math 4 graphing rational functions

Graphing Rational Functions1

Mathematics 4

January 3, 2012

Mathematics 4 () Graphing Rational Functions1 January 3, 2012 1 / 1

Review of Previous Concepts on Rational Functions

Rational Functions

A rational function can be expressed as f(x) =p(x)

q(x)where p(x)

and q(x) are polynomial functions.

The domain of f(x) is R excluding the values where the denominatorq(x) is zero.

The zeros of the numerator p(x) are the zeros of the rational functionf(x).

If f(x) is in lowest terms, the zeros of the denominator q(x)determine the vertical asymptotes of the function.

Rational Functions

q(x)where p(x)

Rational Functions

q(x)where p(x)

Rational Functions

q(x)where p(x)

Rational Functions

q(x)where p(x)

The Horizontal Asymptote

Horizontal Asymptotes

The horizontal asymptote is an imaginary line that a rationalfunction approaches as x→ −∞ and x→∞.

The graph of a rational function CAN cross its horizontal asymptotes.

Consider the function below:

f(x) =3x+ 5

x− 5

f(−100) =

f(−1000) = 2.98009

f(−10000) = 2.998

f(100) = 3.2105

f(1000) = 3.0201

f(10000) = 3.002

As x→ −∞ and as x→∞, the value of f(x) becomes closer to 3.

The horizontal asymptote is the line y = 3.

