. After typing the exponent, press the right arrow to return to the main line before typing the next term.","previewExpressions":["2^5+8^(2/3)"],"window":{"left":-5,"right":10,"bottom":-4,"top":12},"steps":[{"kind":"expression","row":0,"expression":"2^5+8^(2/3)","uses":["^"],"usesLabel":"the ^ key","keys":["2^5","→","+8^2/3","→","→"],"title":"Type both powers in input box 1","instruction":"Type 2^5, then press the right arrow to leave the exponent. Type +8^, then 2/3 inside the new exponent. Press the right arrow twice: once to leave the fraction and once to leave the exponent.","hint":"The exponent of 8 is the whole fraction $\\frac{2}{3}$.","success":"$$2^{5}=32$ and $8^{\\frac{2}{3}}=4$, so the value is $36$."},{"kind":"answer","values":[36],"coordinate":"value","answerLabel":"Sum =","title":"Give the value of $2^{5}+8^{\\frac{2}{3}}$","instruction":"Enter the result shown under input box 1.","hint":"$$32+4=36$.","success":"$$32+4=36$."}],"takeaway":"Press the right arrow after an exponent before typing the next term. A fraction inside an exponent needs two right arrows: one to leave the fraction and one to leave the exponent.","result":"$$2^{5}+8^{\\frac{2}{3}}=36$","guideHref":"/reference/sat_math/desmos-mastery/typing-shortcuts"},{"id":"nth-roots","title":"Type any root with nthroot","skill":"typing","category":"typing","workspace":"calculation","problem":"What is the value of $\\sqrt[4]{81}+\\sqrt[5]{32}$?","introduction":"For a root other than a square root or a cube root, type nthroot. The cursor starts in the small index, where the root's degree goes.","previewExpressions":["nthroot(81,4)","nthroot(32,5)"],"window":"$13:props:lessons:25:window","steps":[{"kind":"expression","row":0,"expression":"nthroot(81,4)","uses":["nthroot"],"usesLabel":"a root sign","keys":["nthroot","4","→","81"],"title":"Type the fourth root of 81","instruction":"In input box 1, type nthroot. Type 4 in the small index, press the right arrow to move under the root sign, and type 81.","hint":"The index holds 4, and 81 goes under the root sign.","success":"The fourth root of $81$ is $3$, because $3^{4}=81$."},{"kind":"expression","row":1,"expression":"nthroot(32,5)","uses":["nthroot"],"usesLabel":"a root sign","keys":["nthroot","5","→","32"],"title":"Type the fifth root of 32","instruction":"In input box 2, type nthroot again, then 5 in the small index. Press the right arrow to move under the root sign, and type 32.","hint":"The index holds 5, and 32 goes under the root sign.","success":"The fifth root of $32$ is $2$, because $2^{5}=32$."},{"kind":"answer","values":[5],"coordinate":"value","answerLabel":"Sum =","title":"Add the two roots","instruction":"Add the fourth root of $81$ and the fifth root of $32$, the two results Desmos shows.","hint":"$$3+2=5$.","success":"$$3^{4}=81$ and $2^{5}=32$, so the roots are $3$ and $2$."}],"takeaway":"Type nthroot for any root. The degree goes in the small index first, then the right arrow moves you under the root sign.","result":"$$\\sqrt[4]{81}+\\sqrt[5]{32}=5$","guideHref":"/reference/sat_math/desmos-mastery/typing-shortcuts"},{"id":"percent-of","title":"Calculate a percent of a number with % of","skill":"typing","category":"typing","workspace":"calculation","problem":"What is $35\\%$ of $240$?","introduction":"Pressing % inserts “% of” in Desmos. Type the percent, press %, then type the amount to calculate a percent of a number.","previewExpressions":["35%of240","0.35*240"],"window":"$13:props:lessons:25:window","steps":[{"kind":"expression","row":0,"expression":"35%of240","marks":["$$35\\%$ of $240$"],"uses":["%of"],"usesLabel":"“% of”","keys":["35%240"],"title":"Type 35%, then 240","instruction":"In input box 1, type 35%, then 240. The % key inserts “of” for you, so the display reads 35% of 240.","hint":"To type %, hold Shift and press 5.","success":"$$35\\%$ of $240$ is $84$."},{"kind":"expression","row":1,"expression":"0.35*240","marks":["$$35\\%$ of $240$"],"uses":["0.35"],"usesLabel":"the decimal 0.35","keys":["0.35*240"],"title":"Check $35\\%$ of $240$ against $0.35$ times $240$","instruction":"A percent counts hundredths, so $35\\%$ of a number is $0.35$ times it.\n\nIn input box 2, type 0.35*240 and compare the two rows.","hint":"Desmos shows * as a multiplication dot.","success":"Both rows show $84$. “% of” gives the same value as multiplying by the decimal, without the conversion step."},{"kind":"answer","values":[84],"coordinate":"value","answerLabel":"$$35\\%$ of $240=$","title":"Give $35\\%$ of $240$","instruction":"Enter the result shown under input box 1.","hint":"$$0.35\\times240=84$.","success":"Typing 35%240 matched multiplying by the decimal $0.35$."}],"takeaway":"To calculate 35% of 240, type 35%240. The % key inserts “of” automatically.","result":"$$35\\%$ of $240$ is $84$","guideHref":"/reference/sat_math/desmos-mastery/percents-rates-and-units"},{"id":"store-a-value","title":"Store a given value in a letter and reuse it","skill":"typing","category":"typing","workspace":"calculation","problem":"If $k=\\frac{7}{4}$, what is the value of $8k^{2}-6k$?","introduction":"Typing k=7/4 in its own input box stores that value. Every other input box that uses k then calculates with it, so you can type the expression exactly as the question prints it.","previewExpressions":["k=7/4","8k^2-6k"],"window":"$13:props:lessons:25:window","steps":[{"kind":"expression","row":0,"expression":"k=7/4","keys":["k=7/4"],"title":"Store the value of k","instruction":"In input box 1, type k=7/4. Desmos now uses $\\frac{7}{4}$ everywhere k appears.","hint":"Type the letter, an equals sign, and the fraction.","success":"k now holds $\\frac{7}{4}$, which is $1.75$."},{"kind":"expression","row":1,"expression":"8k^2-6k","uses":["k"],"usesLabel":"the stored letter k","keys":["8k^2","→","-6k"],"title":"Type the expression as printed","instruction":"In input box 2, type 8k^2, press the right arrow to leave the exponent, then type -6k.","hint":"Use the letter k instead of retyping the fraction.","success":"$$8\\left(\\frac{7}{4}\\right)^{2}-6\\left(\\frac{7}{4}\\right)=24.5-10.5=14$."},{"kind":"answer","values":[14],"coordinate":"value","answerLabel":"$$8k^{2}-6k=$","title":"Give the value of $8k^{2}-6k$","instruction":"Enter the result shown under input box 2.","hint":"$$24.5-10.5=14$.","success":"With $k=\\frac{7}{4}$ stored, Desmos did the squaring and subtracting for you."