Answer:
To determine which graph represents the rational function f(x) = (x^2 - 16)/(x^2 - 2x - 8), we can analyze the behavior of the function as x approaches infinity and negative infinity, as well as the location and behavior of any vertical asymptotes, horizontal asymptotes, x-intercepts, and y-intercepts.
First, let's factor the denominator of the rational function:
x^2 - 2x - 8 = (x - 4)(x + 2)
Therefore, the rational function can be written as:
f(x) = (x^2 - 16)/((x - 4)(x + 2))
To find any vertical asymptotes, we need to look for values of x that make the denominator of the rational function equal to zero. Since the denominator is a product of two linear factors, the values that make it equal to zero are x = 4 and x = -2. Therefore, the rational function has vertical asymptotes at x = 4 and x = -2.
To find any horizontal asymptotes, we can look at the behavior of the function as x approaches infinity and negative infinity. As x becomes very large (either positive or negative), the highest degree term in the numerator and denominator of the rational function will dominate the expression. In this case, both the numerator and denominator have a highest degree of x^2, so we can apply the horizontal asymptote rule and divide the leading coefficient of the numerator by the leading coefficient of the denominator. This gives us:
y = 1
Therefore, the rational function has a horizontal asymptote at y = 1.
To find any x-intercepts, we need to look for values of x that make the numerator of the rational function equal to zero. Since the numerator is a difference of two squares, we can factor it as:
x^2 - 16 = (x - 4)(x + 4)
Therefore, the rational function has x-intercepts at x = 4 and x = -4.
To find the y-intercept, we can set x = 0 in the rational function:
f(0) = (-16)/(-8) = 2
Therefore, the rational function has a y-intercept at y = 2.
Based on this information, we can sketch the graph of the rational function as follows:
- The function has vertical asymptotes at x = 4 and x = -2.
- The function has a horizontal asymptote at y = 1.
- The function has x-intercepts at x = 4 and x = -4.
- The function has a y-intercept at y = 2.
Out of the provided graphs, only graph (C) matches this description. Therefore, graph (C) represents the rational function f(x) = (x^2 - 16)/(x^2 - 2x - 8).