Unit 1: Polynomial and Rational Functions

AP Precalculus: 50 practice questions with detailed explanations.

Unit Study Guide

Core idea

Unit 1 studies functions through their RATES of change: how output moves as input moves, and what the graph reveals about that motion.

Rates of change

The average rate of change of f on [a, b] is (f(b) - f(a))/(b - a) - the secant lines slope. For linear functions this rate is constant (the slope); for quadratics it changes at a constant rate of its own. Reading increasing/decreasing intervals from graphs and tables is tested constantly.

Polynomials

Degree and leading coefficient control end behavior: even degree means both tails agree (up if positive), odd degree means tails disagree. Real zeros come from factoring; multiplicity tells the story at each zero - odd multiplicity crosses the axis, even multiplicity touches and turns.

Rational functions

For r(x) = p(x)/q(x):

  • Factor BOTH top and bottom. Common factors cancel and leave a hole.
  • Unmatched zeros of q give vertical asymptotes (unbounded behavior).
  • Compare degrees for horizontal asymptotes: bottom heavier gives y = 0; degrees equal gives the ratio of leading coefficients; top heavier means none (end behavior follows the polynomial part).

    • Compositions and inverses

    (f o g)(x) = f(g(x)) - apply the inner function first and watch domains. An inverse swaps input and output: reflect across y = x, restricting domains when f is not one-to-one. Verify with f(g(x)) = x.

    Top 5 Concepts to Master

    1. 1Compute and interpret average rates of change from tables, graphs, and formulas.
    2. 2Predict end behavior from degree and leading coefficient.
    3. 3Use multiplicity to describe behavior at each real zero.
    4. 4Fully characterize a rational function: holes, vertical and horizontal asymptotes.
    5. 5Compose functions and find inverses with attention to domain restrictions.

    Key Terms & Definitions

    Practice with Flashcards
    Average rate of change

    (f(b) - f(a))/(b - a): slope of the secant over an interval.

    End behavior

    Tail direction determined by degree (even/odd) and leading coefficient sign.

    Multiplicity

    Repeated-zero count: odd crosses the x-axis, even bounces off it.

    Vertical asymptote

    Line x = c where the function is unbounded, from an uncanceled denominator zero.

    Horizontal asymptote

    End-behavior line from comparing numerator and denominator degrees.

    Hole

    Removable point where a common factor cancels from numerator and denominator.

    Composite function

    (f o g)(x) = f(g(x)); output of g feeds into f.

    Inverse function

    Function reversing inputs and outputs; exists when f is one-to-one.

    Common Misconceptions: Exam Traps

    f^-1(x) means 1/f(x).

    Correct: f^-1 is the INVERSE function (undo mapping); the reciprocal 1/f(x) is different.

    Every denominator zero gives a vertical asymptote.

    Correct: Only UNCANCELED factors; a shared factor cancels into a hole.

    Even-multiplicity zeros cross the axis.

    Correct: Odd multiplicities cross; even multiplicities touch and reverse direction.

    Horizontal asymptotes describe middle-of-graph behavior.

    Correct: They describe END behavior only; the curve may cross the line elsewhere.

    Constant average rate of change fits any polynomial.

    Correct: Constant ARoC characterizes LINEAR functions; quadratics change at linearly changing rates.

    Featured Practice Questions

    Question Bank Breakdown

    By difficulty

    easy 28medium 22

    By topic

    Zeros and End Behavior of Polynomials 9Rational Functions 9Composite and Inverse Functions 9Quadratic Functions 7Cubic and Higher-Order Polynomial Functions 6Linear Functions 5Rates of Change 5

    All Questions in this Unit