}],"takeaway":"Store a given value in a letter once, then type the expression exactly as the question prints it.","result":"$$8k^{2}-6k=14$","guideHref":"/reference/sat_math/desmos-mastery/typing-shortcuts"},{"id":"subscripts","title":"Name related values with subscripts","skill":"typing","category":"typing","workspace":"calculation","problem":"A shirt cost $p_{1}=24.50$ dollars last month and costs $p_{2}=31.75$ dollars now. By how many dollars did the price increase?","introduction":"A digit typed right after a letter becomes a subscript, so typing p1 gives $p_{1}$. Subscripts let you store several related values without running out of letters, and the statistics lessons use the same idea to name lists.","previewExpressions":["p_1=24.5","p_2=31.75","p_2-p_1"],"window":"$13:props:lessons:25:window","steps":[{"kind":"expression","row":0,"expression":"p_1=24.5","uses":["_"],"usesLabel":"the subscript in $p_{1}$","keys":["p1=24.5"],"title":"Store last month's price as $p_{1}$","instruction":"In input box 1, type p1=24.5. Desmos lowers the 1 into a subscript, $p_{1}$, and moves past it on its own, so you need no underscore and no right arrow.","hint":"Type the 1 right after the p. There is no need for the underscore key.","success":"$$p_{1}$ now holds $24.50$."},{"kind":"expression","row":1,"expression":"p_2=31.75","uses":["_"],"usesLabel":"the subscript in $p_{2}$","keys":["p2=31.75"],"title":"Store the current price as $p_{2}$","instruction":"In input box 2, type p2=31.75. The 2 becomes a subscript, as the 1 did.","hint":"Use the same letter with the digit 2.","success":"$$p_{2}$ now holds $31.75$."},{"kind":"expression","row":2,"expression":"p_2-p_1","marks":["did the price increase"],"uses":["p_2","p_1"],"usesLabel":"the two stored names","keys":["p2-p1"],"title":"Subtract $p_{1}$ from $p_{2}$","instruction":"The price increase is the current price minus last month's price, $p_{2}-p_{1}$.\n\nIn input box 3, type p2-p1.","hint":"The newer price comes first, since the question asks for an increase.","success":"$$p_{2}-p_{1}=31.75-24.50=7.25$."},{"kind":"answer","values":[7.25],"coordinate":"value","answerLabel":"Increase =","title":"Give the increase in dollars","instruction":"Enter the result shown under input box 3.","hint":"$$31.75-24.50=7.25$.","success":"The price increased by $7.25$ dollars."}],"takeaway":"Type a letter and then a digit, such as p1, and Desmos writes $p_{1}$. Names such as $p_{1}$ and $p_{2}$ keep related numbers apart, and lists named $L_{1}$ and $L_{2}$ work the same way.","result":"$$p_{2}-p_{1}=7.25$","guideHref":"/reference/sat_math/desmos-mastery/typing-shortcuts"},{"id":"mean-and-median","title":"Store a list, then calculate its median","skill":"table","category":"statistics","workspace":"calculation","problem":"What is the median of the data set $4, 4, 7, 9, 16$?","introduction":"Desmos can store a whole data set under one name. Type the values in square brackets, then ask for the median of that name. Enter every repeated value, because each one counts.","previewExpressions":["L_1=[4,4,7,9,16]","median(L_1)"],"window":{"left":-5,"right":10,"bottom":-5,"top":15},"steps":[{"kind":"think","marks":["$$4, 4, 7, 9, 16$"],"title":"The data set has $4$ twice.\n\nShould your list keep both?","instruction":"You'll type the data into Desmos as a list before asking for its median.","options":[{"text":"Yes, keep both $4$s","correct":true,"feedback":"Each $4$ is its own value, so the set has five values. Dropping one would leave four and move the middle."},{"text":"No, one $4$ is enough","feedback":"A repeated value counts every time it appears. With one $4$ the list holds four values instead of five, and its middle moves."}],"hint":"You'll type the data into Desmos as a list before asking for its median.","success":"Each $4$ is its own value, so the set has five values. Dropping one would leave four and move the middle."},{"kind":"expression","row":0,"expression":"L_1=[4,4,7,9,16]","uses":["L_1=["],"usesLabel":"a list named $L_1$","title":"Store the data as a list named $L_{1}$","instruction":"A list stores the whole data set under one name, $L_{1}$.\n\nIn input box 1, type L1=[4,4,7,9,16]: Desmos lowers the 1 into a subscript, and the square brackets hold the values.","hint":"Type the list name with its digit, such as L1, then an equals sign and the values inside square brackets, separated by commas.","success":"$$L_{1}$ now holds all five values."},{"kind":"expression","row":1,"expression":"median(L_1)","marks":["median"],"uses":["median","l_1"],"usesLabel":"the list names","title":"Calculate the median of $L_{1}$","instruction":"In input box 2, type median(L1). Desmos sorts the list and takes the middle value.","hint":"Type the function name; Desmos opens its parentheses for you.","success":"The median is $7$, the third of the five sorted values."},{"kind":"answer","values":[7],"coordinate":"value","answerLabel":"Median =","title":"Give the median","instruction":"Enter the median shown under input box 2.","hint":"The median of $4, 4, 7, 9, 16$ is $7$.","success":"The median of $4, 4, 7, 9, 16$ is $7$."}],"takeaway":"Name the data set once as a list, then find any statistic with one short function call.","result":"Median: $7$","guideHref":"/reference/sat_math/desmos-mastery/statistics-and-lists"},{"id":"compare-list-means","title":"Compare the means of two lists","skill":"table","category":"statistics","workspace":"calculation","problem":"Class A's quiz scores are $78, 85, 91, 70, 86$. Class B's quiz scores are $88, 72, 95, 80$. How many points greater is Class B's mean score than Class A's?","introduction":"Store each class in its own list. With both lists named, one input box can subtract one mean from the other.","previewExpressions":["L_1=[78,85,91,70,86]","L_2=[88,72,95,80]","mean(L_2)-mean(L_1)"],"window":"$13:props:lessons:30:window","steps":[{"kind":"expression","row":0,"expression":"L_1=[78,85,91,70,86]","uses":["L_1=["],"usesLabel":"a list named $L_1$","title":"Store Class A as $L_{1}$","instruction":"In input box 1, type L1=[78,85,91,70,86].","hint":"Type the list name with its digit, such as L1, then an equals sign and the values inside square brackets, separated by commas.","success":"$$L_{1}$ holds Class A's five scores."},{"kind":"expression","row":1,"expression":"L_2=[88,72,95,80]","uses":["L_2=["],"usesLabel":"a list named $L_2$","title":"Store Class B as $L_{2}$","instruction":"In input box 2, type L2=[88,72,95,80]. Class B has only four scores.","hint":"Type the list name with its digit, such as L1, then an equals sign and the values inside square brackets, separated by commas.","success":"$$L_{2}$ holds Class B's four scores."