PSAT 8/9 Math: Linear Functions
Linear functions describe quantities that change at a constant rate, and the PSAT 8/9 tests this idea through function notation, real-world context, tables, and graphs. This free, complete lesson covers function notation and evaluating functions, interpreting slope and intercept in context, writing equations from tables and points, and reading linear graphs. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback and full explanations. Everything here is free, and you can keep practicing afterward with the linked quiz below.
1. Function Notation and Evaluating Functions
A linear function written as f(x) = mx + b takes an input x and produces an output f(x). To evaluate a function at a specific input, substitute that value in for x. If you're given an output and asked to find an unknown constant, substitute what you know and solve.
Worked Example: For f(x) = 5x + b, f(6) = 23. Find the value of b.
Substitute x = 6 and f(6) = 23: 23 = 5(6) + b.
23 = 30 + b.
Subtract 30: b = −7
1. For f(x) = 5x + b, f(6) = 23. Find the value of b. (see worked example above)
Explanation: Substitute: 23 = 5(6) + b → 23 = 30 + b → b = −7.
2. For g(x) = 3x − 4, find g(7).
Explanation: Substitute x = 7: g(7) = 3(7) − 4 = 21 − 4 = 17.
3. For h(x) = −2x + k, h(4) = 1. Find the value of k.
Explanation: Substitute: 1 = −2(4) + k → 1 = −8 + k → k = 9.
2. Interpreting Slope and Intercept in Context
In a real-world linear model, the slope represents the rate of change (how much the output changes per unit of input), and the y-intercept represents the starting value (the output when the input is 0).
Worked Example: A spring's length, in centimeters, is modeled by L = 20 + 3w, where w is the weight attached, in newtons. What does the 3 represent?
The 3 is the slope: the spring stretches 3 cm for every 1 newton of weight added.
1. A spring's length, in centimeters, is modeled by L = 20 + 3w, where w is the weight attached, in newtons. What does the 3 represent? (see worked example above)
Explanation: The 3 is the coefficient of w, the slope, representing the rate of change: 3 cm of stretch per newton of weight.
2. A gym charges a one-time $15 registration fee plus $20 per month. Write the total cost C after m months.
Explanation: The $20 per month is the rate (multiplied by m), and the one-time $15 fee is a constant added separately: C = 20m + 15.
3. A tank's water usage drops at a constant rate: it holds 5 million gallons at year 0 and 2 million gallons at year 6. Write a function f(t) for the amount of water, in millions of gallons, after t years.
Explanation: Slope = (2 − 5)/(6 − 0) = −3/6 = −0.5. The starting value (intercept) is 5, so f(t) = 5 − 0.5t.
3. Writing Equations from Tables and Points
To write a linear equation from a table, first find the slope using two rows: slope = (change in output) / (change in input). Then use any point to solve for the y-intercept.
Worked Example: A table shows that when x = 2, g(x) = 15, and when x = 5, g(x) = 24. Write the equation for g(x) = mx + b.
Slope: m = (24 − 15)/(5 − 2) = 9/3 = 3.
Substitute (2, 15): 15 = 3(2) + b → 15 = 6 + b → b = 9.
g(x) = 3x + 9
1. A table shows that when x = 2, g(x) = 15, and when x = 5, g(x) = 24. Write the equation for g(x) = mx + b. (see worked example above)
Explanation: Slope = (24 − 15)/(5 − 2) = 3. Using (2, 15): 15 = 3(2) + b → b = 9. So g(x) = 3x + 9.
2. A linear function passes through (0, 21) and (3, 33). Find its y-intercept.
Explanation: The y-intercept is the output when x = 0, which is already given directly in the point (0, 21): the y-intercept is 21.
3. A labor cost table shows: 2 hours costs \$180, and 5 hours costs \$330. Find the hourly rate (the slope).
Explanation: Slope = (330 − 180)/(5 − 2) = 150/3 = $50 per hour.
4. Reading Linear Graphs
On a graph, the y-intercept is where the line crosses the y-axis, and the slope is the "rise over run" between any two points on the line. Vertical shifts of a function, like f(x) + c, move every point on the graph up (for positive c) or down (for negative c) by c units, without changing the shape.
Worked Example: For f(x) = −4x + 12, find the y-coordinate of the y-intercept.
In y = mx + b form, b is the y-intercept, so the y-coordinate is 12.
1. For f(x) = −4x + 12, find the y-coordinate of the y-intercept. (see worked example above)
Explanation: In y = mx + b form, b is the y-intercept. Here b = 12.
2. The graph of y = g(x) is shifted up by 19 units to create y = g(x) + 19. If the point (3, 5) is on the graph of g(x), what point is on the graph of g(x) + 19?
Explanation: Adding 19 to the function shifts every output (y-value) up by 19: 5 + 19 = 24, so the point becomes (3, 24).
3. A line on a graph passes through (0, 4) and (2, 10). Find its slope.
Explanation: Slope = (10 − 4)/(2 − 0) = 6/2 = 3.
Common Mistakes to Avoid
- Substituting into the wrong variable when evaluating a function, always replace every instance of x (not f(x)) with the given input.
- Confusing slope and y-intercept in a word problem. The slope is always attached to the changing quantity (the variable), while the y-intercept is the standalone constant.
- Using the wrong order when calculating slope from two points, always keep the same point first in both the numerator and denominator: (y2 − y1)/(x2 − x1).
- Mixing up a vertical shift with a horizontal shift. f(x) + c shifts the graph vertically, while f(x + c) shifts it horizontally, these behave differently.
- Forgetting to solve for the y-intercept after finding the slope when writing an equation from a table, the slope alone isn't a complete equation.
Frequently Asked Questions
What's the difference between f(x) and f(3)?
f(x) is the general rule or formula for the function, using the variable x as a placeholder. f(3) is a specific output, the result of substituting 3 in for x everywhere it appears in that formula.
How do I find the slope from a table if I'm not given two "nice" rows?
Pick any two rows in the table, it doesn't matter which, since the slope between any two points on a line is always the same. Use slope = (change in output)/(change in input) with whichever pair of rows is most convenient.
Where can I practice more problems like these?
The Linear Functions quizzes in the PSAT 8/9 Math Question Bank include additional original problems on this topic, along with quizzes covering every other Heart of Algebra, Advanced Math, Problem-Solving and Data Analysis, and Geometry and Trigonometry skill tested on the PSAT 8/9. You can also browse the full PSAT 8/9 Math Question Bank.