Interpret slope as rate of change in context. | Linear functions | SAT Math

Slope in context is a rate of change with units attached.

Core Idea

The slope of a linear function tells you how much the output changes for each one-unit increase in the input — in context, it's the rate of change with units attached.

Understanding

When a question says "what does the 15 represent in the equation $𝐶 = 15 ℎ + 40$ ?", they're asking you to interpret the slope in context.

The slope is the change in the output per one-unit change in the input. In $𝐶 = 15 ℎ + 40$ , if $ℎ$ is hours and $𝐶$ is cost in dollars, the slope 15 means the cost increases by $15 for each additional hour. That's it — attach units and direction.

Watch the sign. A negative slope means the quantity is decreasing. If a lake loses 3 inches of water per week, the slope is $- 3$ , and you'd say "the water level decreases by 3 inches per week."

The SAT loves pairing a word-problem equation with answer choices that subtly swap the meaning of slope and intercept, or flip the units. Read carefully: the slope is always "per one unit of the input variable."

Step by Step

Identify which variable is the input (independent) and which is the output (dependent).
Locate the coefficient of the input variable — that's the slope.
State the meaning: "For each additional [input unit], the [output] increases/decreases by [slope value] [output units]."

Misconceptions

Confusing slope with the y-intercept: the slope is the coefficient of the variable, not the constant term.
Forgetting to include direction: a slope of $- 5$ means a decrease of 5, not an increase.
Saying "the slope is 15 hours" — the slope has the output's units per input unit (e.g., dollars per hour), not the input's units.

Question

Worked Example

A plumber charges a flat fee plus an hourly rate. The total cost $𝐶$ , in dollars, for $ℎ$ hours of work is given by $𝐶 = 65 ℎ + 120$ . What does the number 65 represent?

Select an answer to see the explanation