},{"kind":"expression","row":2,"expression":"mean(L_2)-mean(L_1)","marks":["How many points greater is Class B's mean score than Class A's"],"uses":["mean","l_2","l_1"],"usesLabel":"the list names","title":"Subtract Class A's mean from Class B's","instruction":"Class B's mean minus Class A's mean is $\\operatorname{mean}(L_{2})-\\operatorname{mean}(L_{1})$.\n\nIn input box 3, type mean(L2)-mean(L1). Class B comes first because the question asks how much greater its mean is.","hint":"Type the function name; Desmos opens its parentheses for you.","success":"Class B's mean is $83.75$ and Class A's is $82$, so the difference is $1.75$."},{"kind":"answer","values":[1.75],"coordinate":"value","answerLabel":"Difference =","title":"Give the difference","instruction":"Enter the difference shown under input box 3.","hint":"Class B's mean is $1.75$ points greater than Class A's.","success":"Class B's mean is $1.75$ points greater than Class A's."}],"takeaway":"Name each group's data as its own list, and one input box can compare any statistic of the two.","result":"Difference: $1.75$","guideHref":"/reference/sat_math/desmos-mastery/statistics-and-lists"},{"id":"range-with-max-and-min","title":"Calculate a range with max and min","skill":"table","category":"statistics","workspace":"calculation","problem":"The daily high temperatures for one week were $61, 58, 67, 72, 64, 55,$ and $69$ degrees Fahrenheit. What is the range of these temperatures, in degrees?","introduction":"The range is the largest value minus the smallest value. Desmos has no “range” function, but max and min find both ends of a list.","previewExpressions":["L_1=[61,58,67,72,64,55,69]","max(L_1)-min(L_1)"],"window":"$13:props:lessons:30:window","steps":[{"kind":"expression","row":0,"expression":"L_1=[61,58,67,72,64,55,69]","marks":["$$61, 58, 67, 72, 64, 55,$ and $69$"],"uses":["L_1=["],"usesLabel":"a list named $L_1$","title":"Store the temperatures as $L_{1}$","instruction":"In input box 1, type L1=[61,58,67,72,64,55,69].","hint":"Type the list name with its digit, such as L1, then an equals sign and the values inside square brackets, separated by commas.","success":"$$L_{1}$ holds all seven temperatures."},{"kind":"expression","row":1,"expression":"max(L_1)-min(L_1)","marks":["range"],"uses":["max","min"],"usesLabel":"the list names","title":"Subtract the smallest value from the largest","instruction":"The range is the largest value minus the smallest, $\\max(L_{1})-\\min(L_{1})$. Desmos has no range function, but max and min find the two ends of the list.\n\nIn input box 2, type max(L1)-min(L1).","hint":"Type the function name; Desmos opens its parentheses for you.","success":"The largest value is $72$ and the smallest is $55$, so the range is $17$."},{"kind":"answer","values":[17],"coordinate":"value","answerLabel":"Range =","title":"Give the range","instruction":"Enter the range shown under input box 2.","hint":"The range is $72-55=17$ degrees.","success":"The range is $72-55=17$ degrees."}],"takeaway":"Find a range by typing max of the list minus min of the list.","result":"Range: $17$ degrees","guideHref":"/reference/sat_math/desmos-mastery/statistics-and-lists"},{"id":"compare-spread","title":"Compare spread with standard deviation","skill":"table","category":"statistics","workspace":"calculation","problem":"Data set A is $10, 20, 30, 40, 50$. Data set B is $28, 29, 30, 31, 32$. Which statement about the two data sets is true?","choices":["Data set A has the greater standard deviation.","Data set B has the greater standard deviation.","The two data sets have equal standard deviations.","The two data sets have different means."],"introduction":"Standard deviation measures how far the values typically sit from the mean. Desmos calculates it with stdev, so two data sets can be compared directly.","previewExpressions":["L_1=[10,20,30,40,50]","L_2=[28,29,30,31,32]","stdev(L_1)","stdev(L_2)"],"window":"$13:props:lessons:30:window","steps":[{"kind":"expression","row":0,"expression":"L_1=[10,20,30,40,50]","uses":["L_1=["],"usesLabel":"a list named $L_1$","title":"Store data set A as $L_{1}$","instruction":"In input box 1, type L1=[10,20,30,40,50].","hint":"Type the list name with its digit, such as L1, then an equals sign and the values inside square brackets, separated by commas.","success":"$$L_{1}$ holds data set A."},{"kind":"expression","row":1,"expression":"L_2=[28,29,30,31,32]","uses":["L_2=["],"usesLabel":"a list named $L_2$","title":"Store data set B as $L_{2}$","instruction":"In input box 2, type L2=[28,29,30,31,32].","hint":"Type the list name with its digit, such as L1, then an equals sign and the values inside square brackets, separated by commas.","success":"$$L_{2}$ holds data set B."},{"kind":"expression","row":2,"expression":"stdev(L_1)","marks":["standard deviation"],"uses":["stdev","l_1"],"usesLabel":"the list names","title":"Calculate the standard deviation of $L_{1}$","instruction":"Standard deviation measures how far the values typically sit from the mean.\n\nIn input box 3, type stdev(L1).","hint":"Type the function name; Desmos opens its parentheses for you.","success":"Data set A's standard deviation is about $15.8$."},{"kind":"expression","row":3,"expression":"stdev(L_2)","marks":["standard deviation"],"uses":["stdev","l_2"],"usesLabel":"the list names","title":"Calculate the standard deviation of $L_{2}$","instruction":"In input box 4, type stdev(L2).","hint":"Type the function name; Desmos opens its parentheses for you.","success":"Data set B's standard deviation is about $1.6$, much smaller, because its values sit close to $30$."},{"kind":"choice","answer":0,"title":"Choose the true statement","instruction":"Compare the results in input boxes 3 and 4, then choose the statement they support.","hint":"The larger standard deviation belongs to the set whose values sit farther from the mean.","retry":"Compare the two stdev results. The larger one belongs to the set whose values sit farther from the mean.","success":"Both sets have a mean of $30$, so choice D is false as well."