warm up find a polynomial function with integer coefficient that has the given zero. find the domain...

Warm Up

Find a polynomial function with integer coefficient that has the given zero.

Find the domain of:

ii 4,4,2

f ( x) 2x 1

x2 4

Announcements

Assignment

◦p. 278 ◦# 3-12, 23, 26◦Study Guide

Notebook Quiz WednesdayReview Session Tuesday after school

2.7 Rational FunctionsHow to find the domains of rational functions

How to find horizontal and vertical asymptotes of graphs of rational functions

Introduction to Rational Functions

A rational function is a function of the form f(x) = N(x)/D(x), where N and D are both polynomials. The domain of f is all x such that D(x) 0.◦Example:

1

12)(

x

xxf

Example 1.

Find the domain of

All Reals ≠ -2, 2

f ( x) 2x 1

x2 4x2 4 0

x 2 x 2 0

x 2 0 x 2

x 2 0 x 2

Horizontal and Vertical Asymptotes

1

12

x

xy

Vertical Asymptote

X – values where there are no y – valuesFind vertical asymptotes by finding the

domain

1

01

x

x1

12

x

xy

Horizontal asymptotes

The graph of f has one horizontal asymptote or no horizontal asymptote, depending on the degree of n and m.

a. If n < m, then y = 0 is the horizontal asymptote of the graph of f.

b. If n = m, then y = an/bm is the horizontal asymptote of the graph of f.

c. If n > m, then there is no horizontal asymptote of the graph of f.

...

...)(

mm

nn

xb

xaxf

Hint, hint, note, note

Graphs CAN touch a horizontal asymptote

Graphs CAN’T touch a vertical asymptote

12 x

x

1

12

x

x

64

52

x

x

Example Example

Example

Horizontal Asymptote

a. If n < m, then y = 0 is the horizontal asymptote of the graph of f.

xx

xy

11

0

m

n

x

xxf )(

Horizontal Asymptote

b. If n = m, then y = an/bm is the horizontal asymptote of the graph of f.

1

12

x

xy

1

2

m

n

b

ay

...

...)(

mm

nn

xb

xaxf

Horizontal Asymptote

c. If n > m, then there is no horizontal asymptote of the graph of f.

53

22

3

x

xy

m

n

x

xxf )(

Find any horizontal and vertical asymptotes of the following.

The horizontal asymptote is at y= 1/2, and the vertical asymptote is at x = 3/2.

g( x) 2x 5

4x 6

What x-values will make the function undefined?

What is the relationship between the highest powers in the numerator and denominator?

Find any horizontal and vertical asymptotes of the following.

No horizontal asymptote and a vertical asymptote at x = -1

h(x) x2

x 1 What x-values will make the function undefined?

What is the relationship between the highest powers in the numerator and denominator?

Domain of a rational function

To find the domain of a rational function of x, . .

set the denominator of the rational function equal to zero and solve for x. These values of x must be excluded from the domain of the function.

Warm Up

Find the domain of the function and identify any horizontal and vertical asymptotes.

1)(

2

3

x

xxh

Announcements

Assignment

◦p. 281◦# 69 – 74◦Study Guide

Notebook Quiz tomorrowReview Session today after school

Objectives

How to analyze and sketch graphs of rational functions

How to sketch graphs of rational functions that have slant asymptotes

Steps for finding the Graph of a Rational Functions

1st Guideline for graphing rational functions

1. Find and plot the y-intercept (if any) by evaluating f(0)

2

3

20

3)0(

2

3)(

f

xxf

2nd Guideline for graphing rational functions

1. Find the zeros of the numerator (if any) by setting the numerator = 0. Then plot them as x – intercepts

2

1

12

012

12)(

x

x

xx

xxf

3rd Guideline for graphing rational functions

1. Find the zeros of the denominator (if any) by setting the denominator = 0. Then sketch the corresponding vertical asymptotes

2

022

3)(

x

xx

xf

4th Guideline for graphing rational functions

1. Find and sketch the horizontal asymptote (if any) by using the rules for finding the horizontal asymptote

21

22

12)(

x

x

x

x

x

xxf

m

n

5th Guideline for graphing rational functions

1. Plot at least one point between and at least one point beyond each x intercept and vertical asymptote

x

xxf

12)(

X Y

-1 3

(1/4) -2

1 1

X – int. = (1/2)

Vert. Asym. = 0

6th Guideline for graphing rational functions

1. Use smooth curves to complete the graph between and beyond the vertical asymptotes

x

xxf

12)(

Example 1. Sketch the graph of the following function.

y-Intercept: Nonex-Intercept: (-1, 0)Vertical asymptote: x = 0Horizontal asymptote: y = 1Additional points: (-2, 0.5), (-1.5, 1/3),

(1, 2)

f ( x) x 1

x

Sketch the graph of each of the following functions.

y-Intercept: (0, 0)x-Intercept: (0, 0)Vertical asymptote: noneHorizontal asymptote: y = 0Additional points: (-2,-0.4), (-1, -1/2), (1,

1/2)

h(x) x

x2 1

Slant Asymptotes

y = -3x – 3Is our slant asymptote

If n is exactly one more than m, then the graph of f has a slant asymptote at y = q(x), where q(x) is the quotient from the division algorithm.

Decide whether each of the following rational functions has a slant asymptote. If so, find the equation of the slant asymptote.

(a) Yes, y = x 3 (b) No

53

1)(

2

3

xx

xxf 52

23)(

3

x

xxf

Example 2. Sketch the graph of

y-Intercept: (0, 0)x-Intercept: (0, 0)Vertical asymptote: x = 2Slant asymptote: y = x + 2

Additional points: (-1/2,-0.1), (1, -1), (3, 9)

y x 2

x 2x 2

4

x 2

Slant Asymptotes

If n is exactly one more than m, then the graph of f has a slant asymptote at y = q(x), where q(x) is the quotient from the division algorithm.

2

125)(

x

xxf

.5

12..............

105.......

25.......

2

232

2

23)(

2

2

2

x

x

x

xx

xxx

x

xxxf

Sketch the graph of each of the following functions.

y-Intercept: (0, -0.25)x-Intercept: (2, 0)Vertical asymptote: x = -2 and x = 4Horizontal asymptote: y = 0Additional points: (-4, -0.375), (0, 1/4),

(3, -1/5), (6, 1/4)

g( x) x 2

x2 2x 8

Example 2. Find any horizontal and vertical asymptotes of the following.

The horizontal asymptote is y = 0. The only vertical asymptote is x = 1. There will be a hole in the graph at x = -1.

f ( x) x 1

x2 1

1st Guideline for graphing rational functions

2

3

20

3)0(

2

3)(

f

xxf

2nd Guideline for graphing rational functions

2

1

12

012

12)(

x

x

xx

xxf

3rd Guideline for graphing rational functions

2

022

3)(

x

xx

xf

4th Guideline for graphing rational functions

21

22

12)(

x

x

x

x

x

xxf

m

n

5th Guideline for graphing rational functions

x

xxf

12)(

X Y

-1 3

(1/4) -2

1 1

X – int. = (1/2)

Vert. Asym. = 0

6th Guideline for graphing rational functions

x

xxf

12)(