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Linear functions show up all over the Digital SAT Math section: as equations, tables, word problems, and graphs. The good news: they all come back to one tidy form, and once you can read that form you can read all of them. The form is also what separates them from their usual foil, since linear and exponential growth differ by whether you add or multiply each step.
Key takeaways
- Slope-intercept form is y = mx + b: m is the slope, b is the y-intercept.
- Slope = rise over run = the rate of change. In a word problem it’s the "per" number.
- The y-intercept is the starting value (where x = 0).
- On a graph, find b where the line crosses the y-axis, then count slope as rise/run.
The one form to know
Almost every linear-function question is easier once you write it as y = mx + b. The slope m tells you how fast y changes as x increases by 1; the intercept b tells you where the line starts. It’s also the form that makes systems of two lines easy to read.
In a word problem, the slope is the rate: the "$5 per hour", "3 points per week" number. The y-intercept is the starting amount before anything changes.
Translate the words
Word problems don’t hand you m and b; they hide them behind a small, very repetitive vocabulary. Learn the phrases once and the translation becomes automatic.
| The phrase in the problem | What it means |
|---|---|
| "costs $5 per hour", "grows 3 inches each week" | Slope: m = 5 or m = 3. "Per" and "each" are slope words. |
| "a one-time fee", "an initial balance", "starts at" | Y-intercept: b. It happens once, at x = 0. |
| "decreases by 40 per month", "loses", "drains" | Negative slope: m = -40. Direction lives in the verb. |
| "the value when x = 0" | That’s b, by definition. |
| "how much y changes for one more x" | That’s m, the rate of change. |
Read it off the graph
Take the line y = 2x - 1. It crosses the y-axis at -1 (that’s b), and for every 1 step right it climbs 2 (that’s m). Here it is:
Try one
A gym charges a $30 sign-up fee plus $20 per month. Which function gives the total cost c, in dollars, after m months?
The "meaning of the slope" question
The other flavor of linear-function question hands you the model and asks what a number means. The answer is almost always the same sentence with the blanks filled in: "for each additional [x-unit], [y] increases (or decreases) by [m]." Memorize that sentence and these become free points.
The function P = 8t + 120 models the number of subscribers P to a newsletter t weeks after launch. Which is the best interpretation of the 8 in this context?
Wrong choices love to reuse the right number with the wrong job: the slope dressed up as a total (choice A) or as the starting value (choice C). Before answering, say what one unit of x is. If t is in weeks, the slope is per week; if the question asks for the change over 5 weeks, that’s 5 \times 8 = 40, not the raw coefficient.
The traps
The trap
Swapping slope and intercept (putting the one-time fee as the per-month rate), or reading the slope as run-over-rise.
The move
Label the rate (per) and the starting value (once) in words first, then drop them into y = mx + b.
Work linear-function problems with an AI tutor that checks your reasoning, not just your answer.
Practice Linear Functions →Frequently asked questions
- What is slope-intercept form?
- It’s y = mx + b, where m is the slope (the rate of change) and b is the y-intercept (the value of y when x = 0). It’s the most useful way to write a linear function on the SAT.
- How do I find slope from a graph?
- Pick two clear points on the line and compute rise over run: the change in y divided by the change in x. On the SAT, counting grid squares between two lattice points is usually fastest.
- What does the y-intercept mean in a word problem?
- It’s the starting value before any change, like a one-time fee or an initial amount. The slope is the repeating rate (the 'per' quantity).
- Is slope the same as rate of change?
- For a line, yes: the slope is the rate of change. When the SAT asks for the 'rate of change' of a linear function from an equation, table, or context, it’s asking for m, expressed in y-units per one x-unit.