SAT Nonlinear Functions: Practice Questions & Study Guide
Understanding polynomial, radical, and rational functions—evaluating them, identifying their key features, and interpreting them in context.
Understanding Nonlinear Functions
Nonlinear functions include quadratics, rational functions, square-root functions, and higher-degree polynomials. Questions can ask you to evaluate an input, identify zeros and end behavior, describe a transformation, or connect an algebraic form with a graph.
A zero is an x-value where f(x) = 0. For a polynomial, zeros correspond to x-intercepts, and multiplicity affects whether the graph crosses or touches the axis. In f(x) = (x - 2)^2(x + 3), x = 2 has even multiplicity and the graph touches the axis there, while x = -3 has odd multiplicity and the graph crosses.
Transformations follow consistent rules: f(x) + k shifts up by k; f(x + h) shifts left by h; -f(x) reflects across the x-axis; and a f(x) scales vertically by a. Track whether a change is applied to the input or output before describing its effect.
For a rational function, factor and simplify before classifying discontinuities. A denominator zero from a factor that remains after cancellation is a vertical asymptote; a canceled factor creates a hole. When the numerator and denominator have the same degree, the horizontal asymptote is the ratio of leading coefficients; when the denominator's degree is larger, it is y = 0. If the numerator degree is exactly one greater, polynomial division gives a slant asymptote; larger degree differences can produce higher-degree polynomial asymptotes.
Key Rules & Formulas
Memorize these rules — they come up directly in practice questions.
The zeros (roots) of a function are the x-values where f(x) = 0; they are the x-intercepts of the graph.
For f(x) = x^2 - 5x + 6 = (x-2)(x-3), the zeros are x = 2 and x = 3.
Multiplicity of a zero: even multiplicity → graph touches the x-axis; odd multiplicity → graph crosses.
f(x) = (x-1)^2(x+2): at x = 1 (mult. 2) the graph bounces; at x = -2 (mult. 1) it crosses.
f(x) + k shifts the graph up by k; f(x) - k shifts it down by k.
If f(x) = x^2, then g(x) = x^2 + 3 is the same parabola shifted 3 units up.
Vertical asymptote of a rational function occurs at values making the denominator zero (after canceling common factors).
f(x) = (x+1)/(x-4) has a vertical asymptote at x = 4.
The end behavior of a polynomial is determined by the leading term: even degree → both ends same direction; odd degree → opposite ends.
f(x) = -2x^3 + ... → as x → ∞, f → -∞ and as x → -∞, f → +∞ (odd, negative leading coefficient).
Nonlinear Functions Practice Questions
Select an answer and click Check Answer to reveal the full explanation. Questions go from easiest to hardest.
If f(x) = x^2 - 4x + 3, what is f(0)?
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Correct answer: B. 3
Explanation
f(0) = 0^2 - 4(0) + 3 = 0 - 0 + 3 = 3.
The function g(x) = (x - 2)(x + 5). For what values of x does g(x) = 0?
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Correct answer: A. x = 2 and x = -5
Explanation
Set each factor to zero: x - 2 = 0 gives x = 2, and x + 5 = 0 gives x = -5.
If h(x) = x^3 - 1, what is h(2)?
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Correct answer: D. 7
Explanation
h(2) = 2^3 - 1 = 8 - 1 = 7.
The function f(x) = -x^2 + 6x - 5. At which values of x does f(x) cross the x-axis?
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Correct answer: C. x = 1 and x = 5
Explanation
Set f(x) = 0: -x^2 + 6x - 5 = 0 → x^2 - 6x + 5 = 0 → (x-1)(x-5) = 0 → x = 1 or x = 5.
The graph of y = f(x) has zeros at x = -3 and x = 4, and passes through (0, -12). Which of the following could be the rule for f?
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Correct answer: B. f(x) = (x + 3)(x - 4)
Explanation
Zeros at x = -3 and x = 4 mean factors (x+3) and (x-4). Check (0, -12): (0+3)(0-4) = (3)(-4) = -12 ✓. Choice D: 2(3)(-4) = -24 ≠ -12. So B is correct.
