Asymptote Review

Asymptote Review

Degree of a function

highest exponent

Leading Coefficient

number in front of the highest exponent

Rational functions

-can have horizontal, oblique, and vertical asymptotes

Vertical Asympotes

the "invisible" vertical lines that a function can NEVER cross

Vertical Asymptotes cut the function into pieces

How to find the vertical asymptote when given only a function?

• set the denominator equal to 0

• solve for x

For Example: Find the vertical asymp. from the example shown

• set denominator equal to 0:   x - 4 = 0

• solve for x:    x  =  4

• Thus there is a vertical asymptote at x = 4

Fill in the Blank

 $f\left(x\right)=\frac{x+5}{x-6}$  ; Find the vertical asymptote

Factoring the denominator

• When I set the denominator equal to 0 I run into a problem

• How do I solve when it is x squared?

• FACTOR!  x (x  -  3)

• then set each factor equal to 0:   x = 0 and x = 3

Fill in the Blank

Find the vertical asymptotes by factoring

Horizontal Asymptotes

Invisible horizontal lines that a function can SOMETIMES cross

Horizontal asymptotes force the function to end a certain way

How to find the horizontal asymptote when given only a function?

• There are 3 rules when finding horizontal asymptotes

• 1.  If the degree of the numerator is greater than the degree of the denominator; then there are NO HORIZONTAL ASYMPTOTES

• 2.  If the degree of the numerator is less than the degree of the denominator; then the horizontal asymptote MUST EXIST AT Y = 0

• 3.  If the degree of the numerator is equal to the degree of the denominator; then the horizontal asymptote is the ratio of the leading coefficients

Ex. 2 Find the Horizontal Asymptotes

• The degree of the numerator is 0

• The degree of the denominator is 2

• The den > num: therefore there is a horizontal asymptote at y = 0

Ex. 3 Find the Horizontal asymptote

• the degree of the numerator is 1

• the degree of the denominator is 1

• Since the degrees are equal the horizontal asymptote is at y = 1/2

Multiple Choice

Find the horizontal asymptotes:

 $f\left(x\right)=\frac{x-4}{-4x-16}$  

•  $y=-4$ 

•  $y=4$  

•  $y\ =\frac{-1}{4}$  

•  $y\ =\frac{1}{4}$  

Multiple Choice

Find the horizontal asymptotes

 $f\left(x\right)=\frac{x^3-9x}{3x^2-6x=9}$  

•  $y=0$  

•  $y=3$  

•  $y=-3$  

• DNE