rational functions and their graphs section 3.5 jmerrill,2005 revised 08

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Rational Functions and Their Graphs Section 3.5 JMerrill,2005 Revised 08

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Rational Functions and Their GraphsSection 3.5

JMerrill,2005

Revised 08

Why Should You Learn This? Rational functions are used to model and

solve many problems in the business world.

Some examples of real-world scenarios are:Average speed over a distance (traffic

engineers)Concentration of a mixture (chemist)Average sales over time (sales manager)Average costs over time (CFO’s)

Introduction to Rational Functions

What is a rational number?

So just for grins, what is an irrational number?

A rational function has the form ( )

( )( )

p xf x

q x

where p and q are polynomial functions

A number that can be expressed as a fraction:

A number that cannot be expressed as a fraction: , 2

5, 3, 4.5

2

Parent Function The parent function is

The graph of the parent rational function looks like…………………….

The graph is not continuous and has asymptotes

1

x

Transformations

The parent function How does this move?

1

x 13

x

Transformations

The parent function How does this move?

1

x

1

( 3)x

Transformations

The parent function And what about this?

1

x

14

( 2)x

Transformations

The parent function

How does this move?

1

x

2

1

x

Transformations

2

1

x 2

12

x

2

14

( 3)x

2

1

( 3)x

Domain

Find the domain of 2x1f(x)

Denominator can’t equal 0 (it is undefined there)

2 0

2

x

x

Domain , 2 2,

Think: what numbers can I put in for x????

You Do: Domain

Find the domain of 2)1)(x(x

1-xf(x)

Denominator can’t equal 0

1 2 0

1, 2

x x

x

Domain , 2 2, 1 1,

You Do: Domain

Find the domain of 2

xf(x)x 1

Denominator can’t equal 02

2

1 0

1

x

x

Domain ,

Vertical AsymptotesAt the value(s) for which the domain is undefined, there will be one or more vertical asymptotes. List the vertical asymptotes for the problems below.

2x1f(x)

2x

2)1)(x(x1-xf(x)

1, 2x x

2

xf(x)x 1

none

Vertical Asymptotes

The figure below shows the graph of 2x1f(x)

The equation of the vertical asymptote is 2x

Vertical Asymptotes

Definition: The line x = a is a vertical asymptote of the graph of f(x) if f x

or f x as x approaches “a” either from the left or from the right.

Look at the table of values for 2x

1f(x)

Vertical Asymptotesx f(x)

-3 -1

-2.5 -2

-2.1 -10

-2.01 -100

-2.001 -1000

As x approaches____ from the _______,

f(x) approaches _______.

-2

left

x f(x)

-1 1

-1.5 2

-1.9 10

-1.99 100

-1.999 1000

As x approaches____ from the _______,

f(x) approaches _______.

-2right

Therefore, by definition, there is a vertical asymptote at

2x

Vertical Asymptotes - 4Describe what is happening to x and determine if a vertical asymptote exists, given the following information:

x f(x)-4 -1.333

-3.5 -2.545

-3.1 -12.16

-3.01 -120.2

-3.001 -1200

x f(x)-2 1-2.5 2.2222

-2.9 11.837

-2.99 119.84

-2.999 1199.8

As x approaches____ from the _______, f(x) approaches _______.

As x approaches____ from the _______, f(x) approaches _______.

-3 -3

left right

Therefore, a vertical asymptote Therefore, a vertical asymptote occurs at x = -3.occurs at x = -3.

Vertical Asymptotes

Set denominator = 0; solve for x Substitute x-values into numerator. The

values for which the numerator ≠ 0 are the vertical asymptotes

Example

What is the domain? x ≠ 2 so

What is the vertical asymptote? x = 2 (Set denominator = 0, plug back into

numerator, if it ≠ 0, then it’s a vertical asymptote)

( , 2) (2, )

22 3 1( )

2

x xf x

x

You Do

Domain: x2 + x – 2 = 0 (x + 2)(x - 1) = 0, so x ≠ -2, 1

Vertical Asymptote: x2 + x – 2 = 0 (x + 2)(x - 1) = 0 Neither makes the numerator = 0, so x = -2, x = 1

( , 2) ( 2,1) (1, )

2

2

2 7 4( )

2

x xf x

x x

The graph of a rational function NEVER crosses a vertical asymptote. Why?

Look at the last example:

Since the domain is , and the vertical asymptotes are x = 2, -1, that means that if the function crosses the vertical asymptote, then for some y-value, x would have to equal 2 or -1, which would make the denominator = 0!

( , 1) ( 1,2) (2, )

2

2

2 7 4( )

2

x xf x

x x

Points of Discontinuity (Holes)

Set denominator = 0. Solve for x Substitute x-values into numerator. You

want to keep the x-values that make the numerator = 0 (a zero is a hole)

To find the y-coordinate that goes with that x: factor numerator and denominator, cancel like factors, substitute x-value in.