If f(x) = x^2 + 3 and g(x) = 2x - 1, what is f(g(2))?
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Correct answer: A. 12
Explanation
First compute g(2) = 2(2) - 1 = 3. Then f(g(2)) = f(3) = 3^2 + 3 = 9 + 3 = 12.
The function p(x) = (x - 1)^2(x + 3). At x = 1, which of the following best describes the behavior of the graph of y = p(x)?
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Correct answer: D. The graph touches the x-axis without crossing it
Explanation
The factor (x-1) has multiplicity 2 (even), which means the graph touches (is tangent to) the x-axis at x = 1 without crossing it. Odd multiplicity roots produce crossings.
The function f(x) = (x^2 - 1)/(x - 1) has which of the following key features?
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Correct answer: C. A hole at x = 1 and no asymptote
Explanation
Factor: (x^2-1)/(x-1) = (x+1)(x-1)/(x-1) = x+1 for x ≠ 1. The (x-1) factor cancels, creating a hole at x = 1 (the point (1, 2) is missing from the graph), not an asymptote.
If f(x) = 2x^2 - 8 and g(x) = f(x + 1), which of the following is equivalent to g(x)?
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Correct answer: B. 2x^2 + 4x - 6
Explanation
g(x) = f(x+1) = 2(x+1)^2 - 8 = 2(x^2 + 2x + 1) - 8 = 2x^2 + 4x + 2 - 8 = 2x^2 + 4x - 6.
A function f is defined as f(x) = k * x^2 for some constant k. If f(3) = 45, what is f(-2)?
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Correct answer: D. 20
Explanation
From f(3) = 45: k(3)^2 = 45 → 9k = 45 → k = 5. So f(x) = 5x^2. Then f(-2) = 5(-2)^2 = 5(4) = 20.
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Common Mistakes to Avoid
Use this checklist to catch common traps while working through Nonlinear Functions questions, then add your own patterns as you review missed answers.
- !Confusing x-intercepts (zeros of the function) with the y-intercept (value at x = 0)—these are different features with different significance.
- !Forgetting that a horizontal shift f(x + h) moves the graph LEFT when h is positive (opposite of the sign inside).
- !Identifying the vertical asymptote of a rational function without first simplifying—if a common factor cancels, there's a hole there, not an asymptote.
- !Assuming every polynomial has as many distinct real zeros as its degree—some zeros may be complex or repeated.
- !Misreading end behavior: for even-degree polynomials with negative leading coefficients, both ends go DOWN (not up).
Strategy Tips: Nonlinear Functions
Use the graphing calculator aggressively for nonlinear function questions—graphing the function instantly shows zeros, maximums, and intercepts without any algebraic work.
For 'which graph matches this equation' questions, identify two or three key features (zeros, y-intercept, end behavior) and eliminate answer choices that violate any one feature.
When a question changes the function (e.g., 'if g(x) = f(x) - 5, what are the zeros of g?'), translate the transformation's effect on the feature before computing.
Evaluate the function at x = 0 first (to find the y-intercept)—this is usually fast and often eliminates multiple wrong answer choices immediately.
Other Advanced Math Subtopics
Equivalent Expressions
Rewriting algebraic expressions into equivalent forms through factoring, expanding, and applying algebraic identities.
Nonlinear Equations in One Variable
Solving quadratic, radical, and rational equations, and understanding the conditions under which extraneous solutions arise.
Quadratic Equations and Parabolas
Solving quadratics, identifying the vertex and intercepts of parabolas, and interpreting these features in applied contexts.
Exponential Functions
Modeling growth and decay with exponential functions, interpreting the base and exponent, and comparing exponential to linear growth.
Master Nonlinear Functions
Continue with this skill in the question bank, or test how it holds up under section timing before returning to the surrounding Adv. Math skills.