AP Exam Prep · AP Precalculus

AP Precalculus Unit 1 Practice Test: Polynomial and Rational Functions

Unit 1 builds the language of the whole course: how inputs and outputs change together, average rates of change, and the behaviour of polynomial and rational functions, including zeros, end behaviour, asymptotes and holes. It finishes with transformations and building models from data and context.

Exam weighting:: 30–40%
Questions:: 24 multiple choice
Time:: about 60 minutes
Cost:: Free, no sign-up

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Question 1 of 24 · 0 answered

Topic 1.1 · Data analysis

Graph of y = f(x)

On which interval is f decreasing?

−1.73 < x < 1.73−1 < x < 1−2 < x < 2x < −1 and x > 1

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Topic 1.1 · Data analysis

Values of a function f
x	0	1	2	3
f(x)	2	6	9	11

Which statement best describes how f changes for 0 ≤ x ≤ 3?

f is increasing, and its rate of change is decreasing, so f is concave down.f is increasing, and its rate of change is increasing, so f is concave up.f is decreasing, and its rate of change is decreasing, so f is concave down.f is increasing at a constant rate.

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Topic 1.2 · Data analysis

Outdoor temperature T on a spring day
Hours after 6 a.m., t	0	2	4	6
T(t) in °F	49	58	65	70

What is the average rate of change of T from t = 2 to t = 6, and what does it mean?

1/3 °F per hour: the temperature rose 1 °F every 3 hours.64 °F per hour: the average temperature from 8 a.m. to noon was 64 °F.3 °F per hour: on average the temperature rose 3 °F each hour from 8 a.m. to noon.12 °F per hour: the temperature rose 12 °F each hour from 8 a.m. to noon.

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Topic 1.3 · Data analysis

Values of a function g
x	0	1	2	3	4
g(x)	1	4	11	22	37

Which type of function is most appropriate for g, and why?

Cubic, because the outputs increase by more each timeQuadratic, because the second differences are constantLinear, because g increases for every step in xExponential, because the outputs grow faster and faster

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What AP Precalc Unit 1 covers

Unit 1, Polynomial and Rational Functions, makes up 30–40% of the multiple-choice section and usually takes ~30–40 class periods. These are the topics in the current course framework, and every question in the test above is tagged with one of them.

1.1Change in Tandem
1.2Rates of Change
1.3Rates of Change in Linear and Quadratic Functions
1.4Polynomial Functions and Rates of Change
1.5Polynomial Functions and Complex Zeros
1.6Polynomial Functions and End Behavior
1.7Rational Functions and End Behavior
1.8Rational Functions and Zeros
1.9Rational Functions and Vertical Asymptotes
1.10Rational Functions and Holes
1.11Equivalent Representations of Polynomial and Rational Expressions
1.12Transformations of Functions
1.13Function Model Selection and Assumption Articulation
1.14Function Model Construction and Application

Unit 1 key ideas

Average rate of change is a slope

The average rate of change of f on [a, b] is (f(b) − f(a)) ÷ (b − a), the slope of the secant line. Linear functions have a constant rate of change; quadratics have rates of change that change at a constant rate, so their second differences over equal steps are constant.

Zeros, multiplicity and end behaviour

A factor (x − a)ⁿ gives a zero at x = a: odd multiplicity crosses the x-axis, even multiplicity touches and turns. A degree-n polynomial has exactly n complex zeros counting multiplicity, and non-real zeros come in conjugate pairs. The leading term decides end behaviour.

Rational functions: compare degrees, then factor

For end behaviour compare the degrees of the numerator and denominator: lower on top gives y = 0, equal degrees give y = ratio of leading coefficients, and higher on top means no horizontal asymptote (one degree higher gives a slant asymptote). After cancelling common factors, a zero of the denominator gives a vertical asymptote; a cancelled factor gives a hole.

Transformations act on inputs or outputs

g(x) = a·f(b(x − h)) + k: changes inside the function (b and h) act horizontally and feel 'backwards' (x − 3 shifts right 3, b = 2 compresses horizontally by a factor of 1/2), while a and k act vertically.

Common Unit 1 mistakes

Calling every zero of the denominator a vertical asymptote without first cancelling common factors (a cancelled factor is a hole).
Treating average rate of change as the change in output only, and forgetting to divide by the change in input.
Shifting the wrong way: f(x + 2) moves the graph 2 units left, not right.

How to study this unit

For any rational function, write it fully factored first, then list zeros, holes, vertical asymptotes and end behaviour in that order.
Check end-behaviour answers by plugging in a large value such as x = 1000.

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More AP Precalculus practice tests

← All AP Precalculus units Unit 2: Exponential and Logarithmic Functions →

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StudyCoach wrote these questions for this test. Each one is mapped to a topic in the College Board's current AP Precalculus course framework. Every answer key was also checked by a separate AI review, and any disagreement was rechecked by hand. They have not yet been reviewed by an AP Precalculus teacher. If you think an answer is wrong, email support@studycoach.io.

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