f(x) =3x+ 5

x− 5

f(−100) =

f(−1000) = 2.98009

f(−10000) = 2.998

f(100) = 3.2105

f(1000) = 3.0201

f(10000) = 3.002

f(x) =3x+ 5

x− 5

f(−100) = 2.809

f(−1000) =

2.98009

f(−10000) = 2.998

f(100) = 3.2105

f(1000) = 3.0201

f(10000) = 3.002

f(x) =3x+ 5

x− 5

f(−100) = 2.809

f(−1000) =

2.98009

f(−10000) = 2.998

f(100) = 3.2105

f(1000) = 3.0201

f(10000) = 3.002

f(x) =3x+ 5

x− 5

f(−100) = 2.809

f(−1000) = 2.98009

f(−10000) =

f(100) = 3.2105

f(1000) = 3.0201

f(10000) = 3.002

f(x) =3x+ 5

x− 5

f(−100) = 2.809

f(−1000) = 2.98009

f(−10000) =

f(100) = 3.2105

f(1000) = 3.0201

f(10000) = 3.002

f(x) =3x+ 5

x− 5

f(−100) = 2.809

f(−1000) = 2.98009

f(−10000) = 2.998

f(100) =

3.2105

f(1000) = 3.0201

f(10000) = 3.002

f(x) =3x+ 5

x− 5

f(−100) = 2.809

f(−1000) = 2.98009

f(−10000) = 2.998

f(100) =

3.2105

f(1000) = 3.0201

f(10000) = 3.002

f(x) =3x+ 5

x− 5

f(−100) = 2.809

f(−1000) = 2.98009

f(−10000) = 2.998

f(100) = 3.2105

f(1000) =

3.0201

f(10000) = 3.002

f(x) =3x+ 5

x− 5

f(−100) = 2.809

f(−1000) = 2.98009

f(−10000) = 2.998

f(100) = 3.2105

f(1000) =

3.0201

f(10000) = 3.002

f(x) =3x+ 5

x− 5

f(−100) = 2.809

f(−1000) = 2.98009

f(−10000) = 2.998

f(100) = 3.2105

f(1000) = 3.0201

f(10000) =

f(x) =3x+ 5

x− 5

f(−100) = 2.809

f(−1000) = 2.98009

f(−10000) = 2.998

f(100) = 3.2105

f(1000) = 3.0201

f(10000) =

f(x) =3x+ 5

x− 5

f(−100) = 2.809

f(−1000) = 2.98009

f(−10000) = 2.998

f(100) = 3.2105

f(1000) = 3.0201

f(10000) = 3.002

f(x) =3x+ 5

x− 5

f(−100) = 2.809

f(−1000) = 2.98009

f(−10000) = 2.998

f(100) = 3.2105

f(1000) = 3.0201

f(10000) = 3.002

f(x) =3x+ 5

x− 5

f(−100) = 2.809

f(−1000) = 2.98009

f(−10000) = 2.998

f(100) = 3.2105

f(1000) = 3.0201

f(10000) = 3.002

Consider the function f(x) =3x3 + 4x2 + 7x+ 2

4x3 − 5x2 + 3x+ 1

Let’s analyze the function as x→ −∞: (please refer to the blackboard)

Now analyze the function as x→∞: (please refer to the blackboard)

The horizontal asymptote is y = 34 .

4x3 − 5x2 + 3x+ 1

degree of p(x) = degree of q(x)

If the degree of the numerator p(x) is equal to the degree of thedenominator q(x), the horizontal asymptote is

y =anbn

where an is the leading coefficient of p(x) and q(x) is the leadingcoefficient of q(x).

Consider the function f(x) =5x+ 2

3x3 − x2 − 8x− 2

The horizontal asymptote is y = 0.

3x3 − x2 − 8x− 2

degree of p(x) < degree of q(x)

If the degree of the numerator p(x) is less than the degree of thedenominator q(x), the horizontal asymptote is

Summary

The graph of f(x) =anx

n + ...+ a1x+ a0bmxm + ...+ b1x+ b0

its HA at the x-axis n < m

its HA at y =anbm

no horizontal asymptote n > m

Summary

n + ...+ a1x+ a0bmxm + ...+ b1x+ b0

its HA at y =anbm

Summary

n + ...+ a1x+ a0bmxm + ...+ b1x+ b0

its HA at y =anbm

Summary

n + ...+ a1x+ a0bmxm + ...+ b1x+ b0

its HA at y =anbm

Graphing Rational Functions

Graphing

The following information is needed to graph a rational function f(x):

1 Domain of f(x)

2 Zeros of f(x)

3 The vertical asymptotes (if any)

4 The horizontal asymptote (if any)

5 The y-intercept

6 The table of signs

Graphing

1 Domain of f(x)

2 Zeros of f(x)

5 The y-intercept

Graphing

1 Domain of f(x)

2 Zeros of f(x)

5 The y-intercept

Graphing

1 Domain of f(x)

2 Zeros of f(x)

5 The y-intercept

Graphing

1 Domain of f(x)

2 Zeros of f(x)

5 The y-intercept

Graphing

1 Domain of f(x)

2 Zeros of f(x)

5 The y-intercept

Graphing

1 Domain of f(x)

2 Zeros of f(x)

5 The y-intercept

Example: f(x) =x+ 1

x− 1

−5 −4 −3 −2 −1 1 2 3 4

Domain:

x 6= 1

Example: f(x) =x+ 1

x− 1

−5 −4 −3 −2 −1 1 2 3 4

Domain:

x 6= 1

Example: f(x) =x+ 1

x− 1

−5 −4 −3 −2 −1 1 2 3 4

Domain: x 6= 1

Example: f(x) =x+ 1

x− 1

−5 −4 −3 −2 −1 1 2 3 4

Zeros:

Example: f(x) =x+ 1

x− 1

−5 −4 −3 −2 −1 1 2 3 4

Zeros: x = −1

Example: f(x) =x+ 1

x− 1

−5 −4 −3 −2 −1 1 2 3 4

Vertical asymptotes:

Example: f(x) =x+ 1

x− 1

−5 −4 −3 −2 −1 1 2 3 4

Vertical asymptotes: x = 1

Example: f(x) =x+ 1

x− 1

−5 −4 −3 −2 −1 1 2 3 4

Horizontal asymptote:

Example: f(x) =x+ 1

x− 1

−5 −4 −3 −2 −1 1 2 3 4

Horizontal asymptote: y = 1

Example: f(x) =x+ 1

x− 1

−5 −4 −3 −2 −1 1 2 3 4

y-intercept:

