Interpret rate of change and slope in context; identify proportional vs non-proportional relationships. | Functions | ACT Mathematics

Slope is the rate of change, and proportional relationships must pass through the origin.

核心知识

Slope tells how much the output changes for each 1-unit increase in the input. A relationship is proportional only if it has a constant rate and passes through the origin.

深入理解

Rule: In a context, the slope is not just a number. It has units and meaning. If a cost equation is $𝐶 = 12 ℎ + 8$ , the slope 12 means the cost goes up by $12 for each extra hour. The 8 is a starting fee, so the relationship is linear but not proportional.

That distinction matters on ACT. Students often see a constant rate and immediately say proportional. That is only true when the starting amount is 0, which means the equation has the form $𝑦 = 𝑘 𝑥$ .

分步讲解

Read the coefficient of the variable as the rate of change.
Read the constant term as the starting value or fixed amount.
If the starting value is not 0, the relationship is not proportional.
Match the slope to its units so you know what changes per one input unit.

题目

示例解析

The total cost $𝐶$ , in dollars, of renting a kayak is modeled by $𝐶 = 12 ℎ + 8$ , where $ℎ$ is the number of hours. Which statement is true?

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