Question

Use the given graph to create the equation for the rational function. The function is written in factored form to help you see how the given information shapes our equation.vert asymp x=-3 opposite end behavior, vert asymp x=4 same end behavior, x int x=2 crosses axis, y int y=1The numerator is: Answer (x-Answer )The denominator is: (x+Answer )(x-Answer )(x-Answer )

Answer 1

From the given graph

The standard form of a rational function is

`y=(a)/(x-h)+k`

The zero(s) of the graph is at x = 2, the factored form is

`x-2=0`

The y-intercept of the given function is at y(0) = 1

The vertical asymptote is at x = -3 (odd function) and x = 4 (even function)

The factored form is

`(x+3)(x-4)`

The rational function becomes,

`y=(A(x-2))/((x+3)(x-4)^2)`

Where x = 0 and y(0) = 1

`1=(A(0-2))/((0+3)(0-4)^2)`

Solve for A

`\begin{gathered} 1=(A(0-2))/((0+3)(0-4)^2) \\ 1=(-2A)/((3)(-4)^2) \\ 1=(-2A)/(3(16)) \\ 1=(-2A)/(48) \\ \text{Crossmultiply} \\ -2A=48 \\ \text{Divide both sides by -2} \\ (-2A)/(-2)=(48)/(-2) \\ A=-24 \end{gathered}`

The rational function is

`y=\frac{-24(x-2)}{(x+3)(x-4)(x-4)^{}}`

Hence, the numerator is

`-24(x-2)`

Hence, the denominator is

`(x+3)(x-4)(x-4)`