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

Question

Ethan Adkins · Accepted Answer

Below

#f(x)=x/(x^2-9)# #f(x)=x/((x-3)(x+3))# Therefore, the vertical asymptotes are #x=3# and #x=-3# since the denominator cannot equal to 0. ie #x^2-9=0# is used to find our asymptote. For our horizontal asymptote, we look at the degree of our numerator and denominator. Since the degree of the numerator is less than the denominator, then #y=0# is our horizontal asymptote. Another way you can think of it is it you sub random numbers into the #x# of your numerator and denominator, you will find that the numerator will have a smaller value than your denominator. If you divide a smaller number by a larger number, your resulting value will be quite small, tending to 0 ie the line #y=0# For our intercepts, When #x=0#, #y=0# When #y=0#, #x=0# Therefore, #(0,0)# is our only intercept Drawing up our asymptotes (horizontal and vertical) and our intercept, we should be able to draw our graph. Remember that for our asymptotes, it only affects the ends of the graph as it is approaching the asymptotes from above or below. Our graph can actually cross the asymptotes. graph{x/(x^2-9) [-10, 10, -5, 5]}

Jace Anderson · Answer

undefined To graph the function f(x) = x/(x^2-9), we can start by identifying the vertical and horizontal asymptotes, holes, x-intercepts, and y-intercept.

Vertical asymptotes occur when the denominator of the function equals zero. In this case, x^2-9 = 0, which gives us x = -3 and x = 3 as the vertical asymptotes.

To find the horizontal asymptote, we examine the degrees of the numerator and denominator. Since the degree of the numerator (1) is less than the degree of the denominator (2), the horizontal asymptote is y = 0.

To determine any holes in the graph, we factor the denominator. x^2-9 can be factored as (x-3)(x+3). We notice that (x-3) cancels out with the numerator x, leaving a hole at x = 3.

To find the x-intercepts, we set the numerator equal to zero. Since x can be any real number, there are no x-intercepts.

To find the y-intercept, we substitute x = 0 into the function. This gives us f(0) = 0/(0^2-9) = 0/(-9) = 0.

To summarize:
- Vertical asymptotes: x = -3 and x = 3
- Horizontal asymptote: y = 0
- Hole: x = 3
- X-intercepts: None
- Y-intercept: (0, 0)

Using this information, you can plot the graph of f(x) = x/(x^2-9) accurately.