Graph the rational function
x/(x-6)
To graph the function, draw the horizontal and vertical asymptotes (if any) and plot at least two points on each piece of the graph.
as x ---> 6 the denominator becomes zero
on the right it is + oo (like 6.001
/ .001) and on the left of x = 6 it is -oo (like 5.999 /-.001
as x gets large positive it is 1
as x gets large negative it is 1
To graph the rational function f(x) = x/(x-6), we can follow these steps:
1. Determine the vertical asymptote(s):
The vertical asymptote(s) occur where the denominator of the rational function equals zero. In this case, x-6 = 0, so x = 6 is the vertical asymptote.
2. Determine the horizontal asymptote (if it exists):
To find the horizontal asymptote, compare the degree of the numerator (which is 1 in this case) with the degree of the denominator (which is also 1). Since they have the same degree, we divide the leading coefficient of the numerator by the leading coefficient of the denominator. The result (1/1) is the value of the horizontal asymptote, which is y = 1.
3. Find the x-intercept(s):
Set the numerator of the rational function equal to zero and solve for x: x = 0. So, the x-intercept is (0, 0).
4. Find the y-intercept:
Plug in x = 0 into the function to find the y-intercept: f(0) = 0/(0-6) = 0.
5. Choose additional points:
To plot more points, you can choose other values for x. Let's choose x = 1 and x = 10.
When x = 1: f(1) = 1/(1-6) = -1/5. So, (1, -1/5) is a point on the graph.
When x = 10: f(10) = 10/(10-6) = 10/4 = 5/2. So, (10, 5/2) is another point on the graph.
6. Plot the points:
Plot the points (0, 0), (1, -1/5), and (10, 5/2) on a coordinate plane.
7. Draw the asymptotes:
Draw a vertical line at x = 6 as the vertical asymptote. Draw a horizontal line at y = 1 as the horizontal asymptote.
8. Connect the points and draw the graph:
Using a smooth curve, connect the points and extend it towards the asymptotes. The graph should approach but never touch the vertical asymptote at x = 6 and the horizontal asymptote at y = 1.
The final graph should show a curve approaching the vertical asymptote at x = 6 and the horizontal asymptote at y = 1, passing through the points (0, 0), (1, -1/5), and (10, 5/2).