Question

Use the given information 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. If the leading coefficient is not an integer enter the value as a fraction.Vertical asymptote at x=-1, double zero at x=2, y-intercept at (0,2).The numerator is: Answer (x-Answer )(x-Answer )The denominator is: (x+Answer )

Answer 1

Given:

• Vertical asymptote at : x = -1

,

• Double zero at: x = 2

,

• y-intercept at: (0, 2)

Let's create the equation for the rational function using the given properties.

Since the vertical asymptote is at x = -1, to find the deominator of the equation, equate the vertical asymptote to zero.

Add 1 to both sides:

$\begin{gathered} x+1=-1+1 \\ x+1=0 \end{gathered}$

Therefore, the denominator of the function is ==> x + 1

Since it has a double zero at x = 2, we have the factors:

$\Longrightarrow(x-2)(x-2)$

We now have the equation:

$y=\frac{a(x-2)(x-2)}{x+1}$

Also, the y-intercept is at: (0, 2)

To find the value o a, substitute 2 for y and 0 for x then evaluate:

$\begin{gathered} 2=\frac{a(0-2)(0-2)}{0+1} \\ \\ 2=\frac{a(-2)(-2)}{1} \\ \\ 4a=2 \\ \\ a=\frac{2}{4} \\ \\ a=\frac{1}{2} \end{gathered}$

Therefore, the rational function is:

$y=\frac{\frac{1}{2}(x-2)(x-2)}{x+1}$

ANSWER:

$y=\frac{\frac{1}{2}(x-2)(x-2)}{x+1}$$\begin{gathered} \text{Numerator: }\frac{1}{2}(x-2)(x-2) \\ \\ \\ \text{Denominator: (x + 1)} \end{gathered}$