Creating a rational equation

[JSmith]

State the equation of a rational function in the form

if the vertical asymptote is x = 5, the horizontal asymptote is y = 2, the x-intercept is (

, 0) and the y-intercept is (0,

). Explain your thought process.

So I know that the vertical asymptote is a zero of the denominator, so there will be an x-5 there. And that for the horizontal asymptote to be y=2, the leading coefficient of the numerator must be twice that of the denominator. I know that to determine the X-Intercept you plug in 0 for the y value and solve, and vice versa for the Y-Intercept, however I can't figure out how to determine what they are in the equation.

 

[srmichael]

State the equation of a rational function in the form

if the vertical asymptote is x = 5, the horizontal asymptote is y = 2, the x-intercept is (

, 0) and the y-intercept is (0,

). Explain your thought process.

So I know that the vertical asymptote is a zero of the denominator, so there will be an x-5 there. And that for the horizontal asymptote to be y=2, the leading coefficient of the numerator must be twice that of the denominator. I know that to determine the X-Intercept you plug in 0 for the y value and solve, and vice versa for the Y-Intercept, however I can't figure out how to determine what they are in the equation.

Good so far. So we know we have something like:

\(\displaystyle f(x)=\frac{2x+b}{x-5}\).

So for there to be an x-intercept the y value is 0 like you said. The only way that a fraction can equal 0 is for the numerator to equal 0. Therefore, we know that \(\displaystyle x=\frac{-1}{2}\). Plug this in to the numerator, set it equal to 0 and solve for b. When you get b, you can then set x = 0 to make sure the y-intercept is \(\displaystyle \frac{-1}{5}\)

 

[JSmith]

I got b=3.25, and then when I substituted that into the equation, I got -6.5/5 as my answer. Did I make a miscalculation or do we need to alter the formula?

 

[HallsofIvy]

The x intercept is (-1/2, 0) and, as srmichael told you, that means that the numerator, \(\displaystyle 2x+ b\), must be 0 when x= -1/2: 2(-1/2)+ b= 0. That does NOT give b= 3.25!

 

[JSmith]

Yeah, my mistake on that b, I have no idea what I did last time. Anyways, it all works out now when b=1, thanks for your help