SOLUTION: Give an example of a rational function h whose graph has the following properties. vertical asymptote: x = -1 and x = 2 horizontal asymptote: y = 3 x-intercept: (3, 0)

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Question 1146948: Give an example of a rational function h whose graph has the following properties.
vertical asymptote: x = -1 and x = 2
horizontal asymptote: y = 3
x-intercept: (3, 0)
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


(1) vertical asymptote: x = -1 and x = 2

This means there are factors of (x+1) and (x-2) in the denominator:

a%2F%28%28x%2B1%29%28x-2%29%29

(2) x-intercept: (3, 0)

This means there is a factor of (x-3) in the numerator:

%28a%28x-3%29%29%2F%28%28x%2B1%29%28x-2%29%29

(3) horizontal asymptote: y = 3

This means there is a constant factor of 3 in the numerator; and the degrees of the numerator and denominator are the same. So, in addition to the constant factor 3, we need a second linear factor in the numerator. It can be any linear factor; I chose x, meaning there is a second x-intercept at x=0:

%283%28x%29%28x-3%29%29%2F%28%28x%2B1%29%28x-2%29%29

Here is a graph:



If the statement of the problem is supposed to imply that x=3 is the ONLY x-intercept, then we can make the second linear factor in the numerator (x-3) also. Note that means the graph will be tangent to the x-axis at x=3:

%283%28x-3%29%5E2%29%2F%28x%2B1%29%28x-2%29%29

Here is a graph: