Rational Functions and Asymptotes

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

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Example B You tryExample A

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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

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€

€

f (x) =3x 2 −1

x 2

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

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What is an asymptote?

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

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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.

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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

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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.

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Case 2

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f (x) =3x 2 +2x

x − 4HA : none

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

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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:

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Slant Asymptote

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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.

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Example A

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Find all asymptotes of the function g(x).

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Vertical AsymptotesSet the denominator equal to zero and solve.

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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

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Find all asymptotes of the function f(x).

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Vertical Asymptotes:Set the denominator equal to zero and solve

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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

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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

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f (x) =4

x 2 −1

Hmmmm, better try again

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Back to choices

Correct !

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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.

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References

References

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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