Slope and Rate of Change on the SAT
The Story of a Line
Build equations from context, spot patterns fast, and practice with intent.
Why the SAT Emphasizes Slope and Rate of Change
Slope is the math way to describe change. If a car travels $60$ miles in one hour, or a streaming service charges $12$ dollars per month, you are describing a rate of change. The SAT expects you to compute slope and to explain what it means in context.
This lesson focuses on both parts. You will compute slope from points, tables, and graphs, then translate that number into words that match the situation. That is how you avoid the common mistake of treating slope as a random fraction with no meaning.
A Simple Definition Unlocks Slope and Rate of Change
Slope measures how much $y$ changes for each one unit increase in $x$. The formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ is just a formal way of saying, change in output divided by change in input.
A positive slope means the values rise together, and a negative slope means one rises while the other falls. To avoid sign errors, always subtract in the same order, top and bottom, so the direction is consistent.
Work Through Slope and Rate of Change Step by Step
Find the slope between $(1, 2)$ and $(5, 10)$.
Use the slope formula with two points, and keep the subtraction order consistent to avoid sign errors.
Use the slope formula with points $(1, 2)$ and $(5, 10)$
Simplify the numerator and denominator
Reduce the fraction to its simplest form.
Use Desmos to Check Slope and Rate of Change
Find the slope between $(1, 2)$ and $(5, 10)$.
Desmos can compute slope quickly by defining two points and using a formula. This is helpful when the coordinates have decimals or large numbers.
A = (1, 2)
B = (5, 10)
m = (A.y - B.y) / (A.x - B.x)
For clean integers, algebra is faster. Desmos is faster when you need to avoid arithmetic errors or when the points come from a graph you can click.
Expert move: Graph the lines and use intersections for "when do they match?" questions; Desmos also verifies slope and intercepts quickly when coefficients are messy.
When to skip Desmos: For quick slope or parallel/perpendicular checks, algebra is faster; use Desmos for verification.
- Desmos features used: points, coordinate access, expression evaluation.
- Common mistake: reversing the subtraction in one part but not the other.
Practice Slope and Rate of Change with SAT-Style Questions
Compute the slope or interpret its meaning.
What is the slope of the line through and ?
A car travels from miles to miles in hours. What is the rate of change in miles per hour?
A line has slope . Which description matches that slope?
If the slope is , what does that mean in words?
Key Takeaways to Remember for Slope and Rate of Change
- Slope is $\frac{\Delta y}{\Delta x}$, the rate of change.
- Positive slopes rise, negative slopes fall.
- Desmos helps when points are messy or come from a graph.
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