}],"takeaway":"Standard deviation compares spread, not center: these two sets share a mean of 30 but spread out very differently.","result":"Data set A has the greater standard deviation.","guideHref":"/reference/sat_math/desmos-mastery/statistics-and-lists"},{"id":"frequency-table-mean","title":"Calculate a mean from a frequency table","skill":"table","category":"statistics","workspace":"calculation","problem":"The table shows how many books each of $20$ students read last month. What is the mean number of books read?","problemTable":{"type":"table","altText":"4 students read 1 book, 6 read 2, 8 read 3, and 2 read 5","headers":[{"label":"Books read"},{"label":"Number of students"}],"rows":[[1,4],[2,6],[3,8],[5,2]],"size":"compact"},"introduction":"In a frequency table, each value counts as many times as its frequency. Store the values in $L_{1}$ and the frequencies in $L_{2}$; then total(L_1*L_2) adds every book, and total(L_2) counts every student.","previewExpressions":["L_1=[1,2,3,5]","L_2=[4,6,8,2]","total(L_1*L_2)/total(L_2)"],"window":"$13:props:lessons:30:window","steps":[{"kind":"expression","row":0,"expression":"L_1=[1,2,3,5]","marks":["how many books"],"uses":["L_1=["],"usesLabel":"a list named $L_1$","title":"Store the values as $L_{1}$","instruction":"Store the first column's values in input box 1: type L1=[1,2,3,5].","hint":"Type the list name with its digit, such as L1, then an equals sign and the values inside square brackets, separated by commas.","success":"$$L_{1}$ holds the four possible numbers of books."},{"kind":"expression","row":1,"expression":"L_2=[4,6,8,2]","marks":["each of $20$ students"],"uses":["L_2=["],"usesLabel":"a list named $L_2$","title":"Store the frequencies as $L_{2}$","instruction":"Each number in the second column is a frequency: how many students read that many books.\n\nIn input box 2, type L2=[4,6,8,2].","hint":"Type the list name with its digit, such as L1, then an equals sign and the values inside square brackets, separated by commas.","success":"$$L_{2}$ holds how many students read each number of books, $20$ students in all."},{"kind":"expression","row":2,"expression":"total(L_1*L_2)/total(L_2)","marks":["mean number of books read"],"uses":["total","l_1","l_2"],"usesLabel":"the list names","title":"Divide the total number of books by the number of students","instruction":"The mean is the number of books divided by the number of students. $L_{1}\\cdot L_{2}$ gives the books for each row, such as $3\\cdot 8=24$, and total adds up a list.\n\nIn input box 3, enter $\\dfrac{\\operatorname{total}(L_1\\cdot L_2)}{\\operatorname{total}(L_2)}$.","hint":"Type the function name; Desmos opens its parentheses for you.","success":"The students read $4+12+24+10=50$ books in all, and $\\frac{50}{20}=2.5$."},{"kind":"answer","values":[2.5],"coordinate":"value","answerLabel":"Mean =","title":"Give the mean","instruction":"Enter the mean shown under input box 3.","hint":"Averaging only the four values in the first column would give $2.75$, which ignores how many students read each amount.","success":"Averaging only the four values in the first column would give $2.75$, which ignores how many students read each amount."}],"takeaway":"For a frequency table, the mean is total(values times frequencies) divided by total(frequencies).","result":"Mean: $2.5$ books","guideHref":"/reference/sat_math/desmos-mastery/statistics-and-lists"},{"id":"zoom-out-to-a-crossing","title":"Zoom out to find a crossing off the screen","skill":"system","category":"graphing","workspace":"graph","problem":"The solution to the system of equations below is $(x,y)$.\n$y=0.4x+2$\n$21x-50y=-150$\nWhat is the value of $x$?","introduction":"A solution is a point on both lines. The starting window shows only part of each line, so the point they share can be off the screen.","previewExpressions":["y=0.4x+2","21x-50y=-150"],"window":{"left":-10,"right":10,"bottom":-6,"top":10},"steps":[{"kind":"expression","row":0,"expression":"y=0.4x+2","title":"Type $y=0.4x+2$","instruction":"Enter $y=0.4x+2$ in input box 1.","hint":"Type y=0.4x+2.","success":"This line rises $0.4$ for every unit to the right."},{"kind":"expression","row":1,"expression":"21x-50y=-150","title":"Type $21x-50y=-150$ as printed","instruction":"Enter $21x-50y=-150$ in input box 2, exactly as printed. Desmos graphs it without solving for $y$.","hint":"Type 21x-50y=-150, with a minus sign before the 150.","success":"In this window the lines look parallel and no crossing shows. Lines can still meet outside the window, so an empty view doesn't prove there is no solution."},{"kind":"intersection","points":[{"x":-50,"y":-18}],"title":"Zoom out until the lines meet, then click the crossing","instruction":"Zoom out with the minus button at the top right of the graph, or scroll, until the two lines meet. Then click the point where they cross.","hint":"The lines meet far to the lower left, near $x=-50$. Click a line to show the gray point there.","success":"The lines meet at $\\left(-50,-18\\right)$."},{"kind":"answer","values":[-50],"coordinate":"value","answerLabel":"$$x=$","title":"Give $x$, the crossing's first coordinate","instruction":"The question asks for $x$, the first coordinate of the crossing.\n\nEnter it.","hint":"In $\\left(-50,-18\\right)$, $x$ is $-50$.","retry":"Use the first coordinate of $\\left(-50,-18\\right)$. The second coordinate is y.","success":"The point fits both equations: putting in $x=-50$ and $y=-18$ makes each one true."}],"takeaway":"An empty window doesn't mean no solution. Zoom out before you decide that two lines never meet.","result":"$$x=-50$","guideHref":"/reference/sat_math/desmos-mastery/finding-intersections"},{"id":"reject-a-false-solution","title":"Solve a square root equation and reject the false solution","skill":"intersections","category":"graphing","workspace":"graph","problem":"What are all the solutions to the equation below?\n$\\sqrt{x+7}=x-5$","introduction":"Squaring both sides by hand can add a value that doesn't solve the original equation. Graphing each side as printed shows only the true solutions.","previewExpressions":["y=sqrt(x+7)","y=x-5","sqrt(2+7)","2-5"],"window":{"left":-8,"right":14,"bottom":-8,"top":8},"steps":[{"kind":"expression","row":0,"expression":"y=sqrt(x+7)","title":"Graph the left side, $y=\\sqrt{x+7}$","instruction":"Enter $y=\\sqrt{x+7}$ in input box 1. Type sqrt to make the root sign, then type $x+7$ under it.","hint":"Type y=sqrt, then x+7. Everything you type after sqrt stays under the root sign.","success":"The curve starts at $\\left(-7,0\\right)$ and never dips below the x-axis, because a square root is never negative."},{"kind":"expression","row":1,"expression":"y=x-5","title":"Graph the right side, $y=x-5$","instruction":"Enter $y=x-5$ in input box 2.","hint":"Type y=x-5.","success":"The line meets the curve only once."},{"kind":"intersection","points":[{"x":9,"y":4}],"title":"Click the one crossing","instruction":"Click the point where the line crosses the curve.","hint":"The crossing is to the right of the y-axis, above the x-axis.","success":"The graphs cross at $\\left(9,4\\right)$: when $x=9$, both sides equal $4$."