Interpreting Slope as a Rate of Change
On the SAT, slope is rarely just a number — you're asked what it means in a real situation. Slope is always a rate: how much one quantity changes per unit of the other.
Reading the concept isn't enough. The score comes from practicing the real Bluebook-style format and finding the exact mistakes you keep making.
What the SAT tests here
In a model like $C = 15t + 40$, the slope $15$ is the amount added for each one-unit increase in $t$ — dollars per month, miles per hour, and so on. The units are the giveaway: slope is always something per something.
How to solve one, step by step
Example: a gym membership costs $C = 15t + 40$, where $t$ is months.
- The slope is $15$.
- It means the cost rises by $\$15$ for each additional month — a rate of $\$15$ per month.
- The $40$ is a one-time starting cost, not a rate.
The mistakes that cost points
- Reading slope as a value. Slope is a rate of change, not the height of the line at a point.
- Confusing slope with the intercept. The intercept is the starting amount; the slope is the per-unit change.
- Inverting rise over run. Slope is change in output over change in input — not the other way around.
Practice questions
Try these the way you would on test day, then open the solution to check your method.
Easy
The graph shows the amount of water remaining in a tank as it drains. What is the best interpretation of the slope of the graph?
- AThe tank loses 20 gallons of water per minute
- BThe tank loses 1 gallon every 20 minutes
- CThe tank starts with 20 gallons
- DThe tank is empty after 20 minutes
Show solution
Answer: A, The tank loses 20 gallons of water per minute. Slope $= \frac{0 - 200}{10 - 0} = -20$. The magnitude is $20$ gallons per minute draining.
Medium
The slope of $f(x) = mx + 5$ changes from $-1$ to $-5$. What happens to the value of $f(x)$ for negative values of x?
- Af(x) increases
- BWe do not have enough information to know what happens to f(x).
- Cf(x) stays the same
- Df(x) decreases
Show solution
Answer: A, f(x) increases.
Hard
If the slope of $f(x) = -\frac{2}{5}x + 4$ changes to $-\frac{4}{5}$, what happens to $f(x)$ for negative x-values?
- AWe do not have enough information to know what happens to f(x).
- Bf(x) increases
- Cf(x) stays the same
- Df(x) decreases
Show solution
Answer: B, f(x) increases. At $x = -5$: original $f(-5) = 2 + 4 = 6$, new $f(-5) = 4 + 4 = 8$. New f(x) is larger for all negative x.
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