Rational Functions and their Asymptotes
A rational function with vertical asymptotes at x = 5, x = -1 and horizontal asymptote at x = 0
Describe y = 1/(x-5)(x+1)
Domain
Do vertical asymptotes affect the domain or the range?
x=-1, x=3; y=0
Find the asymptotes of
x=-2; y=3
Find the asymptotes of
x=-3, x=1; y=0
Find the asymptotes of
x=0, x=3; y=-1
Find the asymptotes of
x=0; y=-3
Find the asymptotes of
x=3, x=-1; y=0
Find the asymptotes of
x=3, x=-2; y=0
Find the asymptotes of
x=3, x=-3; y=1/3
Find the asymptotes of
x=3, x=-3; y=3
Find the asymptotes of
x=1/3, y=2/3
Find the asymptotes of f(x) = (-2x)/(1-3x)
x=-1/3, y =-2/3
Find the asymptotes of f(x) = (-2x)/(3x+1)
x=3, y=3
Find the asymptotes of f(x) = (3x+5)/(x-3)
x=-1/3
Find the asymptotes of f(x) = (3x³-2x)/(3x+1)
x=1, y=1
Find the asymptotes of f(x) = (x+1)/(x-1)
x=-1, y=0
Find the asymptotes of f(x) = (x+1)/(x²-1)
x=1, y=x-2
Find the asymptotes of f(x) = (x³+x)/(x²+2x+1)
x=1, y=x+1
Find the asymptotes of f(x) = x²/(x-1)
x=1, x=-1, y=1
Find the asymptotes of f(x) = x²/(x²-1)
x=1, x=-1, y=0
Find the asymptotes of f(x) = x⁶/(x⁸-1)
Range
Horizontal asymptotes affect the domain or range?
Vertical Asymptote at x = -3 and x=4
Identify discontinuities in the graph
Vertical asymptote at x=-2
Identify discontinuities in the graph
HA y=0
The type of asymptote that results when the denominator's leading degree is higher than the numerator's
Vertical Asymptotes (VA)
These occur at the values that make the denominator equal 0
The leading degree in the numerator and denominator are the same
This allows us to find the HA by dividing the numerator's and denominator's leading coefficients
Horizontal Asymptotes (HA)
This is determined by comparing the Degrees of Numerator and Denominator in a rational function.
The leading degree in the numerator is greater than the leading degree in the denominator
This trait leads to no HA, but instead a slant/oblique asymptote
y=
What is the equation of a horizontal asymptote?
x=
What is the equation of a vertical asymptote?