},{"kind":"think","marks":["$$\\sqrt{x+7}=x-5$"],"title":"Squaring gives $x=2$ or $x=9$.\n\nWhy does the graph cross only at $9$?","instruction":"Square both sides: $x+7=x^{2}-10x+25$. Move everything to one side: $0=x^{2}-11x+18=\\left(x-2\\right)\\left(x-9\\right)$. So squaring gives two candidates, $2$ and $9$, but the graph shows one crossing.","options":[{"text":"At $x=2$ the two sides are opposites, and squaring hid the sign","correct":true,"feedback":"Squaring makes a number and its opposite look the same, since $3^{2}$ and $\\left(-3\\right)^{2}$ are both $9$. A value where the sides are opposites solves the squared equation but not the original. Test $2$ next."},{"text":"The graph missed a crossing at $x=2$","feedback":"Each crossing is a place where both sides are equal, and the graph shows every one. At $x=2$ the curve and the line are at different heights, so there is no crossing to miss. Test $2$ next to see the two heights."},{"text":"$$2$ is the y-value of the crossing","feedback":"The crossing is $\\left(9,4\\right)$, so its y-value is $4$. The $2$ came from squaring and solving by hand. Test it next."}],"hint":"Square both sides: $x+7=x^{2}-10x+25$. Move everything to one side: $0=x^{2}-11x+18=\\left(x-2\\right)\\left(x-9\\right)$. So squaring gives two candidates, $2$ and $9$, but the graph shows one crossing.","success":"Squaring makes a number and its opposite look the same, since $3^{2}$ and $\\left(-3\\right)^{2}$ are both $9$. A value where the sides are opposites solves the squared equation but not the original. Test $2$ next."},{"kind":"expression","row":2,"expression":"sqrt(2+7)","marks":["$$\\sqrt{x+7}=x-5$"],"title":"Put $x=2$ into the left side, $\\sqrt{x+7}$","instruction":"Enter $\\sqrt{2+7}$ in input box 3.","hint":"Type sqrt, then 2+7.","success":"The left side is $3$ when $x=2$."},{"kind":"expression","row":3,"expression":"2-5","marks":["$$\\sqrt{x+7}=x-5$"],"title":"Put $x=2$ into the right side, $x-5$","instruction":"Enter $2-5$ in input box 4.","hint":"Type 2-5.","success":"The right side is $-3$. Since $3\\ne -3$, $2$ is not a solution. Squaring turned both $3$ and $-3$ into $9$, which is how $2$ slipped in."},{"kind":"answer","values":[9],"coordinate":"x","boxes":2,"answerLabel":"$$x=$","title":"Give every solution of the original equation","instruction":"Enter each value of $x$ that makes the original equation true, one in each box. If there is only one, leave the second box empty.","hint":"Only one crossing appeared on the graph.","retry":"The graphs cross only at $x=9$. At $x=2$ the two sides are $3$ and $-3$, so $2$ is not a solution.","success":"The only solution is $x=9$: $\\sqrt{9+7}=4$ and $9-5=4$."}],"takeaway":"Graph each side as printed. Every crossing is a true solution, and a value that squaring adds never shows up as a crossing.","result":"$$x=9$","guideHref":"/reference/sat_math/desmos-mastery/absolute-radical-and-rational"},{"id":"inequality-system-point","title":"Find the point that makes both inequalities true","skill":"intersections","category":"choices","workspace":"graph","problem":"Which point $(x,y)$ is a solution to the system of inequalities below?\n$y\\le -x+7$\n$x-2y>4$","choices":["$$(1,5)$","$$(8,1)$","$$(4,0)$","$$(5,-1)$"],"introduction":"A point solves the system only if it makes both inequalities true. On the graph, it sits where the two shaded regions overlap.","previewExpressions":["y<=-x+7","x-2y>4","(1,5)","(8,1)","(4,0)","(5,-1)"],"window":{"left":-4,"right":10,"bottom":-6,"top":8},"steps":[{"kind":"think","marks":["solution to the system of inequalities"],"title":"Where on the graph are the points that solve the system?","instruction":"Each inequality shades the points that make it true.\n\nDecide where a solution of both must sit.","options":[{"text":"Where the two shadings overlap","correct":true,"feedback":"A solution makes both inequalities true, so it sits inside both shadings at once. On the graph that is the darker region where they overlap."},{"text":"Anywhere in either shading","feedback":"A point in only one shading makes only one inequality true. The system needs both to be true, so look where the shadings overlap."},{"text":"Where the two boundary lines cross","feedback":"Here the lines cross on the boundary of $x-2y>4$, where $x-2y$ is exactly $4$. That isn't greater than $4$, so the crossing fails. Look where the shadings overlap."}],"hint":"Each inequality shades the points that make it true.\n\nDecide where a solution of both must sit.","success":"A solution makes both inequalities true, so it sits inside both shadings at once. On the graph that is the darker region where they overlap."},{"kind":"expression","row":0,"expression":"y<=-x+7","title":"Type $y\\le -x+7$","instruction":"Enter $y\\le -x+7$ in input box 1. Type <= for the $\\le$ sign.","hint":"Type y<=-x+7.","success":"The line is solid, because points on it make the inequality true, and Desmos shades below it."},{"kind":"expression","row":1,"expression":"x-2y>4","title":"Type $x-2y>4$ as printed","instruction":"Enter $x-2y>4$ in input box 2, exactly as printed. Desmos works out which side to shade.","hint":"Type x-2y>4.","success":"Desmos draws this line dashed and shades below it. The darker region is where both shadings overlap."},{"kind":"think","marks":["$$x-2y>4$"],"title":"Which points on the dashed line make $x-2y>4$ true?","instruction":"On the dashed line, $x-2y$ equals $4$.\n\nCompare that with what $x-2y>4$ asks for.","options":[{"text":"None of them, because $4$ is not greater than $4$","correct":true,"feedback":"Right. The inequality is strict, so its boundary is left out, and Desmos dashes the line to show it. A point on a solid line, like $y\\le -x+7$, does count, because of the \"or equal to\"."},{"text":"All of them, because the line is part of the graph","feedback":"Desmos dashes this line to show it is left out. On it, $x-2y$ is exactly $4$, and $x-2y>4$ needs more than $4$."},{"text":"The ones that are also below the solid line","feedback":"Being below the solid line makes only $y\\le -x+7$ true. A point on the dashed line still has $x-2y=4$, which fails $x-2y>4$."}],"hint":"On the dashed line, $x-2y$ equals $4$.\n\nCompare that with what $x-2y>4$ asks for.","success":"Right. The inequality is strict, so its boundary is left out, and Desmos dashes the line to show it. A point on a solid line, like $y\\le -x+7$, does count, because of the \"or equal to\"."