Simplifying Rational Expressions
Simplify the rational expression.
(x^2+5x-14)/(x^2-4x+4)
(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 is the general form of a reciprocal function f(x)?
y= 1/f(x)
Simplify the rational expression.
(x^2-9x+14)/(x^2-5x-14)
(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.
How can you find the vertical asymptotes of a reciprocal function algebraically?
By setting the denominator equal to zero and solving for x.
Simplify the rational expression.
(-x^3+3x)/(x^2+x-20)
((-x)(x^2-3))/((x+5)(x-4))
What is the x-intercept of the function
f(x) = ((x + 2)(x - 1))/(x - 1).
x=1
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.
As x approaches positive or negative infinity, what happens to the values of a reciprocal function?
The values of a reciprocal function approach zero.
Simplify the rational expression.
(2x^2+2x-4)/(x^2-5x-14)
(2(x-1))/(x-7)
What is the y-intercept of the function
f(x) = ((x + 6)(x + 2))/(x + 2)
y=6
Find both the horizontal and vertical asymptotes.
f(x)=3/(x+8)^2-3
horizontal asymptote: y=-3
vertical asymptote: x=-8
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.
If you multiply a reciprocal function by a constant greater than 1, what happens to its graph?
Its graph is vertically stretched.
Simplify the rational expression.
(8x^2+10x-3)/(6x^2+13x+6)
(4x-1)/(3x+2)
Determine the x and y intercepts of the following function.
(-2(x+1))/(x+4)
(0, -1/2), (-1, 0)
How do we determine the equation of the oblique asymptote of a rational function.
Oblique asymptote is the quotient obtained by using long division to divide the numerator by the denominator.
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
How do we determine invariant points of a reciprocal function?
Set the function equal to 1 and -1 and solve for x.