How do you graph #y=(x-5)/(x+1)# using asymptotes, intercepts, end behavior?

Question

Sadie Ackerman · Accepted Answer

Asymptotes: #x=-1# and #y=1#
Intercepts: #(0,-5)# and #(5,0)#
End behavior: As #x -> oo# and #x -> -oo#, #y ->1# For the asymptote, we have vertical ones and horizontal ones. For vertical ones, we look at the denominator and determine when it would be #0#. Whenever the denominator is #0#, that is where a vertical asymptote will be, except for when it is a hole. Horizontal ones are determined by finding the ratio of the coefficient in front of highest exponential value of the numerator and denominator. The highest exponential value on the numerator is x, and its coefficient is 1. The highest exponential value on the denominator is x as well, with its coefficient being 1. So the horizontal asymptote would be #1/1# which is #1#. The y intercept is when x is zero. So the y intercept would be #(0,-5)#. You can get this by plugging x into the equation. The x intercept is when y is zero. When #x=5#, then y would be 0, so the x intercept would be #(5,0)#. You can determine the end behavior just by looking at the horizontal asymptote. The horizontal asymptote tells you where the y value is going towards, which is #x=1# in this case.

Anna Allen · Answer

undefined To graph $ y = \frac{x-5}{x+1} $, first identify the vertical and horizontal asymptotes. Then find the x-intercept, y-intercept, and determine the end behavior.

Vertical asymptote: Set the denominator equal to zero and solve for x. In this case, $ x + 1 = 0 $ gives $ x = -1 $. So, the vertical asymptote is $ x = -1 $.

Horizontal asymptote: If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at $ y = 0 $. If the degrees are equal, divide the leading coefficients. In this case, the degrees are equal, so the horizontal asymptote is $ y = 1 $.

x-intercept: Set $ y = 0 $ and solve for x. $ \frac{x-5}{x+1} = 0 $ gives $ x = 5 $. So, the x-intercept is $ (5, 0) $.

y-intercept: Set $ x = 0 $ and solve for y. $ \frac{0-5}{0+1} = -5 $. So, the y-intercept is $ (0, -5) $.

End behavior: As $ x $ approaches positive or negative infinity, the function approaches the horizontal asymptote, which is $ y = 1 $.

Plot these points and draw the asymptotes. Then, sketch the graph accordingly.