Rational Functions and Their Graphs

Section 7.2

A rational function, f is a quotient of polynomials. That is,

where P(x) and Q(x) are polynomials and Q(x) ≠ 0.

Definition

( )( )

( )

P xf x

Q x

Determine the domain of the following

a)

b)

c)

Example

23 8( )

4

xf x

x

2

2

7( )

16

xf x

x

2

4( )

9

xf x

x

The line x = k is a vertical asymptote of the graph of f if f(x) g ∞ or f(x) g –∞ as x approaches k fromeither the left or the right.

Vertical Asymptote

x = 2

Finding Vertical Asymptotes

( )( )

( )

P xf x

Q x

Let f be a rational function given by

written in lowest terms.

To find a vertical asymptote, set the denominator, q(x), equal to 0 and solve.

If k is a zero of q(x), then x = k is a vertical

asymptote.

Find the vertical asymptotes of the function and then graph the function on your graphing calculator.

a)

b)

c) 3

3( )

3

xf x

x x

2

5( )

1f x

x

2( )

3

xf x

x

The line y = k is a horizontal asymptote of the graph of f iff(x) g k as xapproacheseither ∞ or –∞.

Horizontal Asymptotes

Horizontal Asymptote

(a) If the degree of the numerator is less than the degree of the denominator, then y = 0 (the x-axis) is a horizontal asymptote.

(b) If the degree of the numerator equals the degree of the denominator, then y = a/b is a horizontal asymptote, where a is the leading coefficient of the numerator and b is the leading coefficient of the denominator.

(c) If the degree of the numerator is greater than the degree of the denominator, then there are no horizontal asymptotes.

Finding Horizontal Asymptotes

Find the horizontal asymptote of the functions below.

a)

b)

c)

2

2( )

3

xf x

x

2

2

3 4( )

5 5

xf x

x

3 3( )

2

xf x

x

To determine whether the graph will intersect its horizontal asymptote at y = k, set the f(x) = k and solve.

If there is no solution the graph will not cross the asymptote.

Function Crosses Horizontal Asymptote?

Determine algebraically if the graph of the function will cross its horizontal asymptote.

a. 𝑓ሺ𝑥ሻ= 2𝑥𝑥2−3

𝑏. 𝑓ሺ𝑥ሻ= 3𝑥2 − 4𝑥2 − 2 𝑐. 𝑓ሺ𝑥ሻ= 𝑥2 − 3𝑥𝑥2 − 5𝑥− 6

To graph a rational function, f (x)=P(x)/Q(x)1. Determine the domain of the function and restrict any x-values

as needed.2. Find and plot the y-intercept (evaluate f (0)). 3. Find and plot any x-intercepts (solve P(x)=0). 4. Find any vertical asymptotes (solve Q(x)=0), if there is any.5. Find the horizontal asymptote, if there is one.

Determine whether the graph will intersect its Horizontal asymptote y = b by solving f(x) = b, where b is the y-value of the horizontal asymptote.

6. Plot at least one point between x-intercepts and vertical asymptotes to determine the behavior of the graph.

7. Complete the sketch.

Sketching the Graph of a Rational Function

Sketch the graph of 3

( )1

xf x

x

Sketch the graph of 2

18( )

9

xf x

x

-20 -18 -16 -14 -12 -10 -8 -6 -4 -2 2 4 6 8 10 12 14 16 18 20

-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

2

4

6

8

10

12

14

16

18

20

x

y

Holes In The Graphs

If f(x) = p(x)/q(x), then it is possiblethat, for some number k, both p(k) = 0 and q(k) = 0. In this case, the graph of f may not have a vertical asymptote at x=k; rather it may have a “hole” at x=k.

Slide

7.2 -

15

Find all asymptotes and/or “holes” for the functionSketch the graph of