Sec 4.2

Rational Functions

Math 1051 - Precalculus I

Rational Functions Sec 4.2

Sec 4.2 Rational Functions

Graph f (x) = (x + 3)2(x − 4)

Rational Functions Sec 4.2

Sec 4.2 Rational Functions

Graph f (x) = (x + 3)2(x − 4)Short Version:

1 Intercepts/zeros2 Even multiplicity touches. Odd multiplicity crosses.3 End behavior4 Turning points5 Behavior near zeros/intercepts6 Plot points

Rational Functions Sec 4.2

Sec 4.2 Rational Functions

Graph f (x) = (x + 3)2(x − 4)

-10 -5 5 10

-50

-40

-30

-20

-10

10

Rational Functions Sec 4.2

Graph f (x) = (x + 2)2(x − 2)2

-6 -4 -2 2 4 6

5

10

15

20

25

Rational Functions Sec 4.2

Graph f (x) = (x + 2)2(x − 2)2

-6 -4 -2 2 4 6

5

10

15

20

25

Rational Functions Sec 4.2

Rational functions

A rational function has the form

R(x) =p(x)q(x)

where p and q are polynomials and q 6= 0.

For example,

f (x) =x2 + 3x − 4

x − 2with x 6= 2.

Rational Functions Sec 4.2

Rational functions

A rational function has the form

R(x) =p(x)q(x)

where p and q are polynomials and q 6= 0.

For example,

f (x) =x2 + 3x − 4

x − 2with x 6= 2.

Rational Functions Sec 4.2

Rational functions

A rational function has the form

R(x) =p(x)q(x)

where p and q are polynomials and q 6= 0.

For example,

f (x) =x2 + 3x − 4

x − 2with x 6= 2.

Rational Functions Sec 4.2

Graphing Rational Functions

f (x) =1x2

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

Rational Functions Sec 4.2

Graphing Rational Functions

f (x) =1x2

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

Rational Functions Sec 4.2

Graphing Rational Functions

f (x) =1x2

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

Rational Functions Sec 4.2

f (x) =−4

(x + 3)2 − 2

is a rational function

We can graph it using transformations...follow a particular point(maybe the point (1,1), for example)...

-6 -4 -2 2 4 6

-10

-8

-6

-4

-2

2

4

Rational Functions Sec 4.2

f (x) =−4

(x + 3)2 − 2

is a rational function

We can graph it using transformations...follow a particular point(maybe the point (1,1), for example)...

-6 -4 -2 2 4 6

-10

-8

-6

-4

-2

2

4

Rational Functions Sec 4.2

f (x) =−4

(x + 3)2 − 2

is a rational function

We can graph it using transformations...follow a particular point(maybe the point (1,1), for example)...

-6 -4 -2 2 4 6

-10

-8

-6

-4

-2

2

4

Rational Functions Sec 4.2

There are some important features to observe in the graph:

-6 -4 -2 2 4 6

-10

-8

-6

-4

-2

2

4

Rational Functions Sec 4.2

Vertical Asymptotes

Vertical Asymptote: If, as x approaches some number c, R(x)approaches infinity, the line x = c is a vertical asymptote.

The graph NEVER crosses a vertical asymptote.

To find vertical asymptotes:

Reduce the fractionFind x that make the denominator = 0If “multiplicity” is even, graph on either side both go up orboth go down (think of 1/x2)If “multiplicity” is odd, graph on either side goes in oppositedirections (think of 1/x)

Rational Functions Sec 4.2

Vertical Asymptotes

Vertical Asymptote: If, as x approaches some number c, R(x)approaches infinity, the line x = c is a vertical asymptote.

The graph NEVER crosses a vertical asymptote.

To find vertical asymptotes:

Reduce the fractionFind x that make the denominator = 0If “multiplicity” is even, graph on either side both go up orboth go down (think of 1/x2)If “multiplicity” is odd, graph on either side goes in oppositedirections (think of 1/x)

Rational Functions Sec 4.2

Vertical Asymptotes

Vertical Asymptote: If, as x approaches some number c, R(x)approaches infinity, the line x = c is a vertical asymptote.

The graph NEVER crosses a vertical asymptote.

To find vertical asymptotes:Reduce the fraction

Find x that make the denominator = 0If “multiplicity” is even, graph on either side both go up orboth go down (think of 1/x2)If “multiplicity” is odd, graph on either side goes in oppositedirections (think of 1/x)

Rational Functions Sec 4.2

Vertical Asymptotes

Vertical Asymptote: If, as x approaches some number c, R(x)approaches infinity, the line x = c is a vertical asymptote.

The graph NEVER crosses a vertical asymptote.

To find vertical asymptotes:Reduce the fractionFind x that make the denominator = 0

If “multiplicity” is even, graph on either side both go up orboth go down (think of 1/x2)If “multiplicity” is odd, graph on either side goes in oppositedirections (think of 1/x)

Rational Functions Sec 4.2

Vertical Asymptotes

Vertical Asymptote: If, as x approaches some number c, R(x)approaches infinity, the line x = c is a vertical asymptote.

The graph NEVER crosses a vertical asymptote.

To find vertical asymptotes:Reduce the fractionFind x that make the denominator = 0If “multiplicity” is even, graph on either side both go up orboth go down (think of 1/x2)

If “multiplicity” is odd, graph on either side goes in oppositedirections (think of 1/x)

Rational Functions Sec 4.2

Vertical Asymptotes

Vertical Asymptote: If, as x approaches some number c, R(x)approaches infinity, the line x = c is a vertical asymptote.

The graph NEVER crosses a vertical asymptote.

