Topics
1B. Polynomial and Rational Functions
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AP Precalculus

1.8 Rational functions and zeros

Lesson
Worksheet
Practice
Worksheet
What do you remember?
1

What does a real zero of a rational function represent?

2

What do the real zeros of a polynomial function represent in terms of a rational function?

3

How does the degree of the numerator and the degree of the denominator of a rational function affect the real zeros of the function?

4

A graph of the function y = \dfrac{x^2 - 1}{x^2 - 4x + 4} is shown below. Determine the zeros of the function.

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5

A graph of the function y = \dfrac{2}{x+4} is shown below.

How many values of x satisfy the equation \dfrac{2}{x+4}=0?

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Let's practice
6

For each rational functions:

i

Determine the zeros of the rational functions.

ii

Identify any restrictions on the domain of the functions.

iii

Identify the zeros and the corresponding values of x.

a
f \left( x \right) = \dfrac{x^2 - 9}{x^2 - 4x + 4}
b
f \left( x \right) = \dfrac{2x^3 - 18x}{x^2 - 6x}
c
f \left( x \right) = \dfrac{x^3 - 8}{x^2 + 2x - 8}
d
f \left( x \right) = \dfrac{3x^2 - 12x}{x^3 - 27}
e
f \left( x \right) = \dfrac{x^3 - 27}{x^2 - 1}
f
f \left( x \right) = \dfrac{2x^3 - 6x^2 + 4x}{x^3 - 3x^2 + 2x}
g
f \left( x \right) = \dfrac{x^4 - 16}{x^3 - 8}
h
f \left( x \right) = \dfrac{x^3 - 8x}{x^2 - 4}
i
f \left( x \right) = \dfrac{x^2 - 1}{x^2 - 4}
j
f \left( x \right) = \dfrac{4x^2 - 16}{x^3 + 8}
k
f \left( x \right) = \dfrac{x^3 - 8}{x^4 - 16x^2}
l
f \left( x \right) = \dfrac{x^2 - 3x + 2}{x^3 - x^2 - 6x}
7

For each function:

i

Identify the real zeros.

ii

Determine if the real zeros are endpoints or asymptotes for the intervals satisfying the inequality r(x) \geq 0.

iii

Determine if the real zeros are endpoints or asymptotes for the intervals satisfying the inequality r(x) \leq 0.

a
r(x) = \dfrac{x^2 - 4}{x - 2}
b
r(x) = \dfrac{x^2 - 9}{x^2 - 4x + 4}
c
r(x) = \dfrac{x^2 - 1}{x^2 - 3x + 2}
d
r(x) = \dfrac{x^2 - 4}{x^2 - 3x + 2}
e
r(x) = \dfrac{x^3 - 8}{x^2 - 6x + 8}
f
r(x) = \dfrac{x^3 - 27}{x^2 - 5x + 6}
8

For each function:

i

Find the zeros.

ii

Determine the domain.

iii

Verify whether the zeros are in the domain.

a
p\left(x\right)=x^3-4x
b
q\left(x\right)=\dfrac{x^2-9}{x+3}
c
f(x) = x^2 - 4
d
p\left(x\right)=x^3-4x
e
f\left(x\right)=\dfrac{x^2-4}{x-2}
f
g\left(x\right)=\sqrt{x-3}
g
h\left(x\right)=\ln(x+1)
h
k\left(x\right)=\dfrac{1}{x^2+1}
i
h(x) = \sqrt{x - 4}
j
p(x) = x^3 - 8
9

Identify the zeros and their multiplicities for these functions:

a
f \left( x \right) = \dfrac{(x - 2)^2 (x + 3)}{x^2 - 5x + 6}
b
f \left( x \right) = \dfrac{3(x + 1)^3 (x - 4)^2}{(x - 1)(x + 2)^2}
10

Determine the zeros and whether they are removable or non-removable for these functions:

a
f \left( x \right) = \dfrac{x^3 - x^2}{x^2 - x}
b
f \left( x \right) = \dfrac{x^2 - 4x + 4}{x^2 - 6x + 9}
c
f \left( x \right) = \dfrac{x^3 - 3x^2 + 2x}{x^2 - 4}
d
f \left( x \right) = \dfrac{x^4 - 16x^2}{x^2 - 4}
11

A company's revenue is modeled by the polynomial R(x) = x^3 - 8x^2 + 15x, and its costs are modeled by the polynomial C(x) = x^2 - 4x + 2. Determine the profit function P(x) = R(x) - C(x), and find its zeros.

12

A company's profit is modeled by the rational function P(x) = \dfrac{x^2 - 2x - 3}{x^2 - 4}. Determine the intervals of the domain where the company is making a profit (P(x) > 0), a loss (P(x) < 0), and breaking even (P(x) = 0).

13

An engineer is analyzing the capacity of a dam based on the water flow rate. The water flow rate is modeled by the rational function F(x) = \dfrac{3x^2 - 10x + 8}{x^2 - 6x + 8}, where x is the number of hours since the last measurement and F(x) is the flow rate in cubic meters per second. The dam's capacity is reached when the flow rate is zero. Find the zeros of the function and determine the number of hours it will take for the dam to reach its capacity.

14

The rational function h(x) = \dfrac{3x^2 - 12x + 9}{x^3 - 6x^2 + 9x} has a real zero at x = 3. Determine whether this zero is an endpoint or an asymptote for the intervals satisfying the inequality h(x) \geq 0.

15

Consider the rational function f(x) = \dfrac{x^4 - 16}{x^2 - 8x + 16}. Determine the real zeros of both the numerator and denominator functions. Identify which of these zeros are endpoints or asymptotes for the intervals satisfying the inequality f(x) \geq 0.

Let's extend our thinking
16

The function f(x) = \dfrac{x^2 - 4x - 5}{x^2 + x - 6} represents the velocity of an object. Determine the intervals when the object is moving forward or backward.

17

The rational function g(x) = \dfrac{x^3 - 8}{x^2 - 4} represents the growth rate of a population, where x is the time in years.

a

Determine the zeros of the function g(x).

b

Identify the intervals of x for which the growth rate is positive ((g(x) > 0).

c

Verify that the zeros of the function g(x) are within the domain of the function.

18

A polynomial function f(x) with a leading coefficient of 2. This function has zeros at x = 1, x = -2, x = 3, and is undefined at x=2.

a
Write the function in factored form.
b
Find the zeros.
c
Determine if the zeros are in the domain of the function
19

The population of a city over time is modeled by the function P(x) = \dfrac{4x^3 - 12x^2 + 9x}{x^2 - 1}, where x is the number of years since the year 2000. Determine the years during which the population is increasing and the years during which it is decreasing. Verify that the zeros of the function are within its domain.

20

The graph below shows the rational function f(x) = \dfrac{x^2 - 4}{x^2 - x - 6}.

a

Identify the zeros of the function f(x).

b

Determine the intervals of x for which f(x) \gt 0.

c

Explain how the zeros of the function affect the graph.

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21

Suppose you have a rational function with real coefficients. If one of its zeros is a complex number (not a real number), what can you say about its other zeros? Explain your reasoning.

22

Consider a rational function with real coefficients. If all of its zeros are real numbers, would it be possible for the function to be positive on some intervals and negative on others? Justify your answer.

23

Consider a rational function r(x) that models the population of a certain species over time. The function has real zeros at x = 2 and x = 5. What could these zeros represent in the context of the population model?

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Outcomes

1.8.A

Determine the zeros of rational functions.

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