PSAT 8/9 Math: Nonlinear Functions
Nonlinear functions cover a wider family of relationships than straight lines, including square roots, absolute value, cubics, and parabolas, and the PSAT 8/9 tests your ability to evaluate them and read their graphs. This free, complete lesson covers evaluating functions with substitution, square root and absolute value functions, reading parabolas and graphs, and finding zeros of factored functions. 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. Evaluating Functions with Substitution
To evaluate any function, including rational and cubic functions, substitute the given input for the variable and simplify. For a cubic equation like p(n) = 5n³, if you're given the output, isolate n³ first, then take a cube root.
Worked Example: Given p(n) = 5n³, find n when p(n) = 40.
Substitute: 40 = 5n³.
Divide by 5: n³ = 8.
Take the cube root: n = 2
1. Given p(n) = 5n³, find n when p(n) = 40. (see worked example above)
Explanation: 40 = 5n³ → n³ = 8 → n = ∛8 = 2.
2. Given f(x) = 2/(5x), find f(4).
Explanation: Substitute x = 4: f(4) = 2/(5 × 4) = 2/20 = 1/10.
3. Given f(x) = 24/x, find f(6).
Explanation: Substitute x = 6: f(6) = 24/6 = 4.
2. Square Root and Absolute Value Functions
For a square root function, simplify what's under the radical first, then take the square root. For an absolute value function, simplify the expression inside the bars first, then apply absolute value (making the result non-negative).
Worked Example: Given g(x) = √(5x + 4), find g(4).
Substitute: g(4) = √(5(4) + 4) = √(20 + 4) = √24.
g(4) = √24
1. Given g(x) = √(5x + 4), find g(4). (see worked example above)
Explanation: g(4) = √(5(4) + 4) = √(20 + 4) = √24.
2. Given g(x) = |x − 7|, find g(10).
Explanation: g(10) = |10 − 7| = |3| = 3.
3. Given h(x) = |2x − 15|, find h(3).
Explanation: h(3) = |2(3) − 15| = |6 − 15| = |−9| = 9.
3. Reading Parabolas and Graphs
A parabola's vertex is its highest or lowest turning point. The y-intercept is where the graph crosses the y-axis (always at x = 0), and in context, it often represents a starting value.
Worked Example: A marble rolls along a track, and its height in inches is graphed against time in seconds, forming a curve that starts at a certain height and dips to a minimum before rising again. What does the y-intercept of this graph represent?
The y-intercept represents the marble's starting height, at time 0.
1. A marble's height over time forms a curve on a graph. What does the graph's y-intercept represent? (see worked example above)
Explanation: The y-intercept occurs at time = 0, the very start, so it represents the marble's starting height.
2. A parabola is graphed with its vertex at the point where x = 0. If the vertex is the lowest point on the graph and has a y-value of −3, what is the vertex?
Explanation: The vertex is written as (x, y). Since x = 0 and the y-value is −3, the vertex is (0, −3).
3. A parabola is defined by f(x) = x² − 2x − 8. Find its y-intercept.
Explanation: The y-intercept occurs at x = 0: f(0) = 0² − 2(0) − 8 = −8. The y-intercept is (0, −8).
4. Finding Zeros of Factored Functions
The zeros (or x-intercepts) of a factored function are the x-values that make the function equal to 0. Using the zero product property, set each factor equal to 0 and solve.
Worked Example: Given f(x) = (x − 7)(x + 2), find the x-coordinates of the x-intercepts.
Set each factor to 0: x − 7 = 0 gives x = 7. x + 2 = 0 gives x = −2.
x-intercepts at x = 7 and x = −2
1. Given f(x) = (x − 7)(x + 2), find the x-coordinates of the x-intercepts. (see worked example above)
Explanation: Setting each factor to 0: x − 7 = 0 gives x = 7, and x + 2 = 0 gives x = −2.
2. Given f(x) = (x − 2)(x + 3)(x − 5), which of the following is NOT an x-intercept of f?
Explanation: The three factors give zeros at x = 2, x = −3, and x = 5. Since 1 is not among these values, (1, 0) is not an x-intercept.
3. Given h(x) = (x + 6)(x − 1), find the x-intercepts.
Explanation: Setting each factor to 0: x + 6 = 0 gives x = −6, and x − 1 = 0 gives x = 1.
Common Mistakes to Avoid
- Forgetting the sign flip when solving a zero from a factor like (x + 6). Setting x + 6 = 0 gives x = −6, not x = 6.
- Simplifying what's inside a square root or absolute value incorrectly before applying the outer operation, always finish the inside first.
- Forgetting that absolute value always produces a non-negative result, even when the expression inside is negative.
- Mixing up a graph's x-intercepts and y-intercept. The y-intercept always has x = 0; x-intercepts always have y = 0.
- Leaving a cube root or square root unevaluated when a decimal or whole-number answer choice is expected, or vice versa, always match the form of the answer choices.
Frequently Asked Questions
What counts as a "nonlinear" function?
Any function whose graph isn't a straight line: this includes quadratics (parabolas), cubics, square root functions, absolute value functions, and rational (fraction) functions, among others.
Why does the zero product property work?
If two or more factors multiply to equal 0, at least one of them must itself be 0 (since any nonzero number times another nonzero number can never equal 0). This lets you solve a factored equation by setting each factor equal to 0 separately.
Where can I practice more problems like these?
The Nonlinear 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.