Linear functions, made visual
Drag points, move sliders and make predictions. You will read slope from any graph, table or word problem, and use function notation with confidence.
Slope is rise over run
Slope tells you how steep a line is and which way it goes. Pick any two points on a line. The run is how far you move across. The rise is how far you move up or down. Slope is the rise divided by the run.
Slope from two points
On the SAT you often get two points instead of a graph. Label them (x₁, y₁) and (x₂, y₂), then subtract.
- Rise = 13 − 5 = 8
- Run = 6 − 2 = 4
- Slope = 8 ÷ 4 = 2. For every 1 step right, the line goes up 2.
- Check by picture: from (2, 5), go right 4 and up 8. You land on (6, 13).
Common trapDoing (y₂ − y₁) on top but (x₁ − x₂) on the bottom flips the sign of your answer. Keep the same order in both.
The star equation: y = mx + b
Most SAT linear questions use slope-intercept form. m is the slope and b is the y-intercept, the point where the line crosses the y-axis at (0, b).
Four kinds of slope
Reading left to right: positive goes uphill, negative goes downhill, zero is flat (y = a number), and undefined is straight up and down (x = a number, where the run is 0).
Reading tables
In a table for a linear function, each equal step in x gives an equal change in y. Slope is the change in y divided by the change in x.
| x | y | |
|---|---|---|
| 0 | 5 | |
| 2 | 11 | x +2, y +6 |
| 4 | 17 | x +2, y +6 |
| 6 | 23 | x +2, y +6 |
Two quick winsIf the table includes x = 0, the y next to it is b. If it does not, find m first, then use b = y − m·x with any row.
What slope means in real life
A question gives you a model like C = 2.5d + 4 and asks what a number means. Use this sentence frame: the slope is how much the output changes for each 1 unit of input, and the intercept is the output when the input is 0.
Function notation: f(x)
f(x) = 3x − 4 means the same as y = 3x − 4. Think of f as a machine: an input goes in, the rule runs, and an output comes out. f(2) means “the output when the input is 2.”
Input or output?
A very common SAT slip is mixing up input and output. f(3) asks: go to x = 3, then read the height. “f(x) = 6” asks the reverse: go to height 6, then read the x. The graph below shows a linear function f.
Spot the trap
SAT wrong answers are built from real mistakes. Each solution below has one faulty step. Find it.
Desmos moves
The Digital SAT gives you the Desmos graphing calculator for the whole math section. For clean numbers, solving by hand is often faster. For messy numbers or a double-check, these moves are worth knowing.