Question

Write an equation for a rational function with:Vertical asymptotes at x = -5 and x = 2x-intercepts at x = 3 and x = -6y-intercept at 9y =

Answer 1

It is required to write the equation of a rational function given vertical asymptotes, x-intercepts, and y-intercepts.

Recall that the vertical asymptotes form the factors in the denominator of the function, while the x-intercepts form the factors in the numerator.

Since the vertical asymptotes are x=-5 and x=2, it follows that the factors are (x+5) and (x-2).

Hence, the denominator will be:

`(x+5)(x-2)`

Since the x-intercepts are x=3 and x=-6, then the factors are (x-3) and (x+6).

Hence, part of the numerator is:

`(x-3)(x+6)`

It implies that the rational function takes the form:

`y=(a(x-3)(x+6))/((x+5)(x-2))`

Where a is a constant to be determined.

It is given that the y-intercept is at 9.

Substitute x=0 and y=9 into the equation to calculate the value of a:

`\begin{gathered} 9=(a(0-3)(0+6))/((0+5)(0-2)) \\ \Rightarrow9=(a(-3)(6))/((5)(-2)) \\ \Rightarrow9=(-18a)/(-10)=(9a)/(5) \\ \Rightarrow(9a)/(5)=9 \\ \Rightarrow9a=45 \\ \Rightarrow a=(45)/(9)=5 \end{gathered}`

Substitute a=5 back into the function:

`y=(5(x-3)(x+6))/((x+5)(x-2))`

Hence, the required rational function.

The rational function is:

`y=(5(x-3)(x+6))/((x+5)(x-2))`