},{"kind":"expression","row":2,"expression":"(1,5)","title":"Plot choice A, $(1,5)$","instruction":"Enter choice A, $(1,5)$, in input box 3.","hint":"Type (1,5) with the parentheses and the comma.","success":"A sits in the first shading only. It is above the dashed line: $1-2\\left(5\\right)=-9$, which is not greater than $4$.","eliminates":[0]},{"kind":"expression","row":3,"expression":"(8,1)","title":"Plot choice B, $(8,1)$","instruction":"Enter choice B, $(8,1)$, in input box 4.","hint":"Type (8,1).","success":"B sits in the second shading only. It is above the solid line: $-8+7=-1$, and $1$ is not at most $-1$.","eliminates":[1]},{"kind":"expression","row":4,"expression":"(4,0)","title":"Plot choice C, $(4,0)$","instruction":"In input box 5, type (4,0). Look at where it lands, then substitute it into $x-2y>4$ to be sure.","hint":"Type (4,0), then work out $4-2\\left(0\\right)$.","success":"C sits exactly on the dashed line: $4-2\\left(0\\right)=4$, and $4$ is not greater than $4$.","eliminates":[2]},{"kind":"expression","row":5,"expression":"(5,-1)","title":"Plot choice D, $(5,-1)$","instruction":"Enter choice D, $(5,-1)$, in input box 6.","hint":"Type (5,-1).","success":"D sits inside the darker overlap. Substituting agrees: $-1\\le 2$, and $5-2\\left(-1\\right)=7$, which is greater than $4$."},{"kind":"choice","answer":3,"title":"Choose the point inside both shadings","instruction":"Choose the point that makes both inequalities true.","hint":"Only one point sat in the darker overlap.","success":"D is right. A fails $x-2y>4$, B fails $y\\le -x+7$, and C sits on the dashed line, so it fails the strict inequality.","retry":"That point fails one of the two inequalities. Choose the point inside both shadings."}],"takeaway":"The solutions of a system of inequalities sit where the shadings overlap. A point on a dashed line is not a solution.","result":"D) $(5,-1)$","guideHref":"/reference/sat_math/desmos-mastery/inequalities-and-domains"},{"id":"equivalent-expression","title":"Find the equivalent expression by graphing the choices","skill":"equation","category":"choices","workspace":"graph","problem":"Which expression is equivalent to $(x-2)^{2}+5x$?","choices":["$$x^{2}+5x+4$","$$x^{2}+9x+4$","$$x^{2}+x+4$","$$x^{2}+x-4$"],"introduction":"Equivalent expressions give the same output for every input, so their graphs lie exactly on top of each other.","previewExpressions":["y=(x-2)^2+5x","y=x^2+5x+4","y=x^2+9x+4","y=x^2+x+4"],"window":{"left":-8,"right":4,"bottom":-18,"top":24},"steps":[{"kind":"expression","row":0,"expression":"y=(x-2)^2+5x","title":"Graph the original expression","instruction":"In input box 1, type y=(x-2)^2, press the right arrow to leave the exponent, then type +5x. The screen shows $y=\\left(x-2\\right)^{2}+5x$.","hint":"Type y=(x-2)^2, press the right arrow to leave the exponent, then type +5x.","success":"This is the curve the right choice has to match at every $x$."},{"kind":"think","marks":["equivalent"],"title":"What will the graph of an equivalent choice do?","instruction":"Equivalent means equal for every value of $x$.\n\nDecide what that looks like on the graph before you type a choice.","options":[{"text":"Lie exactly on top of the original curve","correct":true,"feedback":"Equal outputs at every $x$ put every point of the two graphs in the same place, so you'll see one curve where there were two."},{"text":"Cross the original curve at one point","feedback":"Two different curves can share one point and split apart everywhere else. One shared point shows they agree at one $x$, not at every $x$."},{"text":"Cross the y-axis at the same point","feedback":"That checks one input, $x=0$. Different expressions can agree there and nowhere else, so the whole curve has to match."}],"hint":"Equivalent means equal for every value of $x$.\n\nDecide what that looks like on the graph before you type a choice.","success":"Equal outputs at every $x$ put every point of the two graphs in the same place, so you'll see one curve where there were two."},{"kind":"expression","row":1,"expression":"y=x^2+5x+4","title":"Graph choice A","instruction":"Enter choice A as $y=x^{2}+5x+4$ in input box 2.","hint":"Type y=x^2, press the right arrow to leave the exponent, then type +5x+4.","success":"A meets the original only at $\\left(0,4\\right)$ and separates everywhere else, so A is out. Plugging in $x=0$ alone would have kept it, because A and the original both give $4$ there.","eliminates":[0]},{"kind":"visibility","hide":["y=x^2+5x+4"],"show":["y=(x-2)^2+5x"],"title":"Hide choice A's graph","instruction":"Click the colored circle beside input box 2. The circle turns empty and A's graph disappears, while its expression stays in the list. Keep the original showing, so the next choice has something to match.","hint":"The circle is on the left edge of input box 2.","success":"A's expression is still there; only its graph is hidden. Now each choice meets the original alone."},{"kind":"expression","row":2,"expression":"y=x^2+9x+4","title":"Graph choice B","instruction":"Enter choice B as $y=x^{2}+9x+4$ in input box 3.","hint":"Type y=x^2, press the right arrow to leave the exponent, then type +9x+4.","success":"B also meets the original only at $\\left(0,4\\right)$, so B is out.","eliminates":[1]},{"kind":"visibility","hide":["y=x^2+9x+4"],"show":["y=(x-2)^2+5x"],"title":"Hide choice B's graph","instruction":"Click the colored circle beside input box 3, so only the original curve is left.","hint":"The circle is on the left edge of input box 3.","success":"Only the original curve is showing again."},{"kind":"expression","row":3,"expression":"y=x^2+x+4","title":"Graph choice C","instruction":"Enter choice C as $y=x^{2}+x+4$ in input box 4.","hint":"Type y=x^2, press the right arrow to leave the exponent, then type +x+4.","success":"C lands exactly on the original, so you see one curve where there were two."},{"kind":"visibility","hide":["y=(x-2)^2+5x"],"show":["y=x^2+x+4"],"title":"Hide the original and watch C's curve","instruction":"Click the colored circle beside input box 1 to hide the original. If C's curve sits exactly on it, the curve on screen won't move.","hint":"The circle is on the left edge of input box 1.","success":"Nothing moved: C's graph was drawn in exactly the same place as the original's, at every $x$ on screen."