Rational Functions and Asymptotes

Rational Functions and Asymptotes

Let’s find: vertical, horizontal, and slant asymptotes when given a rational function.

Get Started

Example B You tryExample A

MAINMENU

All done?

What is a Rational Function?A function that is the ratio of two polynomials. A few examples……

€

f (x) =2x − 7

x 3 − 4x 2 −5x€ €

f (x) =x 2 +5

x −2

€

€

€

f (x) =3x 2 −1

x 2

It is “Rational” because one is divided by the other, like a ratio. Return t

o Main Menu

What is an asymptote?

A line that a curve approaches but never reaches, or “touches”

Return to main menu

Vertical Asymptotes

To find the vertical asymptote of a function, we must set the denominator equal to zero and solve for x.

€

f (x) =x 2 +5

x −2

€

x −2 = 0

x = 2

There will be a vertical asymptote, x = 2.

Back to Main Menu

Horizontal Asymptotes

Compare the degrees of the numerator and denominator. We will let the denominator degree be “d” and the numerator degree be “n.”

€

€ There are 3 cases

Case 1n < d

Case 2n > d

Case 3n = d

Case 1

€

f (x) =5x 2 +2x

7x 2 −6

HA: y = 0

If n < d, then y = 0 is a horizontal asymptote(HA) of the function.

Return to main menu

Case 2

€

f (x) =3x 2 +2x

x − 4HA : none

If n > d, then there is no horizontal asymptote (HA).

Return to main menu

Case 3

€

f (x) =5x 2 +2x

7x 2 −6

€

y =5

7

If n = d, then the horizontal asymptote is the ratio of the numerator leading coefficient over the denominator leading coefficient.

HA:

Return to main menu

Slant Asymptote

Return to Main Menu

If the degree of the numerator (n) is EXACTLY one greater that the degree of

the denominator (d), there will be a slant asymptote.

Since the degree of the numerator is 2 and the degree of the denominator is 1, there will be a slant asymptote.

Lets find the slant asymptote€

f (x) =x 2 − x −2

x +1

Let’s find the slant asymptote

We must divide the denominator into the

numerator

€

x +1 x 2 − x −2x −2

) €

f (x) =x 2 − x −2

x +1

The line y = x – 2 is the slant asymptote of the rational function.

Please note, if there is a remainder upon dividing, we discard it as it will have no affect on the rational function as x approaches infinity.

Return to Main Menu

Example A

Return to Main

Menu

Find all asymptotes of the function g(x).

€

Vertical AsymptotesSet the denominator equal to zero and solve.

€

x 2 − 4 = 0

x 2 = 4

x = ±2€

g(x) =4x

x 2 − 4

VA: x =2, x=-2

Horizontal AsymptotesCompare degrees….Numerator-degree of 1Denominator-degree of 2Since n < d,

HA: y = 0

Example B

Return to Main Menu

Find all asymptotes of the function f(x).

€

Vertical Asymptotes:Set the denominator equal to zero and solve

€

x 2 + x −6 = 0

x + 3( ) x −2( ) = 0

x + 3 = 0

x −2 = 0

x = −3,x = 2

VA: x = -3, x = 2

Horizontal Asymptotes:Compare degrees….Numerator-degree of 2Denominator-degree of 2Since n = d,

HA: y = 2

€

f (x) =2x 2 + 3x

x 2 + x −6

You Try

Return to Main Menu

Find all asymptotes for

€

f (x) =4

x 2 +1

A.) VA: none, HA: y = 1

B) VA: x = -1, HA: y=0

C) VA: x = 1, x = -1, HA: y = 0

€

f (x) =4

x 2 −1

Hmmmm, better try again

Return to Main Menu

Back to choices

Correct !

Return to Main Menu

Back to choices

All Done? Let’s Review!•To find vertical asymptotes - set the denominator equal to zero and solve.

•To find horizontal asymptotes - compare the degree of the numerator to the degree of the denominator.

• To find slant asymptotes - divide the numerator by the denominator.

Return to Main Menu

References

References

Return to Main Menu

Retrieved 9/22/13 Dreamstimehttp://www.animationlibrary.com animation/22939/Yellow_dog_thinks/

Retrieved 9/22/13 Dreamstimehttp/www.animationlibrary.com/animation/22938/Yellow_dog_sings/

Larson, R. (2011). Algebra and Trigonometry. Belfast, CA: Cengage

Asymptotes. Retrieved from Mathisfun.com