How do you graph #f(x)=x^2/(x-1)# using holes, vertical and horizontal asymptotes, x and y intercepts?

Question

Hazel Anglin · Accepted Answer

See explanation...

Alright, So for this question we are looking for six items - holes, vertical asymptotes, horizontal asymptotes, #x# intercepts, and #y# intercepts - in the equation #f(x)=x^2/(x-1)# First lets graph it graph{x^2/(x-1 [-10, 10, -5, 5]} Right off the bat you can see some strange things happening to this graph. Lets really break it down. To begin, lets find the # x# and #y# intercept. you can find the #x# intercept by setting #y=0# and vise versa #x=0# to find the #y# intercept. For the #x# intercept: #0=x^2/(x-1)# #0=x# Therefore, #x=0# when #y=0#. So without even knowing that information, we have just found BOTH the #x# and #y# intercept. Next, lets work on the asymptotes. To find the vertical asymptotes, set the denominator equal to #0#, then solve. #0=x-1# #x=1# So we just found that there is a vertical asymptote at #x=1#. You can visually check this by looking at the above graph. Next, lets find the horizontal asymptote. There are three general rules when talking about a horizontal asymptote. 1) If both polynomials are the same degree,divide the coefficients of the highest degree term. 2) If the polynomial in the numerator is a lower degree than the denominator, then #y=0# is the asymptote. 3) If the polynomial in the numerator is a higher degree than the denominator, then there is no horizontal asymptote. It is a slant asymptote. Knowing these three rules, we can determine that there is no horizontal asymptote, since the denominator is a lower degree than the numerator. Finally, lets find any holes that might be in this graph. Now, just from past knowledge, we should know that no holes will appear in a graph with a slant asymptote. Because of this, lets go ahead and find the slant. We need to do long division here using both polynomials: #=x^2/(x-1)# #=x-1# I'm sorry that there isn't a great way to show you the long divition there, but if you have anymore questions about that, click here. So there you go, I really hope this helped, and I apologize for the length! ~Chandler Dowd

Eli Assouline · Answer

undefined To graph the function f(x) = x^2/(x-1), we can start by analyzing its properties:

1. Holes: The function has a hole at x = 1, since the denominator becomes zero at that point. To find the y-coordinate of the hole, substitute x = 1 into the function and simplify.

2. Vertical Asymptotes: The function has a vertical asymptote at x = 1, as the denominator approaches zero but the numerator does not. To determine the behavior of the function as x approaches positive or negative infinity, divide the leading terms of the numerator and denominator.

3. Horizontal Asymptotes: To find the horizontal asymptote(s), compare the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. If the degrees are equal, divide the leading coefficients of the numerator and denominator to find the horizontal asymptote.

4. x-intercepts: To find the x-intercepts, set f(x) = 0 and solve the resulting equation.

5. y-intercept: To find the y-intercept, substitute x = 0 into the function and simplify.

Using these properties, we can plot the graph of f(x) = x^2/(x-1) accurately.