math 4 lecture on graphing rational functions

Graphing Rational Functions

Mathematics 4

July 26, 2011

Mathematics 4 () Graphing Rational Functions July 26, 2011 1 / 61

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.

Rational Functions

q(x)where p(x)

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

Rational Functions

q(x)where p(x)

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

Rational Functions

q(x)where p(x)

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.

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(x) =3x+ 5

x− 5

f(−100) =

f(x) =3x+ 5

x− 5

f(−100) = 2.809

f(−1000) =

f(x) =3x+ 5

x− 5

f(−100) = 2.809

f(−1000) =

f(x) =3x+ 5

x− 5

f(−100) = 2.809

f(−1000) = 2.98009

f(−10000) =

f(x) =3x+ 5

x− 5

f(−100) = 2.809

f(−1000) = 2.98009

f(−10000) =

f(x) =3x+ 5

x− 5

f(−100) = 2.809

f(−1000) = 2.98009

f(−10000) = 2.998

f(100) =

f(x) =3x+ 5

x− 5

f(−100) = 2.809

f(−1000) = 2.98009

f(−10000) = 2.998

f(100) =

f(x) =3x+ 5

x− 5

f(−100) = 2.809

f(−1000) = 2.98009

f(−10000) = 2.998

f(100) = 3.2105

f(1000) =

f(x) =3x+ 5

x− 5

f(−100) = 2.809

f(−1000) = 2.98009

f(−10000) = 2.998

f(100) = 3.2105

f(1000) =

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

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

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

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

The horizontal asymptote is the line y = 3.

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)

4x3 − 5x2 + 3x+ 1

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

4x3 − 5x2 + 3x+ 1

The horizontal asymptote is y = 3

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

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.

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+ a0

bmxm + ...+ b1x+ b0has:

its HA at the x-axis n < m

Summary

n + ...+ a1x+ a0

Summary

n + ...+ a1x+ a0

its HA at y =an

bmn = m

Summary

n + ...+ a1x+ a0

its HA at y =an

bmn = m

no horizontal asymptote n > m

Graphing

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

Graphing

1 Domain of f(x)

Graphing

1 Domain of f(x)

2 Zeros of f(x)

Graphing

1 Domain of f(x)

2 Zeros of f(x)

3 The vertical asymptotes (if any)

Graphing

1 Domain of f(x)

2 Zeros of f(x)

4 The horizontal asymptote (if any)

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

6 The table of signs

Example: f(x) =x+ 1

x− 1

−5 −4 −3 −2 −1 1 2 3 4

Domain:

Example: f(x) =x+ 1

x− 1

−5 −4 −3 −2 −1 1 2 3 4

Domain:

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 +

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 +

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+ 1)(3− x)

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

Domain:

(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 -

(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 -

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 −

(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 −

Assignment: Homework 12 due tomorrow.

Graphing Rational Functions with Oblique Asymptotes

Oblique Asymptote

A rational function f(x) = anxn+...+a1x+a0

bmxm+...+b1x+b0has an oblique asymptote if it

is in lowest terms and n = m+ 1.

Oblique Asymptote

A rational function f(x) = anxn+...+a1x+a0

bmxm+...+b1x+b0has an oblique asymptote if it

is 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.

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+1

Oblique Asymptote

x− 2

x2 − 2x+ 1− x2

− 2x2x

The oblique asymptote is y = x− 2.

Oblique Asymptote

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

Oblique Asymptote

x2 + 6x+ 8− x2 − 3x

3x+ 8− 3x− 9

The oblique asymptote is y = x+ 3.

Oblique Asymptote

Find the oblique asymptote of f(x) = x3+7x2

−20x−96

x2−10x+16

Oblique Asymptote

−20x−96

x2−10x+16

Oblique Asymptote

−20x−96

x2−10x+16

x + 17

x2 − 10x+ 16)

x3 + 7x2 − 20x − 96− x3 + 10x2 − 16x

17x2 − 36x − 96− 17x2 + 170x− 272

134x− 368

The oblique asymptote is y = x+ 17.

Oblique Asymptote

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

x2+3x+3:

Oblique Asymptote

x2+3x+3:

Oblique Asymptote

x2+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.

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

−7 −6 −5 −4 −3 −2 −1 1 2

Domain:

−7 −6 −5 −4 −3 −2 −1 1 2

Domain:

−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: 8

−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

−7 −6 −5 −4 −3 −2 −1 1 2

Table of signs:

−4 −3 −2

x + 2 - - - - - 0

x + 4 - 0

x + 3 - - - 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

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 lecture on graphing rational functions

domain of f x

value of f x

rational functionf x

horizontal asymptoteconsider

polynomial functions

review of previous concepts

px qxand qx

pxqxand qx

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