Write an equation for a rational function with: Vertical asymptotes at x = − 5 and x = 1 x -intercepts at ( − 4 , 0 ) and ( − 1 , 0 ) y -intercept at ( 0 , 7 )

To construct a rational function with the given properties, let’s break it down step by step.

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### 1. General form of a rational function:
A rational function \( f(x) \) has the general form:

\[
f(x) = \frac{N(x)}{D(x)}
\]

where \( N(x) \) is the numerator (containing the x-intercepts) and \( D(x) \) is the denominator (containing the vertical asymptotes). Let's build these based on the conditions.

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### 2. Vertical asymptotes:
- Vertical asymptotes occur when the **denominator** becomes zero.  
- Since the vertical asymptotes are at \( x = -5 \) and \( x = 1 \), the factors of the denominator must be:

\[
D(x) = (x + 5)(x - 1)
\]

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### 3. x-intercepts:
- The **numerator** determines the x-intercepts.
- Since the x-intercepts are at \( (-4, 0) \) and \( (-1, 0) \), the numerator must contain these as factors:

\[
N(x) = (x + 4)(x + 1)
\]

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### 4. Putting it together:
So far, the rational function is:

\[
f(x) = \frac{A (x + 4)(x + 1)}{(x + 5)(x - 1)}
\]

where \( A \) is a constant that ensures the function passes through the y-intercept at \( (0, 7) \).

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### 5. Finding the constant \( A \):
- Use the **y-intercept** at \( (0, 7) \) to find \( A \).  
- Substituting \( x = 0 \) and \( f(0) = 7 \):

\[
7 = \frac{A (0 + 4)(0 + 1)}{(0 + 5)(0 - 1)}
\]

\[
7 = \frac{A \cdot 4 \cdot 1}{5 \cdot (-1)}
\]

\[
7 = \frac{-4A}{5}
\]

\[
-35 = -4A \quad \Rightarrow \quad A = \frac{35}{4}
\]

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### 6. Final equation:
Thus, the rational function is:

\[
f(x) = \frac{\frac{35}{4} (x + 4)(x + 1)}{(x + 5)(x - 1)}
\]

Or equivalently:

\[
f(x) = \frac{35 (x + 4)(x + 1)}{4 (x + 5)(x - 1)}
\]

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Would you like further explanation or a graph of this function?

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Here are 5 related questions:
1. What happens to the function as \( x \to \infty \)?
2. How do horizontal asymptotes differ from vertical asymptotes?
3. How do you find slant asymptotes for rational functions?
4. How would the equation change if the x-intercepts had multiplicity 2?
5. How can you determine whether the function has any holes?

**Tip:** When solving for constants using known points, be mindful of negative signs—small mistakes can affect the solution significantly.

Write an equation for a rational function with: Vertical asymptotes at x = −5 and x = 1, x-intercepts at (−4, 0) and (−1, 0), y-intercept at (0, 7).

Learn to construct a rational function with vertical asymptotes at x = −5 and x = 1, x-intercepts at (−4, 0) and (−1, 0), and a y-intercept at (0, 7). This problem demonstrates key concepts in rational functions, including asymptotes, intercepts, and solving for constants.

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