Simplifying Rational Functions to find Holes
Simplify the rational function and identify any holes.
f(x)=(x^2+5x-14)/(x^2-4)
Hole at x = 2
f(x)=(x+7)/(x+2)
How is the x-intercept determined?
Set the function equal to 0 and solve for x.
What is the equation of the vertical asymptote?
f(x)=1/(x-3) +2
x=3
As x approaches positive or negative infinity, what determines the end behavior of a rational function?
Horizontal or Oblique Asymptote
What causes a hole (AKA removable discontinuity) in a function's graph?
A hole happens when a function in factored form, has a common factor in the numerator and the denominator that cancel.
Simplify the rational function and identify any holes.
f(x)=(x^2-9x+14)/(x^2-5x-14)
Hole at x = 7
f(x)=(x-2)/(x+2)
How is the y-intercept determined?
Evaluate the function at x=0.
What is the equation of the horizontal asymptote?
f(x)=4/(x+2)-1
y=-1
What happens to the graph of a rational function at the vertical asymptotes in terms of the behavior of the function's values?
The function approaches positive or negative infinity.
This is the horizontal asymptote of the rational function
f(x)=(15x-9)/(21x+4)
What is
y=5/7?
Simplify the rational function and identify any holes.
g(x)=(-x^3+3x)/(x^2+x-20)
No holes
g(x)=((-x)(x^2-3))/((x+5)(x-4))
Identify any x-intercept(s) of the function
f(x) = ((x + 2)(x - 1))/(x - 5).
x = 1 and x = -2
f(x)=3/(x-1)-2
State the domain and range.
domain:
x!=1
range:
y!=-2
What is the end behavior of a rational function when the degree of the numerator is equal to the degree of the denominator?
Approaches a non-zero horizontal asymptote.
How do you find the y coordinate of a hole?
After cancelling common factors, you substitute the x coordinate of the hole into the remaining part of the rational function and simplify.
Simplify the rational function and identify any holes.
h(x)=(2x^2+2x-4)/(x^2-8x-20)
Hole at x = -2
h(x)=(2(x-1))/(x-10)
What is the y-intercept of the function
f(x) = ((x + 6)(x + 2))/(x + 2)
y=6
Find the slant asymptote.
f(x)=\frac{x^3-x+1}{x^2 + 1}
The slant asymptote is y = x - 2
What is the end behavior of a rational function when the degree of the numerator is less than the degree of the denominator?
Approaches the horizontal asymptote, y=0.
Find the hole of the rational function, as well as any HA or VA.
f(x)=(x^2 - 1)/(x^2 - 3x +2)
The hole is located at the point (1, -2).
The HA is y = 1 and the VA is x = 2
Simplify the rational function and identify any holes.
q(x)=(8x^2+10x-3)/(6x^2+13x+6)
Hole at x=-3/2
q(x)= (4x-1)/(3x+2)
Determine the x and y intercepts of the following function.
p(x)=(-2(x+1))/(x+4)
(0, -1/2), (-1, 0)
Determine the slant asymptote of the rational function
f(x)=\frac{2x^3+x^2-3}{x^2+2}
The slant asymptote is y = 2x
What is the end behaviour of a rational function when the degree of the numerator is one greater than the degree of the denominator?
Approaches Oblique (slant) asymptote
What is the simplified version of this function?
g(x)=(9x)/(x^2+2x+1)-(7)/(x+1)
What is
g(x)=(2x-7)/(x+1)^2 or (2x-7)/(x^2+2x+1)?