SOLUTION: The figure below shows the graph of a rational function with vertical asymptotes x=1,x=5, and horizontal asymptote y=0. The graph does not have an x-intercept, and it passes throu

Question 844183: The figure below shows the graph of a rational function with vertical asymptotes x=1,x=5, and horizontal asymptote y=0. The graph does not have an x-intercept, and it passes through the point (3,1). Write in simple form
Answer by josgarithmetic(39617) (Show Source): You can put this solution on YOUR website!
Some help on the analysis:

The denominator contains factors, (x-1) and (x-5). These factors do not occur in the numerator. Between x=1 and x=5, the denominator becomes negative, but to the left of x=1 and to the right of x=5, the denominator is positive. You want a way for the rational expression to stay positive as described, "graph does not have an x-intercept" and contains "point (3,1)". A way to do this is have both binomial factors in the denominator, squared. This way, the rational expression is always positive; and the extreme left and right behavior will approach zero.

The function can be something, .
You want to find k using the included point of (3,1).
At x=3, the expression must be 1. Find k.
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