To find vertical asymptotes:Reduce the fractionFind x that make the denominator = 0If “multiplicity” is even, graph on either side both go up orboth go down (think of 1/x2)If “multiplicity” is odd, graph on either side goes in oppositedirections (think of 1/x)

Rational Functions Sec 4.2

Find the vertical asymptotes of

f (x) =x2 + 3x + 2

x2 − 1

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

But there should be a hole at x = −1!

Rational Functions Sec 4.2

Find the vertical asymptotes of

f (x) =x2 + 3x + 2

x2 − 1

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

But there should be a hole at x = −1!

Rational Functions Sec 4.2

Find the vertical asymptotes of

f (x) =x2 + 3x + 2

x2 − 1

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

But there should be a hole at x = −1!Rational Functions Sec 4.2

Horizontal Asymptotes

Horizontal Asymptote: If, as x approaches +∞ or −∞, R(x)approaches some fixed number L, then the line y = L is ahorizontal asymptote.

For example, in the function f (x) = x2+3x+2x2−1 we just looked at

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

The line y = 1 is a horizontal asymptote.

Rational Functions Sec 4.2

Horizontal Asymptotes

Horizontal Asymptote: If, as x approaches +∞ or −∞, R(x)approaches some fixed number L, then the line y = L is ahorizontal asymptote.

For example, in the function f (x) = x2+3x+2x2−1 we just looked at

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

The line y = 1 is a horizontal asymptote.Rational Functions Sec 4.2

Two possibilities

Case 1: If (degree of top) < (degree of bottom), then y = 0 is ahorizontal asymptote.

f (x) =x + 2

x2 + 7x + 12

-6 -4 -2 2 4 6

-10

-5

5

10

Rational Functions Sec 4.2

Two possibilities

Case 1: If (degree of top) < (degree of bottom), then y = 0 is ahorizontal asymptote.

f (x) =x + 2

x2 + 7x + 12

-6 -4 -2 2 4 6

-10

-5

5

10

Rational Functions Sec 4.2

Two possibilities

Case 1: If (degree of top) < (degree of bottom), then y = 0 is ahorizontal asymptote.

f (x) =x + 2

x2 + 7x + 12

-6 -4 -2 2 4 6

-10

-5

5

10

Rational Functions Sec 4.2

Two possibilities

Case 2: If (degree of top) = (degree of bottom), then y = anbn

is ahorizontal asymptote.

f (x) =3x2 − 5x − 2

x2 − 1

-10 -5 5 10

-10

-5

5

10

Rational Functions Sec 4.2

Two possibilities

Case 2: If (degree of top) = (degree of bottom), then y = anbn

is ahorizontal asymptote.

f (x) =3x2 − 5x − 2

x2 − 1

-10 -5 5 10

-10

-5

5

10

Rational Functions Sec 4.2

Two possibilities

Case 2: If (degree of top) = (degree of bottom), then y = anbn

is ahorizontal asymptote.

f (x) =3x2 − 5x − 2

x2 − 1

-10 -5 5 10

-10

-5

5

10

Rational Functions Sec 4.2

More possibilities

If (degree of top) = (degree of bottom) + 1

y = mx + b is called an oblique asymptote

f (x) =x2 + 3x − 4

x − 5

Rational Functions Sec 4.2

More possibilities

If (degree of top) = (degree of bottom) + 1

y = mx + b is called an oblique asymptote

f (x) =x2 + 3x − 4

x − 5

Rational Functions Sec 4.2

More possibilities

If (degree of top) = (degree of bottom) + 1

y = mx + b is called an oblique asymptote

f (x) =x2 + 3x − 4

x − 5

Rational Functions Sec 4.2

More possibilities

If (degree of top) = (degree of bottom) + 1

y = mx + b is called an oblique asymptote

f (x) =x2 + 3x − 4

x − 5

-30 -20 -10 10 20 30

-20

-10

10

20

30

40

50

Rational Functions Sec 4.2

More possibilities

If (degree of top) = (degree of bottom) + 1

y = mx + b is called an oblique asymptote

f (x) =x2 + 3x − 4

x − 5

-30 -20 -10 10 20 30

-20

-10

10

20

30

40

50

Rational Functions Sec 4.2

More possibilities

What if (degree of top) > (degree of bottom) + 1 ?

There is no horizontal or oblique asymptote

f (x) =x4 − 50x + 2

x2 + 5

-6 -4 -2 2 4 6

-10

-5

5

10

15

20

Rational Functions Sec 4.2

More possibilities

What if (degree of top) > (degree of bottom) + 1 ?

There is no horizontal or oblique asymptote

f (x) =x4 − 50x + 2

x2 + 5

-6 -4 -2 2 4 6

-10

-5

5

10

15

20

Rational Functions Sec 4.2

More possibilities

What if (degree of top) > (degree of bottom) + 1 ?

There is no horizontal or oblique asymptote

f (x) =x4 − 50x + 2

x2 + 5

-6 -4 -2 2 4 6

-10

-5

5

10

15

20

Rational Functions Sec 4.2

More possibilities

What if (degree of top) > (degree of bottom) + 1 ?

There is no horizontal or oblique asymptote

f (x) =x4 − 50x + 2

x2 + 5

-6 -4 -2 2 4 6

-10

-5

5

10

15

20

Rational Functions Sec 4.2

One more example

f (x) =x3 − 1x − x2

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

With a hole at x = 1!

Rational Functions Sec 4.2

One more example

f (x) =x3 − 1x − x2

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

With a hole at x = 1!

Rational Functions Sec 4.2

Read section 4.3 for Friday.

We will learn techniques to graph rational functions by hand.

Rational Functions Sec 4.2