Interpret a linear function's slope as a rate of change and its intercept as a starting value, build linear models, and read slope from points, tables and graphs (Functions).

What this topic is asking

A linear function $f(x) = mx + b$ models a constant rate of change. The ACT tests reading the slope as a rate, the intercept as a starting value, building a linear model from a description, and finding slope from points, tables or graphs. The algebra is light; the focus is on interpretation in context.

Reading a linear function

Each part of $f(x) = mx + b$ has a fixed meaning.

The most tested idea is that slope is a rate: if output is distance in miles and input is time in hours, the slope is a speed in miles per hour.

Building a linear model

The ACT often asks you to assemble a linear function from words.

For a draining pool, the rate would be negative, as in $V(t) = 500 - 25t$.

Slope as a rate of change

When a question asks what a coefficient "represents", name the rate with its units and its direction. A coefficient of $-3$ on $x$ where the output is temperature in degrees and the input is time in hours means the temperature falls 3 degrees per hour. This interpretation is also how you compute slope from a table: divide any change in output by the matching change in input, and the result is constant for a linear function. If the differences are not constant, the relationship is not linear.

Reading slope from a graph

From a graph, read the $y$-intercept where the line crosses the vertical axis, and the slope as rise over run between two clear lattice points. A line rising to the right has positive slope; falling to the right, negative; horizontal, zero; vertical, undefined. Translating freely among equation, table and graph is one of the most frequently tested ideas in the whole Functions area, and it usually turns a wordy comparison into a quick reading.

Comparing two linear functions

Some questions give two linear models and ask which grows faster or where they meet. The function with the larger slope grows faster and eventually produces the larger output; the one with the larger intercept starts higher. They are equal where their expressions are equal, found by solving a single equation, the same idea as a system. Reading "which plan is cheaper after 10 months" in slope-and-intercept terms makes the comparison fast.

Try this

Q1. What is the slope of the line through $(1, 4)$ and $(3, 10)$? [1 point]

• Cue. $\frac{10 - 4}{3 - 1} = \frac{6}{2} = 3$.

Q2. A gym costs $20 to join plus$30 per month. Write the cost after $m$ months. [1 point]

• Cue. $C(m) = 30m + 20$: rate $30$ times $m$, plus the $20$ fee.