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

Question

Nathan Axel · Accepted Answer

There is a vertical asymptote at #x=2#, a horizontal asymptote at #y=0# (x-axis) , no holes and no x-intercepts.

Analytically: Setting the numerator = #0# gives x-intercepts. There are no factors in the numerator, so there are no x-intercepts. Setting the denominator =#0# yields vertical asymptotes: #x-2=0# yields a vertical asymptote at #x=2# Horizontal asymptotes are found based on the degree of the numerator and denominator. If the degree of the numerator is one less than the degree of the denominator, the horizontal asymptote is the x-axis (#y=0#) Since there is no variable in the numerator, the degree = 0. The degree of the denominator is 1. From the graph: graph{-1/(x-2) [-10, 10, -5, 5]}

Avery Alexander · Answer

undefined To graph the function f(x) = -1/(x-2), we can follow these steps:

1. Determine the vertical asymptote: Set the denominator (x-2) equal to zero and solve for x. In this case, x = 2. So, the vertical asymptote is x = 2.

2. Identify any holes: Simplify the function by canceling out common factors between the numerator and denominator. If any factors cancel out, there will be a hole at that x-value. In this case, there are no common factors to cancel out, so there are no holes.

3. Find the x-intercept: Set f(x) equal to zero and solve for x. In this case, -1/(x-2) = 0. Since the numerator is -1, there is no solution for x that makes the function equal to zero. Therefore, there is no x-intercept.

4. Determine the y-intercept: Substitute x = 0 into the function f(x) = -1/(x-2) and solve for y. In this case, f(0) = -1/(0-2) = -1/(-2) = 1/2. So, the y-intercept is (0, 1/2).

5. Plot the points: Plot the vertical asymptote at x = 2 and the y-intercept at (0, 1/2).

6. Determine the behavior of the function as x approaches positive and negative infinity: As x approaches positive or negative infinity, the function approaches zero. Therefore, the x-axis (y = 0) is the horizontal asymptote.

7. Sketch the graph: Based on the information above, draw a curve that approaches the vertical asymptote at x = 2, passes through the y-intercept at (0, 1/2), and approaches the x-axis (y = 0) as x approaches positive and negative infinity.