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Rational Functions and Asymptotes. Let’s find: vertical , horizontal, and slant asymptotes when given a rational function. Get Started. MAIN MENU. All done?. Example A. Example B. You try. What is a Rational Function?. A function that is the ratio of two polynomials. A few examples……. - PowerPoint PPT Presentation
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Rational Functions and Asymptotes
Let’s find: vertical, horizontal, and slant asymptotes when given a rational function.
Get Started
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”
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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
€
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
€
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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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
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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
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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
€
f (x) =4
x 2 −1
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