Example: f(x) =x+ 1

x− 1

−5 −4 −3 −2 −1 1 2 3 4

y-intercept: −1

Example: f(x) =x+ 1

x− 1

−5 −4 −3 −2 −1 1 2 3 4

Table of signs:

−1 1

x + 1 - 0 + + +

x − 1 - - - 0 +

f(x) + 0 − VA +

Example: f(x) =x+ 1

x− 1

−5 −4 −3 −2 −1 1 2 3 4

Table of signs:

−1 1

x + 1 - 0 + + +

x − 1 - - - 0 +

f(x) + 0 − VA +

Example: f(x) =x+ 1

x− 1

−5 −4 −3 −2 −1 1 2 3 4

Table of signs:

−1 1

x + 1 - 0 + + +

x − 1 - - - 0 +

f(x) + 0 − VA +

Example: f(x) =x+ 1

x− 1

−5 −4 −3 −2 −1 1 2 3 4

Table of signs:

−1 1

x + 1 - 0 + + +

x − 1 - - - 0 +

f(x) + 0 − VA +

Example: f(x) =x+ 1

x− 1

−5 −4 −3 −2 −1 1 2 3 4

Table of signs:

−1 1

x + 1 - 0 + + +

x − 1 - - - 0 +

f(x) + 0 − VA +

Example: f(x) =x+ 1

x− 1

−5 −4 −3 −2 −1 1 2 3 4

Table of signs:

−1 1

x + 1 - 0 + + +

x − 1 - - - 0 +

f(x) + 0 − VA +

Example: f(x) =(x+ 3)(x− 1)

(x+ 1)(3− x)

−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9

Domain:

x 6= −1, 3

(x+ 1)(3− x)

−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9

Domain:

x 6= −1, 3

(x+ 1)(3− x)

−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9

Domain: x 6= −1, 3

(x+ 1)(3− x)

−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9

(x+ 1)(3− x)

−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9

Zeros −3, 1

(x+ 1)(3− x)

−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9

y-intercept

(x+ 1)(3− x)

−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9

y-intercept: −1

(x+ 1)(3− x)

−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9

vertical asymptote:

(x+ 1)(3− x)

−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9

vertical asymptote: x = −1, x = 3

(x+ 1)(3− x)

−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9

horizontal asymptote:

(x+ 1)(3− x)

−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9

horizontal asymptote: y = −1

(x+ 1)(3− x)

−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9

Table of signs:

−3 −1 1 3

x + 3 - 0 + + + + + + +

x − 1 - - - - - 0 + + +

x + 1 - - - 0 + + + + +

3 − x + + + + + + + 0 -

f(x) − 0 + VA − 0 + VA −

(x+ 1)(3− x)

−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9

Table of signs:

−3 −1 1 3

x + 3 - 0 + + + + + + +

x − 1 - - - - - 0 + + +

x + 1 - - - 0 + + + + +

3 − x + + + + + + + 0 -

f(x) − 0 + VA − 0 + VA −

(x+ 1)(3− x)

−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9

Table of signs:

−3 −1 1 3

x + 3 - 0 + + + + + + +

x − 1 - - - - - 0 + + +

x + 1 - - - 0 + + + + +

3 − x + + + + + + + 0 -

f(x) − 0 + VA − 0 + VA −Mathematics 4 () Graphing Rational Functions1 January 3, 2012 33 / 1

(x+ 1)(3− x)

−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9

Table of signs:

−3 −1 1 3

x + 3 - 0 + + + + + + +

x − 1 - - - - - 0 + + +

x + 1 - - - 0 + + + + +

3 − x + + + + + + + 0 -

(x+ 1)(3− x)

−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9

Table of signs:

−3 −1 1 3

x + 3 - 0 + + + + + + +

x − 1 - - - - - 0 + + +

x + 1 - - - 0 + + + + +

3 − x + + + + + + + 0 -

(x+ 1)(3− x)

