Linear functions
Algebra is about 35% of SAT Math (13β15 of 44 questions). Linear functions appear in both modules, as multiple choice and as typed answers, very often as a short story about a rate (cost per hour, liters per minute) and a starting amount.
A linear function has a constant rate
A linear function has the form f(x) = mx + b. The letter x is the input and f(x) is the output (it means the same as y). The number m is the slope: the change in the output when the input goes up by 1. The number b is the y-intercept: the output when x = 0, or the starting value. The graph is always a straight line.
Slope: rise over run
To find the slope from two points (xβ, yβ) and (xβ, yβ), divide the change in y by the change in x: m = (yβ β yβ) Γ· (xβ β xβ). Subtract in the same order on the top and on the bottom. A line that goes up from left to right has a positive slope. A line that goes down has a negative slope. In the picture, the line goes through (0, β1) and (2, 3), so m = (3 β (β1)) Γ· (2 β 0) = 2.
Reading a table
In a linear function, equal steps in x always give equal steps in f(x). In the table, f(x) goes up by 4 each time x goes up by 1, so the slope is 4. The row with x = 0 shows the y-intercept, 5. So f(x) = 4x + 5. If x does not go up by 1, divide: slope = (change in f(x)) Γ· (change in x).
| x | f(x) | change in f(x) |
|---|---|---|
| 0 | 5 | β |
| 1 | 9 | +4 |
| 2 | 13 | +4 |
| 3 | 17 | +4 |
What the numbers mean in a story
Many SAT questions give a function for a real situation and ask what m or b means. The slope is βperβ something: the amount the output changes for each 1 input unit, with units βoutput units per input unitβ. The y-intercept is the output at the very beginning, when x = 0. Always answer in the words of the story, not just βthe slopeβ.
| Words in the problem | Usually means |
|---|---|
| per, each, every, rate, increases by | slope m |
| starting amount, fixed fee, initial, at time 0 | y-intercept b |
| the output is 0 (nothing left, empty, break-even) | x-intercept |
Finding intercepts
To find the y-intercept, put x = 0: it is f(0), which is b. To find the x-intercept, put f(x) = 0 and solve for x. For f(x) = 2x β 10: f(0) = β10, so the y-intercept is (0, β10); 2x β 10 = 0 gives x = 5, so the x-intercept is (5, 0). In a story, an x-intercept often tells you when something runs out.
| x | 1 | 3 | 5 |
|---|---|---|---|
| f(x) | 4 | 10 | 16 |
- From x = 1 to x = 3, x goes up by 2 and f(x) goes up by 6.
- Slope: 6 Γ· 2 = 3.
- So f(x) = 3x + b. Use the point (1, 4): 4 = 3(1) + b, so b = 1.
- f(x) = 3x + 1, and f(0) = 1.
- Check with x = 5: 3(5) + 1 = 16 β
- The slope is β2.5. It is the change in height for each 1 hour: the candle gets 2.5 cm shorter every hour.
- Put h(t) = 10: 30 β 2.5t = 10.
- β2.5t = β20, so t = 8.
- Check: 30 β 2.5(8) = 30 β 20 = 10 β
- The y-intercept is b = β4.
- Slope: (0 β (β4)) Γ· (6 β 0) = 4/6 = 2/3.
- f(x) = (2/3)x β 4.
- f(9) = (2/3)(9) β 4 = 6 β 4 = 2.
- Check: f(6) = 4 β 4 = 0 β
- Slope: (26 β 11) Γ· (5 β 2) = 15 Γ· 3 = 5.
- The input goes from 5 to 8, which is 3 more, so the output goes up by 3 Γ 5 = 15.
- f(8) = 26 + 15 = 41.
- Losing the sign of a negative slopeIf the line goes down from left to right, or the quantity decreases (water draining, a battery running down), the slope is negative. Check the sign against the picture or the story.
- Confusing slope with the y-interceptSlope = βperβ, rate, change for each 1. Y-intercept = value at the start (x = 0). In f(x) = 12x + 40, the 40 is the start, not the rate.
- Subtracting in a different orderIn (yβ β yβ) Γ· (xβ β xβ) use the same point first on the top and the bottom. Mixed order flips the sign.
- Misreading the axis scaleLook at the numbers on the axes before you count squares. One square may be 5 or 10 units, not 1.
- Mixing up the x-intercept and the y-interceptThe y-intercept is f(0). The x-intercept is where f(x) = 0. Do not swap the two coordinates when you write the point.
- Answering with a bare number in a meaning questionSay what the number means with units: βthe candle gets 2.5 cm shorter each hourβ, not just βthe slopeβ.
Type the rule as y = 3x + 1 and Desmos draws the line. Click the line to see the intercepts, or click any point to see its coordinates. If a table gives you points, plot them with the table tool and check that your ruleβs line passes through every point.
- Type y = 3x + 1
- Click the line: gray dots appear where it crosses the axes. Click them to see (0, 1) and (β0.333, 0); Desmos shows decimals, and β0.333 is β1/3
- To test a rule against a table, plot the table points and see if the line goes through all of them
- To find the x-value for a given output, type y = 10 on a second line and click the crossing point
Use Desmos when you must test several answer choices against a table or a graph, or when the numbers are ugly. For a simple slope between two nice points, the formula is faster.
Open Desmos β