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SAT Math Linear Functions: The Complete Guide
Everything the SAT tests about linear functions in one place: slope-intercept form, writing equations from points and tables, interpreting slope and intercept in context, function notation, and parallel and perpendicular lines — with step-by-step examples and 3 free practice quizzes.
Practice these quizzes
Free · No signupWhat is a linear function?
A linear function is a function whose graph is a straight line — every input increases or decreases the output by the same fixed amount. That fixed amount is the rate of change, or slope. Linear functions are one of the most heavily tested topics on the SAT, appearing in equations, tables, graphs, and word problems.
| Representation | What to look for |
|---|---|
| Equation | Written as y = mx + b, or f(x) = mx + b |
| Table | Equal jumps in x always produce equal jumps in y |
| Graph | A perfectly straight line, no curves |
The fastest way to confirm a table represents a linear function: check that the ratio of the change in y to the change in x is constant between every pair of consecutive points. If that ratio ever changes, the function is not linear.
Slope-intercept form
Most SAT linear function questions use slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept.
Find the slope of the line through (2, 5) and (6, 17).
Keep the order of subtraction consistent in the numerator and denominator. If you compute y2 − y1 on top, you must compute x2 − x1 on bottom — using the same point order both times. Mixing the order flips the sign of the slope.
Writing equations from points & tables
Many SAT questions give you two points, a table, or a graph and ask you to build the full equation. The process is the same every time: find the slope first, then solve for the y-intercept.
A table shows (x, y) pairs: (0, 4), (2, 10), (4, 16). Write the equation of the line.
Write the equation of the line through (3, 11) and (5, 19).
Interpreting slope & intercept in context
A large share of SAT linear function questions are word problems that give you an equation modeling a real situation and ask what a specific number means. These questions don't require solving anything — they require connecting each part of the equation to the scenario.
Slope (m)
The rate of change — how much the output changes for each 1-unit increase in the input. Units: (output unit) per (input unit).
Y-intercept (b)
The starting value — the output when the input is 0. Often a flat fee, initial amount, or starting position.
A tutoring service models its cost with C = 25h + 40, where C is total cost in dollars and h is hours of tutoring. What does the 25 represent?
SAT answer choices often swap the meanings of slope and intercept, or attach the right number to the wrong quantity. Read every answer choice carefully — don't just match a number, match the full meaning of that number in context.
Function notation f(x)
f(x) is just another name for y — it means "the output of function f when the input is x." Evaluating f(3) means substituting 3 everywhere x appears.
If f(x) = 5x − 8, what is f(4)?
If the SAT asks for the value of x when f(x) equals a given number, set the whole expression equal to that number and solve for x — the reverse of a normal evaluation.
Parallel & perpendicular lines
| Relationship | Slope rule |
|---|---|
| Parallel | Same slope, different y-intercept |
| Perpendicular | Slopes are negative reciprocals (m and −1/m) |
A line has slope −2/3. What is the slope of any line perpendicular to it?
Linear function word problems
The SAT frequently describes a real-world situation in words and asks you to build the linear function yourself, rather than handing you the equation.
- Find the starting value first. Look for a flat fee, initial amount, or starting position — that's your y-intercept.
- Find the constant rate. Look for a phrase like "per," "each," or "every" — that number is your slope.
- Match units carefully. The slope's units should match (output unit) per (input unit) exactly as described.
- Check the direction. A quantity that increases uses a positive slope; a quantity that decreases uses a negative slope.
A candle is 12 inches tall when lit and burns down at a constant rate of 0.5 inches per hour. Write a function for height H after t hours.
Test day strategy for linear functions
| Question signal | Fastest approach |
|---|---|
| Two points given | Find slope first, then plug a point into y = mx + b to solve for b |
| Table of values | Check for a constant rate of change before assuming linear |
| "What does ___ represent" question | Identify slope vs. intercept, then match full meaning — not just the number |
| f(a) = ? question | Substitute directly; if solving for x instead, set f(x) equal to the target value |
| Parallel or perpendicular line | Same slope for parallel; negative reciprocal for perpendicular |
| Word problem with no equation given | Identify the starting value and constant rate before writing anything |
Now put it to work
Three quiz sets, each building on the last — start with Quiz 1 and work through in order, or jump straight to the topic you need.
Linear Functions
Slope-intercept form, writing equations from two points and tables.
Start quiz → Quiz 2Linear Functions 2
Interpreting slope and intercept, function notation.
Start quiz → Quiz 3Linear Functions 3
Parallel and perpendicular lines, word problems.
Start quiz →