Nonlinear Functions: SAT Practice Questions & Study Guide
Understanding polynomial, radical, and rational functions—evaluating them, identifying their key features, and interpreting them in context.
Understanding Nonlinear Functions on the SAT
Nonlinear functions are those whose graphs are curves rather than straight lines: parabolas (quadratic), hyperbolas (rational), square root curves, and higher-degree polynomials. The Digital SAT tests your ability to evaluate these functions at specific inputs, identify features like zeros (x-intercepts), end behavior, and transformations, and connect algebraic forms to graphical representations.
A foundational skill for this category is finding the zeros of a function—the x-values where f(x) = 0. For a polynomial, zeros correspond to x-intercepts on the graph, and each zero's multiplicity determines whether the graph crosses or just touches the x-axis at that point. If f(x) = (x-2)^2(x+3), the zero x = 2 has multiplicity 2 (graph touches and bounces), while x = -3 has multiplicity 1 (graph crosses). The SAT tests this relationship between factors, zeros, and graphs.
Transformations of nonlinear functions follow the same rules as linear transformations: f(x) + k shifts the graph up by k; f(x + h) shifts left by h; -f(x) reflects over the x-axis; a*f(x) scales vertically by factor a. For the SAT, the most commonly tested transformation is vertical shift (adding a constant to the output), and questions often ask how a transformation changes the zeros, maximum, or vertex of the function.
Rational functions (ratios of polynomials like f(x) = (x+1)/(x-3)) introduce asymptotes: vertical asymptotes where the denominator is zero (x = 3 here) and horizontal asymptotes determined by the degrees of numerator and denominator. The Digital SAT tests these conceptually rather than requiring heavy computation—you should know that a vertical asymptote occurs where the denominator equals zero (after factoring and canceling), and that a horizontal asymptote is y = leading coefficient ratio when degrees match, y = 0 when denominator degree is higher, and none (oblique asymptote) when numerator degree exceeds denominator degree.
Key Rules & Formulas
Memorize these rules — they come up directly in SAT 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)?
The function g(x) = (x - 2)(x + 5). For what values of x does g(x) = 0?
If h(x) = x^3 - 1, what is h(2)?
The function f(x) = -x^2 + 6x - 5. At which values of x does f(x) cross the x-axis?
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?
If f(x) = x^2 + 3 and g(x) = 2x - 1, what is f(g(2))?
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)?
The function f(x) = (x^2 - 1)/(x - 1) has which of the following key features?
If f(x) = 2x^2 - 8 and g(x) = f(x + 1), which of the following is equivalent to g(x)?
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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Common Mistakes to Avoid
These are the most frequent errors students make on Nonlinear Functions questions. Knowing them in advance prevents costly point losses.
- !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).
SAT Strategy Tips: Nonlinear Functions
Use Desmos 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 on the SAT
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