Desmos Tricks for SAT Math

Worked example: find $y$ at $x = 0$ from a table that skips that row

Here is that reading on a table question. It gives three rows and asks for $y$ when $x = 0$, a row the table doesn’t have. A regression finds the whole equation from the rows: you type the table into Desmos, then the equation’s form with a tilde ($\sim$) where the equals sign would go, and Desmos fills in its numbers.

By hand, that takes a subtraction, a division, and a count back to $x = 0$, three steps that each invite a slip once the numbers get messy. The regression takes the same few lines for any table, tidy or not. Step through it, and answer each step’s question to go on.

Each trick below starts the way this one did: what the question prints goes into Desmos as printed, and Desmos does the step you would otherwise work by hand.

How to tell which regression a table needs

The worked example’s question named its form, “linear,” and many table questions do the same, in a word or in an equation such as $f(x) = ax^2 + bx + c$. When the question names a form, type that one, and when it doesn’t, the pattern in the table’s $y$-values shows it.

A regression fills in the numbers of whatever form you type, so a wrong form still gets numbers, for a curve that doesn’t match the table. Read the pattern at equal steps in $x$, where each form changes in its own way.

Uneven jumps in $x$ are the trap in a table worked by hand, and a regression accounts for them on its own.

Type a regression when the answer needs the equation: a value at an $x$ the table doesn’t show, or one of the equation’s numbers. When one subtraction or division between two rows gives the answer, it’s faster in your head.

Fit a table that curves in “Find the quadratic model that matches a table”Chapter 6 of the free walkthrough

Define $f$ once, and Desmos evaluates $f(5) - f(2)$ in one line

A table gives you points on a function. Some questions give you the function itself, such as $f(x) = x^2 - 6x + 11$, and ask for its values. Type the definition into Desmos exactly as printed, and from then on $f$ with a number in parentheses gives the function’s value at that number.

Without Desmos, $f(5) - f(2)$ means substituting twice, with a square and a minus sign to get right each time. Type the function into box 1, then $f(5) - f(2)$ into box 2, and watch its value appear as you type.

Once $f$ is defined, every square and sign in an expression with $f$ is Desmos’s job. A function inside another works the same way: define $g$ as well, and $f(g(3))$ gives its value.

Practice on new functions in “Evaluate a function”Chapter 2 of the free walkthrough

The equivalent answer choice draws the same graph as the question’s expression

The tricks so far end in a number. Many SAT questions end in four expressions instead and ask which one equals the question’s. Equivalent expressions are equal at every $x$, so the right choice draws its graph exactly on top of the question’s graph.

Expanding $(x + 3)^2 - 2x$ yourself works if every term comes out right, and one slip, such as writing $(x + 3)^2$ as $x^2 + 9$, can lead you to a wrong choice. Predict the answer from the expressions first. Then tap each choice to draw its graph, dashed, over the solid curve labeled “question,” and pick the choice whose graph lands exactly on it.

Testing one number can mislead: at $x = 0$, three of the choices equal $9$, the same as the question’s expression, while a graph compares every $x$ at once. The same trick works whenever the choices can be graphed, lines and inequalities included.

Graph a new set of choices in “Find the equivalent expression by graphing the choices”Chapter 4 of the free walkthrough

Which value of $k$ leaves a system with no solution? Find it with a slider

Graphing works when the answer choices are expressions. Some questions put the unknown inside the equations instead, as a letter such as $k$ in $kx + 10y = 3$, and ask which value of $k$ leaves the system with no solution. A slider lets you change $k$ and watch the two lines move until they show the picture the question describes.

The algebra route rests on a rule about the equations’ coefficients, and it’s easy to mix up which ratios must match for no solution and which for infinitely many. With a slider, you type both equations with $k$ still in them, add the slider Desmos offers, and drag.

Try it in the practice calculator below, a lesson from the free “Complete Desmos Walkthrough.” It’s built like the calculator in Bluebook, and it checks each step as you go.

A drag is only as exact as your hand, and lines that nearly match can look parallel on screen. Before you answer, check the value you found against the rule: a system has no solution when its $x$-coefficientsand its $y$-coefficients are in the same ratio and its constants are not. When the answer is a fraction a drag can’t land on, that rule gives it directly.

Mean, median, and standard deviation: type the data set once as a list

Every trick so far works on equations and graphs, and questions about data take a different feature, the list. Type the values in square brackets with a name, such as $L_1 = [70, 72, 75, 78]$, and $\operatorname{mean}(L_1)$, $\operatorname{median}(L_1)$, and $\operatorname{stdev}(L_1)$ each give a value on their own line.

Standard deviation measures how far a data set’s values typically sit from their mean. Values bunched near the mean give a small one, and values spread far from it give a large one.

The practice question below lists nine values, and ordering them for a median, or adding them all for a mean, gives you plenty of chances to miscount by hand. Type the list into box 1, then $\operatorname{mean}(L_1)$ and $\operatorname{median}(L_1)$ into the boxes after it, and the question appears once you have both.

Desmos put the values in order to find the median, so the order they were printed in never mattered. According to the College Board’s test framework, SAT questions about standard deviation compare data sets. When two dot plots look about equally spread, type each one as a list, entering a dot stacked three high as three values, and compare $\operatorname{stdev}(L_1)$ with $\operatorname{stdev}(L_2)$.

Compare two data sets in “Compare spread with standard deviation”Chapter 7 of the free walkthrough

Pick the Desmos trick for each question in a mixed set

Every question so far sat under a heading that named its trick. On the test, these questions come mixed in with every other kind, and none of them names its trick. The cue is always the same pair: what the question gives you and what it asks for. Read each question and pick its trick before you would type anything.