Chapter 4: Polynomial & Rational Functions4.4: Rational Functions

Essential Question: How can you determine the vertical and horizontal asymptotes of an equation?

4.4: Rational Functions Domain of Rational Functions

The domain is the set of all real numbers that are not zeros of its denominator.

Example 1: The Domain of a Rational Function Find the domain of each rational function

2

1( )f x

x

2

2

3 1( )

6

x xg x

x x

All real numbers except x = 0

All real numbers except x2 – x – 6 = 0(x – 3)(x + 2) = 0, so all real numbers except for 3 and -2

4.4: Rational Functions Properties of Rational Graphs

Intercepts As with any graph, the y-intercept is at f(0) The x-intercepts are when the numerator = 0 and the

denominator does not equal 0.

Example Find the intercepts of

y-intercept:

x-intercepts: Neither are solutions of x – 1 = 0

so both are x-intercepts

2 2( )

1

x xf x

x

20 0 2 2

(0) 20 1 1

f

2 2 0

( 2)( 1) 0

2 1

x x

x x

x x

4.4: Rational Functions Properties of Rational Graphs (continued)

Vertical Asymptotes Whenever only the denominator = 0 (numerator is not

0) Vertical asymptotes will either spike up to ∞ or down to

-∞

Big-Little Concept Dividing by a small number results in a large

number Dividing by a large number results in a small

number.

1little

big 1

biglittle

4.4: Rational Functions Behavior near a Vertical Asymptote

Describe the graph of near x = 2

As x gets closer and closer to 2 from the right (2.1, 2.01, 2.001, …) the denominator becomes a really small positive number. Division by a small positive number means the graph

of f(x) approaches ∞ from the rightAs x gets closer and closer to 2 from the left

(1.9, 1.99, 1.999, …) the denominator becomes a really small negative number Division by a small negative number means the graph

of f(x) approaches -∞ from the left

1( )

2 4

xf x

x

4.4: Rational Functions Holes

When a number c is a zero of both the numerator and denominator of a rational function, the function might have a vertical asymptote, or it might have a hole.

Example #1

But this is not the same as the function g(x) = x + 2, as f(2) = while g(2) = 4, so though they may look the same, f(x) has a hole at x = 2

Example #2

The graph of x2/x3 looks the same as 1/x, and has a vertical asymptote, as neither of the functions are defined at x = 0

2 ( 2)

2

4 ( 2)( ) 2

2

x xf x x

x

x

x

0

0

2

3

1( )

xf x

x x

4.4: Rational Functions Holes

If and a number d exists such that g(d) and h(d) = 0 If the degree of the numerator is greater than (or equal

to) the degree of the denominator after simplification, then the function has a hole at x = d = hole @ x = 5

If the degree of the denominator is greater than the degree of the numerator after simplification, then the function has a vertical asymptote at x = d = asymptote @ x = 5

It is far easier to first determine the domain of the function, and then visually inspect to see whether “hiccups” in the domain are holes or asymptotes.

( )( )

( )

g xf x

h x

2( 5)

5

x

x

2

5

( 5)

x

x

4.4: Rational Functions End Behavior (Horizontal Asymptotes)

The horizontal asymptote is found by determining what the function will be when x is extraordinarily large

When x is large, a polynomial function behaves like its highest degree term

Example #1 List the vertical asymptotes and describe the end

behavior.

There is a vertical asymptote at x = 5/2

Both numerator and denominator have the same degree, so the horizontal asymptote is at y = -3/2

3 6( )

5 2

xf x

x

3 6 3 6 3 3( )

5 2 2 5 2 2

x x xf x

x x x

4.4: Rational Functions Asymptotes, Example #2

Vertical asymptotes at: x2 – 4 = 0 (x – 2)(x + 2) = 0 Vertical asymptotes at x = 2 or x = -2

Horizontal Asymptotes at: As x becomes large, 1/x becomes small. So horizontal asymptote at y = 0.

2( )

4

xf x

x

2 2

1( )

4

x xf x

x x x

4.4: Rational Functions Asymptotes, Example #3

Vertical asymptotes at: x3 + 1 = 0 Graphing tells you there is only one root, at x = -1 Vertical asymptotes at x = -1

Horizontal Asymptotes at:

Horizontal asymptote at y = 2.

3

3

2( )

1

x xf x

x

3 3

3 3

2 2 2( )

1 1

x x xf x

x x

4.4: Rational Functions Assignment

Page 290 1 – 49, odd problems

Due tomorrow Show work

Only worry about holes, vertical and horizontal asymptotes. Disregard stuff on slant/parabolic asymptotes Ignore the graphing (you have a graphing calculator for

that)