Questions: Write an equation for a rational function with: Vertical asymptotes of x=1 and x=3 x intercepts at (-7,0) and (-1,0) y intercept at (0,5) Use y as the output variable. You may leave your answer in factored form.

Write an equation for a rational function with:
Vertical asymptotes of x=1 and x=3
x intercepts at (-7,0) and (-1,0)
y intercept at (0,5)
Use y as the output variable. You may leave your answer in factored form.

Transcript text: Write an equation for a rational function with: Vertical asymptotes of $x=1$ and $x=3$ $x$ intercepts at $(-7,0)$ and $(-1,0)$ $y$ intercept at $(0,5)$ Use $y$ as the output variable. You may leave your answer in factored form. $\square$ Question Help: Video Message instructor Submit Question

Solution

Solution Steps

Step 1: Form the Denominator Q(x)

The denominator Q(x) is formed by the vertical asymptotes [1, 3]: \[ Q(x) = (x-1) * (x-3) \]

Step 2: Form the Numerator P(x)

The numerator P(x) is formed by the x-intercepts [-7, -1]: \[ P(x) = 2.14 * ((x+7) * (x+1)) \]

Step 3: Adjust for the y-intercept

To adjust for the y-intercept 5, we evaluate the function at x=0 and find the necessary constant factor to multiply the numerator: \[ y-intercept = \frac{P(0)}{Q(0)} = 5 \] \[ Constant Factor = 2.14 \]

Step 4: Simplify and Factor

Given the assumptions, the function in its simplest form is: \[ y = \frac{2.14 * ((x+7) * (x+1))}{(x-1) * (x-3)} \]