How do you graph #f(x)=(2x^2)/(x-3)# using holes, vertical and horizontal asymptotes, x and y intercepts?

Question

Julia Adams · Accepted Answer

Let's find the holes of this formula. A hole means that the same factor is in the numerator as denominator and they divide out. Such as #(x^2)/(x^3-4x)# or #(cancel(x) xx x)/(cancel(x)(x^2-4)# so there is a hole when #x = 0#. In our case of #(2x^2)/(x-3)#, there are no common factors, so there is no hole. Vertical asymptotes occur when we try to divide a value by #0#. So  let's see what value of #x# makes the denominator equal to #0#: #x-3=0# #x=3# So, there is a vertical asymptote at #x = 3# Now let's see about the Horizontal asymptote.  I like to use this  to help me remember:  BOBO - Bigger on bottom, y=0 BOTN - Bigger on top, none EATS DC - Exponents are the same, divide coefficients So in our case, the numerator (top) has a greater exponent (bigger). So there is no Horizontal asmptote (BOTN) Now let's find the #x#- intercepts and #y#- intercepts: #x#-intercept is the value of #x# when #y# equals #0#: #0 = (2x^2)/(x-3)# #0 = 2x^2# #0 = x^2# #x = 0# The #y#-intercept is the value of #y# when #x# equals #0# #y = (2(0)^2)/(0-3)# #y = 0/-3# #y = 0# Now we have all the information we need To check our answers, let's graph the equation
graph{y=(2x^2)/(x-3)} We have an #x# and #y# intercept at #0#, that's right. There's no horizontal asymptote although there is an asymptote for #x = 3# Our math is correct. Good job

Genesis Allen · Answer

undefined To graph the function f(x) = (2x^2)/(x-3), we can follow these steps:

1. Determine the vertical asymptote:
   Set the denominator (x-3) equal to zero and solve for x. The resulting value of x (in this case, x = 3) gives us the vertical asymptote.

2. Identify any holes in the graph:
   Check if there are any common factors between the numerator (2x^2) and the denominator (x-3). If there are, cancel them out and determine the x-values where the function is undefined. These x-values represent holes in the graph.

3. Find the horizontal asymptote:
   Determine the degree of the numerator and the 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. If the degree of the numerator is greater, there is no horizontal asymptote.

4. Calculate the x-intercepts:
   Set the numerator (2x^2) equal to zero and solve for x. The resulting values of x represent the x-intercepts.

5. Calculate the y-intercept:
   Substitute x = 0 into the function f(x) and solve for y. The resulting value of y represents the y-intercept.

Using these steps, you can graph the function f(x) = (2x^2)/(x-3) accurately, indicating the holes, vertical and horizontal asymptotes, as well as the x and y intercepts.