−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9

Table of signs:

−3 −1 1 3

x + 3 - 0 + + + + + + +

x − 1 - - - - - 0 + + +

x + 1 - - - 0 + + + + +

3 − x + + + + + + + 0 -

(x+ 1)(3− x)

−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9

Table of signs:

−3 −1 1 3

x + 3 - 0 + + + + + + +

x − 1 - - - - - 0 + + +

x + 1 - - - 0 + + + + +

3 − x + + + + + + + 0 -

(x+ 1)(3− x)

−9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9

Table of signs:

−3 −1 1 3

x + 3 - 0 + + + + + + +

x − 1 - - - - - 0 + + +

x + 1 - - - 0 + + + + +

3 − x + + + + + + + 0 -

Assignment: Homework 12 due tomorrow.

Graphing Rational Functions with Oblique Asymptotes

Oblique Asymptote

A rational function f(x) = anxn+...+a1x+a0bmxm+...+b1x+b0

has an oblique asymptote if itis in lowest terms and n = m+ 1.

Functions with oblique asymptotes

f(x) =x2 − 2x+ 1

f(x) =x2 + 6x+ 8

f(x) =x3 + 7x2 − 20x− 96

x2 − 10x+ 16

Oblique Asymptote

A rational function f(x) = anxn+...+a1x+a0bmxm+...+b1x+b0

has an oblique asymptote if itis in lowest terms and n = m+ 1.

Functions with oblique asymptotes

f(x) =x2 − 2x+ 1

f(x) =x2 + 6x+ 8

f(x) =x3 + 7x2 − 20x− 96

x2 − 10x+ 16

Oblique Asymptote

The oblique asymptote is a line of the form y = mx+ b.

Finding the oblique asymptote

Take the polynomial part of the long division of the rational function.

Oblique Asymptote

The oblique asymptote is a line of the form y = mx+ b.

Finding the oblique asymptote

Take the polynomial part of the long division of the rational function.

Oblique Asymptote

Find the oblique asymptote of f(x) = x2−2x+1x :

x− 2

x2 − 2x+ 1− x2

− 2x2x

The oblique asymptote is y = x− 2.

Oblique Asymptote

x− 2

x2 − 2x+ 1− x2

− 2x2x

Oblique Asymptote

x− 2

x2 − 2x+ 1− x2

− 2x2x

Oblique Asymptote

Find the oblique asymptote of f(x) = x2+6x+8x+3 :

x2 + 6x+ 8− x2 − 3x

3x+ 8− 3x− 9

The oblique asymptote is y = x+ 3.

Oblique Asymptote

x2 + 6x+ 8− x2 − 3x

3x+ 8− 3x− 9

Oblique Asymptote

x2 + 6x+ 8− x2 − 3x

3x+ 8− 3x− 9

Oblique Asymptote

Find the oblique asymptote of f(x) = x3+7x2−20x−96x2−10x+16

x + 17

x2 − 10x+ 16)

x3 + 7x2 − 20x − 96− x3 + 10x2 − 16x

17x2 − 36x − 96− 17x2 + 170x− 272

134x− 368

Oblique Asymptote

x + 17

x2 − 10x+ 16)

x3 + 7x2 − 20x − 96− x3 + 10x2 − 16x

17x2 − 36x − 96− 17x2 + 170x− 272

134x− 368

Oblique Asymptote

x + 17

x2 − 10x+ 16)

x3 + 7x2 − 20x − 96− x3 + 10x2 − 16x

17x2 − 36x − 96− 17x2 + 170x− 272

134x− 368

Oblique Asymptote

Find the oblique asymptote of f(x) = 3x3−2x2+4x−3x2+3x+3

3x− 11

x2 + 3x+ 3)

3x3 − 2x2 + 4x − 3− 3x3 − 9x2 − 9x

− 11x2 − 5x − 311x2 + 33x+ 33

28x+ 30

The oblique asymptote is y = 3x− 11.

