Home › AP Precalculus › Practice › Polynomial and Rational FunctionsCollege Board · AP Precalculus
AP Precalculus: Polynomial and Rational Functions — Practice Questions & Answers
Unit 1 of AP Precalculus builds the foundation of function analysis: how polynomial and rational functions behave, where they cross zero, how fast they change, and how they transform. You will learn to read structure from a formula, predict end behavior, locate zeros and asymptotes, solve inequalities, and choose appropriate models from data.
178 practice questions available for this topic — here are 10 with full answers and explanations.
Start an interactive quiz on Polynomial and Rational Functions →
Practice questions with answers
1
Multiple choice · Easy
For $f(x)=x^{2}$, what is the average rate of change on the interval $[1,3]$?
Tap an answer to check it.
Why Average rate of change is $\frac{f(3)-f(1)}{3-1}=\frac{9-1}{2}=4$.
2
Multiple choice · Easy
What is the degree of the polynomial $f(x)=3x^{4}-2x^{2}+x-7$?
Tap an answer to check it.
Why The degree is the highest exponent on the variable, which is $4$.
3
Multiple choice · Easy
Describe the end behavior of $f(x)=2x^{4}-5x+1$.
As $x\to\pm\infty$, $f(x)\to\infty$
As $x\to\pm\infty$, $f(x)\to-\infty$
As $x\to-\infty$, $f(x)\to\infty$; as $x\to\infty$, $f(x)\to-\infty$
As $x\to-\infty$, $f(x)\to-\infty$; as $x\to\infty$, $f(x)\to\infty$
Tap an answer to check it.
Why An even degree with a positive leading coefficient rises on both ends: $f(x)\to\infty$ as $x\to\pm\infty$.
4
Multiple choice · Easy
The function $f(x)=x^{3}$ is best classified as:
Odd
Even
Neither even nor odd
Both even and odd
Tap an answer to check it.
Why Since $f(-x)=(-x)^{3}=-x^{3}=-f(x)$, the function is odd.
5
Multiple choice · Easy
What is the leading term of $f(x)=5-3x^{2}+x^{5}$?
$x^{5}$
$5$
$-3x^{2}$
$x$
Tap an answer to check it.
Why The leading term is the term with the highest degree, which is $x^{5}$.
6
Fill in the blank · Easy
For $f(x)=2x+1$, the average rate of change on the interval $[0,4]$ is
Check answer
Answer:
2
Why $\frac{f(4)-f(0)}{4-0}=\frac{9-1}{4}=2$, which equals the slope of the line.
7
Multiple choice · Easy
Where is the vertical asymptote of $f(x)=\frac{1}{x-2}$?
Tap an answer to check it.
Why The denominator is zero at $x=2$ and the numerator is nonzero there, giving a vertical asymptote $x=2$.
8
Fill in the blank · Easy
The degree of the polynomial $f(x)=(x-1)^{2}(x+3)^{3}$ is
Check answer
Answer:
5
Why Add the multiplicities: $2+3=5$.
9
Multiple choice · Easy
At a real zero of even multiplicity, the graph of a polynomial:
Touches the x-axis and turns around
Crosses straight through the x-axis
Crosses while flattening out
Never reaches the x-axis
Tap an answer to check it.
Why At an even-multiplicity zero the graph touches the x-axis and turns around without crossing.
10
Multiple choice · Medium
For $f(x)=x^{2}-3x$, what is the average rate of change on $[1,4]$?
Tap an answer to check it.
Why $f(4)=4$ and $f(1)=-2$, so $\frac{4-(-2)}{4-1}=\frac{6}{3}=2$.
Key terms in Polynomial and Rational Functions
Function: A relation that assigns to each input exactly one output. It can be expressed as $f(x)$ where $x$ is the input.
Input Value: The independent variable supplied to a function, typically denoted $x$. It belongs to the domain.
Output Value: The dependent value produced by a function for a given input, typically denoted $y$ or $f(x)$.
Domain: The complete set of all permissible input values for a function. It excludes values that cause undefined outputs.
Range: The complete set of all output values a function can produce over its domain.
Independent Variable: The variable representing input, whose value is chosen freely, usually $x$.
Dependent Variable: The variable representing output, whose value depends on the input, usually $y$.
Increasing Function: A function whose output values rise as the input values increase over an interval.
See the full study notes and flashcards for this topic.
Practise other AP Precalculus topics