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Vocabulary
Complete the table.
Word  Definition 
Vertical Asymtote  ___________________________________________________________ 
___________  A horizontal line marking a specific value toward which the graph of a function may approach, but will generally not cross 
Oblique Asymtote  ___________________________________________________________ 
___________  function that is a ratio of two polynomials 
End Behavior  ___________________________________________________________ 
___________  A single point that is not possible on the graph of an otherwise continuous function 
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Remember: Rational functions are defined as
In your own words, what is an asymtote? ________________________________________
Asymtotes
How do you find asymtotes?
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Vertical
To find a vertical asymtote:

Set the denominator to 0

Solve for x
This graph has two vertical asymtotes and one horizontal asymtote:
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For more on vertical asymtotes, click here.
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Horizontal
To find a horizontal asymtote:

Put in standard form

Remove all terms but the one where x has the largest exponent in numerator and denominator

Three possibilities:

If the degree of the numerator is smaller than the degree of the denominator, the horizontal asymptote crosses the
y− axis aty=0 
If the degree of the denominator and the numerator are the same, the horizontal asymptote equals to the ratio of the leading coefficients

If the degree of the numerator is larger than the degree of the denominator, then there is no horizontal asymptote

Oblique
If the degree of the numerator is one greater than the degree of the denominator, then there is an oblique asymtote. To find this asymtote:
 Divide the numerator and the denominator

Rewrite that function in the form
f(x)D(x)=Q(x)+R(x)D(x) whereQ(x) is the quotient andR(x) is the remainder 
Q(x) is the asymtote
Oblique asymtotes are diagonal lines:
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For more on oblique asymtotes, click here.
Holes
Holes occur when one term in the numerator cancels with one term in the denominator. It is not an asymtote, it is merely a missing point.
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For more on holes, click here.
Practice
Identify the asymtotes and holes. Then sketch the graph.

f(x)=x+73x−10 
f(x)=x3+7x3x2−6x+13 
f(x)=x2+2x−4x+8 
f(x)=x+14x−10x2+x+2 
f(x)=4x3+9x 
y=4x3x2+3 
y=(3x)(x+8)x+2 
f(x)=x3+4x2 
y=4x4−22x3
Factor the numerator and denominator, then set the denominator equal to zero and solve to find restrictions on the domain.

y=x2+3x−10x−2 
f(x)=x2+2x−24x−4 
f(x)=x2−12x+32x−4 
y=x2+2120x+45 
y=x2+13x+42x+7