How do you graph #f(x)=(x^2-2x-8)/(x^2-9)# using holes, vertical and horizontal asymptotes, x and y intercepts?

Question

Isaiah Anthony · Accepted Answer

see explanation.

#color(blue)"Asymptotes"# The denominator of f(x) cannot be zero as this would make f(x) undefined. Equating the denominator to zero and solving gives the values that x cannot be and if the numerator is non-zero for these values then they are vertical asymptotes. solve : #x^2-9=0rArrx^2=9rArrx=+-3# #rArrx=-3" and " x=3" are the asymptotes"# Horizontal asymptotes occur as #lim_(xto+-oo),f(x)toc" ( a constant)"# divide numerator/denominator by the highest power of x, that is #x^2# #f(x)=(x^2/x^2-(2x)/x^2-8/x^2)/(x^2/x^2-9/x^2)=(1-2/x-8/x^2)/(1-9/x^2)# as #xto+-oo,f(x)to(1-0-0)/(1-0)# #rArry=1" is the asymptote"# Holes occur when there is a duplicate factor on the numerator/denominator. This is not the case here, hence there are no holes. #color(blue)"Intercepts"# #x=0toy=(-8)/(-9)=8/9# #rArr"y-intercept at " (0,8/9)# #y=0tox^2-2x-8=0to(x-4)(x+2)=0# #rArr"x-intercepts at "(-2,0)" and " (4,0)# graph{(x^2-2x-8)/(x^2-9) [-10, 10, -5, 5]}

Jace Anderson · Answer

undefined To graph the function f(x) = (x^2-2x-8)/(x^2-9), we can follow these steps:

1. Determine the domain of the function by finding the values of x for which the denominator (x^2-9) is equal to zero. In this case, x cannot be equal to ±3 since it would make the denominator zero.

2. Find the vertical asymptotes by determining the values of x for which the function approaches infinity or negative infinity. In this case, the vertical asymptotes occur at x = -3 and x = 3.

3. Identify any holes in the graph by simplifying the function and canceling out common factors between the numerator and denominator. If any factors cancel out, the resulting function will have a hole at that x-value. In this case, we can simplify the function to f(x) = (x+2)/(x+3). Since there are no common factors to cancel out, there are no holes in the graph.

4. Determine the horizontal asymptote by analyzing the behavior of the function as x approaches positive or negative infinity. In this case, the degree of the numerator and denominator is the same (both are quadratic), so the horizontal asymptote is y = 1.

5. Find the x-intercepts by setting the numerator equal to zero and solving for x. In this case, the x-intercept occurs when x + 2 = 0, so x = -2.

6. Find the y-intercept by evaluating the function at x = 0. In this case, the y-intercept is f(0) = (0^2-2(0)-8)/(0^2-9) = -8/(-9) = 8/9.

Using this information, you can plot the graph of f(x) = (x^2-2x-8)/(x^2-9) by marking the vertical asymptotes at x = -3 and x = 3, the hole at x = -2, the horizontal asymptote at y = 1, the x-intercept at x = -2, and the y-intercept at (0, 8/9).