},{"kind":"choice","answer":2,"title":"Choose the expression whose graph matched","instruction":"Choose the expression whose graph landed exactly on the original.","hint":"Only one choice covered the original curve instead of crossing it.","success":"C is right, and expanding confirms it for every $x$, not only the ones on screen: $\\left(x-2\\right)^{2}+5x=x^{2}-4x+4+5x=x^{2}+x+4$. A drops the middle term of $\\left(x-2\\right)^{2}$, B gives that term the wrong sign, and D gives the $4$ the wrong sign, which moves the whole graph down $8$.","retry":"That graph separated from the original. Choose the one that landed exactly on it."}],"takeaway":"Graph the original and each choice, and keep the one that lands exactly on it. One input can't decide, because several choices can agree there.","result":"C) $x^{2}+x+4$","guideHref":"/reference/sat_math/desmos-mastery/graphing-strategies"},{"id":"no-solution-constant","title":"Find the constant that gives a system no solution","skill":"system","category":"constants","workspace":"graph","problem":"In the system of equations below, $k$ is a constant.\n$2x+5y=7$\n$kx+10y=3$\nFor what value of $k$ does the system have no solution?","introduction":"A solution is a point on both lines. A slider for $k$ lets you move the second line and watch when the two lines stop meeting.","previewExpressions":["2x+5y=7","kx+10y=3",""],"window":{"left":-6,"right":10,"bottom":-6,"top":6},"steps":[{"kind":"think","marks":["no solution"],"title":"What do two lines look like when their system has no solution?","instruction":"Decide what the graph must show before you type anything.","options":[{"text":"They run side by side and never meet","correct":true,"feedback":"A solution is a point on both lines, so lines that never meet share no solution. They must be parallel: the same slope with different y-intercepts."},{"text":"They lie on top of each other","feedback":"Lines on top of each other share every point, which gives infinitely many solutions, not none."},{"text":"They cross once","feedback":"One crossing is one shared point, so that system has exactly one solution."}],"hint":"Decide what the graph must show before you type anything.","success":"A solution is a point on both lines, so lines that never meet share no solution. They must be parallel: the same slope with different y-intercepts."},{"kind":"expression","row":0,"expression":"2x+5y=7","title":"Type $2x+5y=7$ as printed","instruction":"Enter $2x+5y=7$ in input box 1, exactly as printed. Desmos graphs it without solving for $y$.","hint":"Type 2x+5y=7.","success":"This line stays where it is for every value of $k$."},{"kind":"expression","row":1,"expression":"kx+10y=3","title":"Type $kx+10y=3$ with $k$ still in it","instruction":"Enter $kx+10y=3$ in input box 2, exactly as the question writes it.","hint":"Type kx+10y=3.","success":"Desmos can’t draw this line until $k$ has a value, so it offers “add slider” under it."},{"kind":"slider","variable":"k","title":"Add a slider for $k$","instruction":"Click $k$ beside “add slider” under input box 2. A new line, $k=1$, appears in input box 3 with a slider under it.","hint":"The button is just under $kx+10y=3$. You can also type $k=1$ in input box 3.","success":"At $k=1$ the lines cross, so the system has one solution."},{"kind":"slider","variable":"k","value":4,"title":"Slide $k$ until the lines run side by side","instruction":"Drag the slider and watch the second line tilt. Stop when it runs beside the first line without meeting it.","hint":"Parallel lines have the same slope. The first line falls $2$ for every $5$ to the right, and the match is between $3$ and $5$.","success":"At $k=4$ the lines run side by side. Solved for $y$, they are $y=-0.4x+1.4$ and $y=-0.4x+0.3$: the same slope with different y-intercepts, so they never meet."},{"kind":"answer","values":[4],"coordinate":"value","answerLabel":"$$k=$","title":"Give the value of $k$","instruction":"Enter the value of $k$ that made the lines parallel.","hint":"Use the value in input box 3 when the lines ran side by side.","retry":"At that value the lines still cross somewhere, so the system has one solution. Use the value where they ran side by side.","success":"Doubling the first equation gives $4x+10y=14$, so the second equation asks the same $4x+10y$ to equal $3$ instead. No point can do both."}],"takeaway":"No solution means parallel lines. Slide the constant until the lines run side by side, then check that the slopes match and the intercepts don't.","result":"$$k=4$","guideHref":"/reference/sat_math/desmos-mastery/parameters-and-equivalence"},{"id":"infinite-solutions-constant","title":"Find the constant that gives a system infinitely many solutions","skill":"system","category":"constants","workspace":"graph","problem":"In the system of equations below, $c$ is a constant.\n$3x-4y=4$\n$6x-8y=c$\nFor what value of $c$ does the system have infinitely many solutions?","introduction":"A slider for $c$ moves the second line up and down without tilting it. Watch for the value that puts it on the first line.","previewExpressions":["3x-4y=4","6x-8y=c",""],"window":{"left":-6,"right":10,"bottom":-6,"top":6},"steps":[{"kind":"think","marks":["infinitely many solutions"],"title":"What do two lines look like when their system has infinitely many solutions?","instruction":"Decide what the graph must show before you type anything.","options":[{"text":"One line on top of the other","correct":true,"feedback":"Every point of one line is then a point of the other, so the two equations share infinitely many solutions."},{"text":"Two lines side by side","feedback":"Parallel lines never meet, so that system has no solution."}],"hint":"Decide what the graph must show before you type anything.","success":"Every point of one line is then a point of the other, so the two equations share infinitely many solutions."},{"kind":"expression","row":0,"expression":"3x-4y=4","title":"Type $3x-4y=4$ as printed","instruction":"Enter $3x-4y=4$ in input box 1, exactly as printed.","hint":"Type 3x-4y=4.","success":"This line stays where it is for every value of $c$."},{"kind":"expression","row":1,"expression":"6x-8y=c","title":"Type $6x-8y=c$ with $c$ still in it","instruction":"Enter $6x-8y=c$ in input box 2.","hint":"Type 6x-8y=c.","success":"Desmos offers “add slider” under it, because $c$ has no value yet."},{"kind":"slider","variable":"c","title":"Add a slider for $c$","instruction":"Click $c$ beside “add slider” under input box 2. A new line, $c=1$, appears in input box 3 with a slider under it.","hint":"You can also type $c=1$ in input box 3.","success":"At $c=1$ the second line runs beside the first without meeting it, so right now the system has no solution."