How do you graph #f(x) = (4x^2-36x) / (x-9)#?

Question

Hazel Anglin · Accepted Answer

see below

you can first simplify this expression: #4x^2 - 36x = 4x(x-9)# #(4x^2-36x)/(x-9) = (4x(x-9))/(x-9) = 4x# therefore, for all points where #(4x^2-36x)/(x-9)# can be defined, you'll get a graph of #f(x)= 4x#. however, not all points can be defined. any number divided by zero is undefined. this means that the #x#-value where the denominator #x-9# is #0# is also undefined. when #x-9 = 0#, #x = 9#. this means that the line cannot touch any point where #x = 9#. however, in all other ways, it will look like the graph of #f(x) = 4x#. this gives a straight line with gradient #4#, and with a hole where the point on the #x#-axis is #9#: graph{(4x^2-36x)/(x-9) [2.7, 22.7, 32.56, 42.56]} if you scroll along the graph, you'll see a directly proportional relationship between #x# and #y#, where the #y#-coordinate is #4# times the #x#-coordinate. if you scroll up to where #x=9#, you'll see that the coordinates are (#9#, undefined).

Hazel Anglin · Answer

undefined To graph the function f(x) = (4x^2-36x) / (x-9), follow these steps:

1. Determine the domain of the function by finding the values of x for which the denominator (x-9) is equal to zero. In this case, x cannot be equal to 9.

2. Find the x-intercepts by setting the numerator (4x^2-36x) equal to zero and solving for x. Factor out common terms if possible and solve the resulting equation.

3. Find the y-intercept by substituting x = 0 into the function and evaluating f(0).

4. Determine the vertical asymptotes by finding the values of x for which the denominator is equal to zero. In this case, x cannot be equal to 9.

5. Determine the horizontal asymptote by analyzing 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.

6. Plot additional points by choosing values of x within the domain and evaluating f(x).

7. Use the obtained information to sketch the graph, connecting the points and asymptotes.

Note: It may be helpful to use a graphing calculator or software to visualize the graph accurately.