Nonlinear functions

Build, read and interpret quadratic and exponential functions and their graphs, vertices, intercepts and growth rates. It is part of Advanced Math in the Math section. The book teaches it in Chapter 22, Nonlinear Functions, with 19 practice questions, and tests it 12 more times across the two full-length practice tests.

About Chapter 22

A function is a rule that gives exactly one output for each input. This chapter covers three nonlinear families: quadratic functions, whose graphs are parabolas; exponential functions, which grow or shrink by a constant percent; and polynomial functions with several zeros. It also covers function notation and the shifts and reflections that move a graph. The College Board names this skill "Nonlinear functions," and like Chapter 21 it belongs to the Advanced Math domain.

What the question looks like

Typical stems include:

  • "The function f is defined by f(x) = …. What is the value of f(−2)?"
  • "What is the minimum value of f?" or "What is the maximum height, in feet, of the ball?"
  • "Which function gives the number of cells h hours after the start?"
  • "Which statement best describes how the population changes, according to the model?"
  • "Which equivalent form of the function displays the minimum value as a constant or coefficient?"
  • "The graph of y = f(x) is shown. Which of the following could define f?"

Common contexts include the height of a ball or a fountain's spray, a population that changes by a percent each year, a machine that loses value, and a bacteria culture that doubles.

How to recognize it

The question asks about a function's features or behavior, not the solutions of one equation. Watch for the words vertex, maximum, minimum, zeros, x-intercepts, initial value, growth rate, percent per year, doubles, half, shifted and reflected. A graph or table of a curve, or notation such as f(x + 3) or g(x) = −f(x), also signals this skill.

Common traps

  • Giving the input when asked for the value: "Minimum value" and "maximum height" are outputs: the y-coordinate of the vertex.
  • Misreading the sign in vertex form: In (x − 3)^2 the vertex x-coordinate is 3; in (x + 3)^2 it is −3.
  • Treating the base as the rate: 0.85 means 85% remains, so the decrease is 15%. A base of 1.06 means 6% growth.
  • Building the exponent upside down: Doubling every 5 years is 2^t/5, not 2^5t. Test t = 5: the factor should be 2.
  • Splitting a percent across periods: 44% growth per 2 years is not 22% per year; the yearly factor is √1.44 = 1.2, or 20%.
  • Shifting the wrong way: f(x + 2) moves the graph 2 units left, not right.
  • Linear thinking about exponential change: A 15% yearly drop does not remove the same dollar amount each year.

Video lessons

Two short lessons from independent YouTube creators, chosen for this skill. The notes are ours; the videos belong to their creators, who are not affiliated with this site.

  • Exponential functions (Penguin SAT Prep , 10:15). Watch for: Growth and decay factors, and percent change per period.
  • Ten must-knows about exponential functions (JensenMath, 38:12). Watch for: A longer review of exponential models and their graphs.

Practice this skill

The full method, worked examples and 19 practice questions with every answer choice explained are in Chapter 22 of the book and in the study workspace, where an AI coach can talk you through each one.

Independence

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