},{"kind":"slider","variable":"c","value":8,"title":"Slide $c$ until the second line lands on the first","instruction":"Changing $c$ moves the second line up and down without tilting it.\n\nDrag the slider and stop when the second line lands exactly on the first.","hint":"Changing $c$ moves the line without tilting it. The match is between $5$ and $10$.","success":"At $c=8$ the two lines lie on top of each other, so you see one line. Every point on it solves both equations."},{"kind":"answer","values":[8],"coordinate":"value","answerLabel":"$$c=$","title":"Give the value of $c$","instruction":"Enter the value of $c$ that put the second line on the first.","hint":"Use the value in input box 3 when you saw one line.","retry":"At that value the lines run side by side and never meet. Use the value where the second line landed on the first.","success":"Doubling $3x-4y=4$ gives $6x-8y=8$, the same equation, so every solution of one is a solution of the other."}],"takeaway":"Infinitely many solutions means the same line twice: one equation is a multiple of the other, constant included.","result":"$$c=8$","guideHref":"/reference/sat_math/desmos-mastery/parameters-and-equivalence"},{"id":"typed-regression-form","title":"Type a regression in the question's own form","skill":"table","category":"tables","workspace":"graph","problem":"The value $V(t)$, in dollars, of a collectible card $t$ years after it was bought is modeled by the function below, where $a$ and $b$ are positive constants.\n$V(t)=a\\cdot b^{t/3}$\nThe table shows the card's value at three times. What is the value of $b$, rounded to the nearest tenth?","problemTable":{"type":"table","altText":"t-values 1, 4, 9; V(t)-values 292.40, 467.84, 1,024.00","headers":[{"label":"t","align":"center"},{"label":"V(t)","align":"center"}],"rows":[[1,"292.40"],[4,"467.84"],[9,"1,024.00"]],"size":"compact"},"introduction":"The question's $b$ multiplies the value once every $3$ years. A regression typed with a tilde, ~, fits whatever form you type, so type the question's form to get the question's $b$.","previewExpressions":["y_1~ab^(x_1)","y_1~ab^(x_1/3)",""],"previewTable":[{"x":"1","y":"292.4"},{"x":"4","y":"467.84"},{"x":"9","y":"1024"}],"window":{"left":-1,"right":11,"bottom":0,"top":1200},"steps":[{"kind":"table","points":[{"x":1,"y":292.4},{"x":4,"y":467.84},{"x":9,"y":1024}],"marks":["The table shows"],"title":"Copy the table into Desmos","instruction":"Open the + menu and choose table, or type table in an empty input box. Put the years in the $x_{1}$ column and the values in the $y_{1}$ column, without commas.","hint":"Press Enter to move down a column; the table adds rows as you type.","success":"The three rows are now three points: the card's value after $1$, $4$, and $9$ years."},{"kind":"expression","row":0,"expression":"y_1~ab^(x_1)","marks":["$$V(t)=a\\cdot b^{t/3}$"],"title":"Type the plain form, $y_{1}\\sim ab^{x_{1}}$","instruction":"In the first empty input box below the table, type $y_{1}\\sim ab^{x_{1}}$ to see what the plain form gives. The tilde, ~, tells Desmos to fit the letters to the table; to type it, hold Shift and press the key just left of 1.","hint":"Type y1~ab^(x1). Each digit typed right after a letter becomes a subscript by itself.","success":"This $b$, about $1.17$, is the factor for $1$ year. The question's $b$ multiplies the value once every $3$ years, so this $b$ is not the answer."},{"kind":"expression","row":1,"expression":"y_1~ab^(x_1/3)","marks":["$$V(t)=a\\cdot b^{t/3}$"],"title":"Type the question's form, $y_{1}\\sim ab^{x_{1}/3}$","instruction":"In the next empty input box, type the question's own form, $y_{1}\\sim ab^{x_{1}/3}$, so each letter Desmos fits is the question's letter. Press the right arrow after the 3 to leave the fraction.","hint":"Type y1~ab^(x1/3, press the right arrow to leave the denominator, then type ).","success":"Now $b$ is about $1.60$, the factor for every $3$ years, and $a$ is about $250$, the value when the card was bought. As a check, $1.17^{3}\\approx1.60$, so both fits draw the same curve."},{"kind":"answer","values":[1.6],"coordinate":"value","answerLabel":"$$b=$","title":"Give the question's $b$","instruction":"Enter $b$ from the second regression, the one with $x_{1}/3$ in its exponent, rounded to the nearest tenth.","hint":"The regression with $x_{1}/3$ fits the question's own form.","retry":"Use $b$ from the regression with $x_{1}/3$. The first regression's $b$ is the factor for one year.","success":"Every $3$ years, the card's value is multiplied by $1.6$."}],"takeaway":"When the question gives its own form of the model, type that form after the tilde, so each letter Desmos fits is the question's letter.","result":"$$b=1.6$","guideHref":"/reference/sat_math/desmos-mastery/regression-tools"}],"nextLibraryLesson":{"title":"Desmos or Not?","href":"/reference/sat_math/desmos-mastery/desmos-or-not"},"initialLessonId":"$undefined"}] 17:[["$","meta","0",{"charSet":"utf-8"}],["$","meta","1",{"name":"viewport","content":"width=device-width, initial-scale=1"}]] 15:null 19:[["$","title","0",{"children":"The Complete Desmos Walkthrough | Deep Test Prep"}],["$","meta","1",{"name":"description","content":"Practice SAT graphing, tables, regression, fractions, roots, and function notation with guided Desmos-style walkthroughs and reference guides."}],["$","link","2",{"rel":"canonical","href":"https://www.deeptestprep.com/reference/sat_math/desmos-mastery/complete-desmos-walkthrough"}],["$","meta","3",{"name":"msvalidate.01","content":"2BBFD73D0B6CBAEF22D81F43A814B3E6"}],["$","meta","4",{"property":"og:title","content":"Deep Test Prep | The Ultimate SAT Study Guide"}],["$","meta","5",{"property":"og:description","content":"10 full-length SAT practice tests to build test-day stamina and confidence, a Desmos simulator to solve math questions faster, and 202 lessons that teach every skill the SAT tests."}],["$","meta","6",{"property":"og:url","content":"https://www.deeptestprep.com"}],["$","meta","7",{"property":"og:site_name","content":"Deep Test Prep"}],["$","meta","8",{"property":"og:type","content":"website"}],["$","meta","9",{"name":"twitter:card","content":"summary"}],["$","meta","10",{"name":"twitter:title","content":"Deep Test Prep | The Ultimate SAT Study Guide"}],["$","meta","11",{"name":"twitter:description","content":"10 full-length SAT practice tests to build test-day stamina and confidence, a Desmos simulator to solve math questions faster, and 202 lessons that teach every skill the SAT tests."}]]

Master the SAT Desmos calculator

Chapter 1 of 8:Type every symbol from your keyboard