Contents

Preface
1Prerequisites
2Equations and Inequalities
3Functions
4Linear Functions
5Polynomial and Rational Functions
6Exponential and Logarithmic Functions
7The Unit Circle: Sine and Cosine Functions
8Periodic Functions
9Trigonometric Identities and Equations
10Further Applications of Trigonometry
11Systems of Equations and Inequalities
12Analytic Geometry
13Sequences, Probability, and Counting Theory
AProofs, Identities, and Toolkit Functions
Answer Key
Index

Practice Test

Give the degree and leading coefficient of the following polynomial function.

$f (x) = x^{3} (3 - 6 x - 2 x^{2}) f (x) = x^{3} (3 - 6 x - 2 x^{2})$

Determine the end behavior of the polynomial function.

$f (x) = 8 x^{3} - 3 x^{2} + 2 x - 4 f (x) = 8 x^{3} - 3 x^{2} + 2 x - 4$

$f (x) = - 2 x^{2} (4 - 3 x - 5 x^{2}) f (x) = - 2 x^{2} (4 - 3 x - 5 x^{2})$

Write the quadratic function in standard form. Determine the vertex and axes intercepts and graph the function.

$f (x) = x^{2} + 2 x - 8 f (x) = x^{2} + 2 x - 8$

Given information about the graph of a quadratic function, find its equation.

Vertex $(2, 0) (2, 0)$ and point on graph $(4, 12) . (4, 12) .$

Solve the following application problem.

A rectangular field is to be enclosed by fencing. In addition to the enclosing fence, another fence is to divide the field into two parts, running parallel to two sides. If 1,200 feet of fencing is available, find the maximum area that can be enclosed.

Find all zeros of the following polynomial functions, noting multiplicities.

$f (x) = {(x - 3)}^{3} (3 x - 1) {(x - 1)}^{2} f (x) = {(x - 3)}^{3} (3 x - 1) {(x - 1)}^{2}$

$f (x) = 2 x^{6} - 12 x^{5} + 18 x^{4} f (x) = 2 x^{6} - 12 x^{5} + 18 x^{4}$

Based on the graph, determine the zeros of the function and multiplicities.

Graph of an odd-degree polynomial with two turning points.

Use long division to find the quotient.

10.

$\frac{2 x^{3} + 3 x - 4}{x + 2} \frac{2 x^{3} + 3 x - 4}{x + 2}$

Use synthetic division to find the quotient. If the divisor is a factor, write the factored form.

11.

$\frac{x^{4} + 3 x^{2} - 4}{x - 2} \frac{x^{4} + 3 x^{2} - 4}{x - 2}$

12.

$\frac{2 x^{3} + 5 x^{2} - 7 x - 12}{x + 3} \frac{2 x^{3} + 5 x^{2} - 7 x - 12}{x + 3}$

Use the Rational Zero Theorem to help you find the zeros of the polynomial functions.

13.

$f (x) = 2 x^{3} + 5 x^{2} - 6 x - 9 f (x) = 2 x^{3} + 5 x^{2} - 6 x - 9$

14.

$f (x) = 4 x^{4} + 8 x^{3} + 21 x^{2} + 17 x + 4 f (x) = 4 x^{4} + 8 x^{3} + 21 x^{2} + 17 x + 4$

15.

$f (x) = 4 x^{4} + 16 x^{3} + 13 x^{2} - 15 x - 18 f (x) = 4 x^{4} + 16 x^{3} + 13 x^{2} - 15 x - 18$

16.

$f (x) = x^{5} + 6 x^{4} + 13 x^{3} + 14 x^{2} + 12 x + 8 f (x) = x^{5} + 6 x^{4} + 13 x^{3} + 14 x^{2} + 12 x + 8$

Given the following information about a polynomial function, find the function.

17.

It has a double zero at $x = 3 x = 3$ and zeros at $x = 1 x = 1$ and $x = - 2 x = - 2$ . Its y-intercept is $(0, 12) . (0, 12) .$

18.

It has a zero of multiplicity 3 at $x = \frac{1}{2} x = \frac{1}{2}$ and another zero at $x = - 3 x = - 3$ . It contains the point $(1, 8) . (1, 8) .$

Use Descartes’ Rule of Signs to determine the possible number of positive and negative solutions.

19.

$8 x^{3} - 21 x^{2} + 6 = 0 8 x^{3} - 21 x^{2} + 6 = 0$

For the following rational functions, find the intercepts and horizontal and vertical asymptotes, and sketch a graph.

20.

$f (x) = \frac{x + 4}{x^{2} - 2 x - 3} f (x) = \frac{x + 4}{x^{2} - 2 x - 3}$

21.

$f (x) = \frac{x^{2} + 2 x - 3}{x^{2} - 4} f (x) = \frac{x^{2} + 2 x - 3}{x^{2} - 4}$

Find the slant asymptote of the rational function.

22.

$f (x) = \frac{x^{2} + 3 x - 3}{x - 1} f (x) = \frac{x^{2} + 3 x - 3}{x - 1}$

Find the inverse of the function.

23.

$f (x) = \sqrt{x - 2} + 4 f (x) = \sqrt{x - 2} + 4$

24.

$f (x) = 3 x^{3} - 4 f (x) = 3 x^{3} - 4$

25.

$f (x) = \frac{2 x + 3}{3 x - 1} f (x) = \frac{2 x + 3}{3 x - 1}$

Find the unknown value.

26.

$y y$ varies inversely as the square of $x x$ and when $x = 3, x = 3,$ $y = 2. y = 2.$ Find $y y$ if $x = 1. x = 1.$

27.

$y y$ varies jointly with $x x$ and the cube root of $z . z .$ If when $x = 2 x = 2$ and $z = 27, z = 27,$ $y = 12, y = 12,$ find $y y$ if $x = 5 x = 5$ and $z = 8. z = 8.$

Solve the following application problem.

28.

The distance a body falls varies directly as the square of the time it falls. If an object falls 64 feet in 2 seconds, how long will it take to fall 256 feet?