Oblique Asymptote

3x− 11

x2 + 3x+ 3)

3x3 − 2x2 + 4x − 3− 3x3 − 9x2 − 9x

− 11x2 − 5x − 311x2 + 33x+ 33

28x+ 30

Oblique Asymptote

3x− 11

x2 + 3x+ 3)

3x3 − 2x2 + 4x − 3− 3x3 − 9x2 − 9x

− 11x2 − 5x − 311x2 + 33x+ 33

28x+ 30

Example: f(x) = x2+6x+8x+3

−7 −6 −5 −4 −3 −2 −1 1 2

Domain:

x 6= −3

−7 −6 −5 −4 −3 −2 −1 1 2

Domain:

x 6= −3

−7 −6 −5 −4 −3 −2 −1 1 2

Domain: x 6= −3

−7 −6 −5 −4 −3 −2 −1 1 2

Zeros:

−7 −6 −5 −4 −3 −2 −1 1 2

Zeros: −4,−2

−7 −6 −5 −4 −3 −2 −1 1 2

y-intercept:

−7 −6 −5 −4 −3 −2 −1 1 2

y-intercept: 83

−7 −6 −5 −4 −3 −2 −1 1 2

vertical asymptotes:

−7 −6 −5 −4 −3 −2 −1 1 2

vertical asymptotes: x = −3

−7 −6 −5 −4 −3 −2 −1 1 2

oblique asymptote:

−7 −6 −5 −4 −3 −2 −1 1 2

oblique asymptote: y = x+ 3

−7 −6 −5 −4 −3 −2 −1 1 2

Table of signs:

−4 −3 −2

x + 2 - - - - - 0

x + 4 - 0

x + 3 - - - 0

f(x) − 0 + VA − 0 +

−7 −6 −5 −4 −3 −2 −1 1 2

Table of signs:

−4 −3 −2

x + 2 - - - - - 0

x + 4 - 0

x + 3 - - - 0

f(x) − 0 + VA − 0 +

−7 −6 −5 −4 −3 −2 −1 1 2

Table of signs:

−4 −3 −2

x + 2 - - - - - 0

x + 4 - 0

x + 3 - - - 0

f(x) − 0 + VA − 0 +

−7 −6 −5 −4 −3 −2 −1 1 2

Table of signs:

−4 −3 −2

x + 2 - - - - - 0

x + 4 - 0

x + 3 - - - 0

f(x) − 0 + VA − 0 +

−7 −6 −5 −4 −3 −2 −1 1 2

Table of signs:

−4 −3 −2

x + 2 - - - - - 0

x + 4 - 0

x + 3 - - - 0

f(x) − 0 + VA − 0 +

−7 −6 −5 −4 −3 −2 −1 1 2

Table of signs:

−4 −3 −2

x + 2 - - - - - 0

x + 4 - 0

x + 3 - - - 0

f(x) − 0 + VA − 0 +

−7 −6 −5 −4 −3 −2 −1 1 2

Table of signs:

−4 −3 −2

x + 2 - - - - - 0

x + 4 - 0

x + 3 - - - 0

f(x) − 0 + VA − 0 +

Examples on the Board

1 f(x) =(x+ 2)(x− 3)

5− x

2 f(x) =(x+ 8)(x+ 3)(x− 4)

(x− 2)(x− 8)

1 f(x) =(x+ 2)(x− 3)

5− x

2 f(x) =(x+ 8)(x+ 3)(x− 4)

(x− 2)(x− 8)

1 f(x) =(x+ 2)(x− 3)

5− x

2 f(x) =(x+ 8)(x+ 3)(x− 4)

(x− 2)(x− 8)

Good luck on tomorrow’s LT 2 and next week’s Periodic Test!

math 4 graphing rational functions

Education

domain of f x

value of f x

rational functionf x

horizontal asymptoteconsider

review of previous concepts

polynomial functions

pxqxand qx

px qxand qx