Questions: Write an equation for a rational function with: Vertical asymptotes at x=-4 and x=-1 x-intercepts at x=-3 and x=1 y-intercept at 8 y=

Write an equation for a rational function with: Vertical asymptotes at x=-4 and x=-1 x-intercepts at x=-3 and x=1 y-intercept at 8 y=
Transcript text: Write an equation for a rational function with: Vertical asymptotes at $x=-4$ and $x=-1$ $x$-intercepts at $x=-3$ and $x=1$ $y$-intercept at 8 \[ y= \]
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Solution

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Solution Steps

Step 1: Construct the Denominator

The denominator is constructed based on the given vertical asymptotes: $y = (x + 4) * (x + 1)$.

Step 2: Construct the Numerator

The numerator is constructed based on the given $x$ intercepts: $y = (x + 3) * (x - 1)$.

Step 3: Determine the Constant $k$

Given a specific $y$ intercept ($y = 8$ when $x = 0$), we substitute $x = 0$ into the equation and solve for $k$. After calculations, $k = -10.67$.

Final Answer:

The constructed rational function is: $y = y = -10.67 \cdot ((x + 3) \cdot (x - 1)) / ((x + 4) \cdot (x + 1))$.

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