Example

Function:

Solve denom.

Factor and cancel

Plug in -2:

2

2

4( )

2 8

xf x

x x

2 2 8 0

( 4)( 2) 0

4, 2

x x

x x

x

( 2)( 2)

( 4)( 2)

x x

x x

( 2) 2 2 4 2

( 4) 2 4 6 3

x

x

22,

3 Hole is

Asymptotes

Some things to note: Horizontal asymptotes describe the behavior at the

ends of a function. They do not tell us anything about the function’s behavior for small values of x. Therefore, if a graph has a horizontal asymptote, it may cross the horizontal asymptote many times between its ends, but the graph must level off at one or both ends.

The graph of a rational function may or may not cross a horizontal asymptote.

The graph of a rational function NEVER crosses a vertical asymptote. Why?

Horizontal Asymptotes

Definition:The line y = b is a horizontal asymptote if f x b as x or x

Look at the table of values for f x 1

x 2

Horizontal Asymptotesx f(x)

1 .3333

10 .08333

100 .0098

1000 .0009

y→_____ as x→________

0

x f(x)

-1 1

-10 -0.125

-100 -0.0102

-1000 -0.001

y→____ as x→________

0

Therefore, by definition, there is a horizontal Therefore, by definition, there is a horizontal asymptote asymptote at y = 0.at y = 0.

Examples

f xx

( )

4

12f x

x

x( )

2

3 12

What similarities do you see between problems?

The degree of the denominator is larger than the degree of the numerator.

Horizontal Asymptote at y = 0

Horizontal Asymptote at y = 0

Examples

h xx

x( )

2 1

1 82x

15xg(x)

2

2

What similarities do you see between problems?

The degree of the numerator is the same as the degree or the denominator.

Horizontal Asymptote at y = 2

Horizontal Asymptote at 5

2y

Examples

13x

54x5x3xf(x)

23

2x

9xg(x)

2

What similarities do you see between problems?

The degree of the numerator is larger than the degree of the denominator.

No Horizontal Asymptote

No Horizontal Asymptote

Asymptotes: Summary1. The graph of f has vertical asymptotes at the _________ of q(x).

 2. The graph of f has at most one horizontal asymptote, as follows:

 a)   If n < d, then the ____________ is a horizontal asymptote.

b)    If n = d, then the line ____________ is a horizontal asymptote (leading coef. over leading coef.)

c)   If n > d, then the graph of f has ______ horizontal asymptote.

zeros

line y = 0

no

ay

b

You DoFind all vertical and horizontal asymptotes of the following function

2 1

1

xf x

x

Vertical Asymptote: x = -1

Horizontal Asymptote: y = 2

You Do AgainFind all vertical and horizontal asymptotes of the following function

2

4

1f x

x

Vertical Asymptote: none

Horizontal Asymptote: y = 0

Oblique/Slant Asymptotes

The graph of a rational function has a slant asymptote if the degree of the numerator is exactly one more than the degree of the denominator. Long division is used to find slant asymptotes.

The only time you have an oblique asymptote is when there is no horizontal asymptote. You cannot have both.

When doing long division, we do not care about the remainder.

ExampleFind all asymptotes.

2 2

1

x xf x

x

Vertical

x = 1

Horizontal

none

Slant

2

2

1 2

-2

x

x x x

x x

y = x

Example

Find all asymptotes: 2 2

( )1

xf x

x

Vertical asymptote at x = 1

n > d by exactly one, so no horizontal asymptote, but there is an oblique asymptote.

2

2

11 2

2

( 1)

1

-

xx x

x x

x

x

y = x + 1

Solving and Interpreting a Given Scenario

The Average Cost of Producing a Wheelchair A company that manufactures wheelchairs

has costs given by the function C(x) = 400x + 500,000, where the x is the number of wheelchairs produced per month and C(x) is measured in dollars. The average cost per wheelchair for the company is given by …

Original: C(x) = 400x + 500,000

C(x) = 400x + 500,000

x

a. Find the interpret C(1000), C(10,000), C(100,000).

C(1000) = 900; the average cost of producing 1000 wheelchairs per month is $900.

C(x) = 400x + 500,000 x Find the interpret C(10,000)

C(10,000) = 450; the average cost of producing 10,000 wheelchairs per month is $450.

Find the interpret C(100,000)

C(100,000) = 405; the average cost of producing 100,000 wheelchairs per month is $405.

C(x) = 400x + 500,000 x

b. What is the horizontal asymptote for the average cost function?

Since n = d (in degree) then y = 400

Describe what this represents for the company.

C(x) = 400x + 500,000 xThe horizontal asymptote

means that the more wheelchairs produced per month, the closer the average cost comes to $400. Lower prices take place with higher production levels, posing potential problems for small businesses.