Use the function to answer the following questions:

Question

Use the function f(x) = x^2-2x-8 / x^2-9 to answer the questions:

 

1.) Find all zeros of f(x):

 

 

2.) Find any y-intercepts of f(x):

 

 

3.) Find the equation(s) of all vertical asymptotes:

 

4.) Find the equation(s) of all horizontal asymptotes:

Answer 1

f(x) = (x²-2x-8)/(x²-9)

 

Factors to:

 

f(x) = (x+2)(x-4) / (x+3)(x-3)

 

1)  The zeros occur when the numerator is equal to zero.  What two values of x make the numerator equal to zero?  On of them is x = -2.  What is the other value of x?

 

2)  The y-intercept occurs when x = 0.  Plug x = 0 into f(x) to compute the answer.

 

3)  The vertical asymptotes occur where the denominator of f(x) equals zero.  One of those x = -3.  What is the other?

 

4)  The horizontal asymptote occurs at the ratio of the x² terms in the numerator and denominator, which is y = x²/x² = 1

 

Answer 2

1. x² - 2x - 8 = 0

(x - 4)(x + 2) = 0

 

x - 4 = 0

x = 4

 

x + 2 = 0

x = -2

 

x = 4 and x = -2

 

2. 0² - 2(0) - 8

-8

 

3. x² - 9 = 0

(x + 3)(x - 3) = 0

 

x + 3 = 0

x = -3

 

x - 3 = 0

x = 3

 

x = -3 and x = 3

 

4. The leading coefficients are the same.

 

y = 1/1

 

y = 1