How do you graph #f(x)=(2x)/(3x-1)# using holes, vertical and horizontal asymptotes, x and y intercepts?

Question

Eva Abramson · Accepted Answer

#x=1/3# is the vertical asymptote.

#y=2/3# is the horizontal asymptote.

#x=0# is the x-intercept.

#y=0# is the y intercept.

The vertical asymptote is found when #f(x)# tends to infinity. #f(x)# normally tends to infinity when the denominator tends to #0#. So here: #3x-1=0# #3x=1# #x=1/3# is the vertical asymptote. For the horizontal asymptote, we use the degrees of the numerator and the denominator. Say #m# is the former and #n# the latter. If: #m>n#, then there is no horizontal asymptote, only a slant. #m=n#, the horizontal asymptote is at the quotient of the leading coefficient of the numerator and denominator #m<##n#, the asymptote is at #y=0#. Here, #m=1# and #n=1#. So #m=n#. We must divide the leading coefficients of the numerator #(2)# and the denominator (#3#). #y=2/3# is the horizontal asymptote. The x-intercept is found when #f(x)=0#. Here, #(2x)/(3x-1)=0# #2x=0# #x=0# is the x-intercept. The y-intercept is the answer to #f(0)#. Inputting: #(2*0)/(3*0+1)# #0/1# #y=0# is the y intercept.

Eva Adamson · Answer

undefined To graph the function f(x) = (2x)/(3x-1), we can start by identifying the vertical and horizontal asymptotes, x and y intercepts, and any holes in the graph.

Vertical asymptote:
The vertical asymptote occurs when the denominator of the function becomes zero. In this case, the denominator 3x-1 becomes zero when x = 1/3. Therefore, the vertical asymptote is x = 1/3.

Horizontal asymptote:
To find the horizontal asymptote, we compare the degrees of the numerator and denominator. In this case, both the numerator and denominator have a degree of 1. Therefore, the horizontal asymptote is y = 2/3.

X-intercept:
To find the x-intercept, we set y (or f(x)) equal to zero and solve for x. In this case, setting (2x)/(3x-1) = 0, we find that x = 0. Therefore, the x-intercept is (0, 0).

Y-intercept:
To find the y-intercept, we set x equal to zero and evaluate the function. In this case, setting x = 0, we find that f(0) = 0. Therefore, the y-intercept is (0, 0).

Holes:
To check for any holes in the graph, we simplify the function and see if any common factors cancel out. In this case, there are no common factors that cancel out, so there are no holes in the graph.

Using this information, we can plot the vertical asymptote at x = 1/3, the horizontal asymptote at y = 2/3, the x-intercept at (0, 0), and the y-intercept at (0, 0).