Write an equation for a rational function with:

Vertical asymptotes at x = 5 and x = -2

x-intercepts at x = -5 and x = -3

y-intercept at 7

To construct a rational function with the given characteristics, we'll proceed step by step:

### 1. **Vertical Asymptotes** at \(x = 5\) and \(x = -2\):  
Vertical asymptotes occur when the denominator of a rational function equals zero. Therefore, the denominator must include the factors \((x - 5)\) and \((x + 2)\), since the function will be undefined at these points.

Thus, the denominator will be:
\[
(x - 5)(x + 2)
\]

### 2. **x-Intercepts** at \(x = -5\) and \(x = -3\):  
x-intercepts occur when the numerator of a rational function equals zero. To achieve intercepts at \(x = -5\) and \(x = -3\), the numerator must include the factors \((x + 5)\) and \((x + 3)\), as these will result in zero when \(x = -5\) and \(x = -3\).

Thus, the numerator will be:
\[
(x + 5)(x + 3)
\]

### 3. **y-Intercept** at \(y = 7\):  
The y-intercept occurs when \(x = 0\). To ensure that the function passes through the point \((0, 7)\), we need to scale the rational function by a constant \(A\). To find \(A\), we substitute \(x = 0\) into the equation and set the output to 7.

The function (with the constant \(A\)) so far is:
\[
f(x) = A \frac{(x + 5)(x + 3)}{(x - 5)(x + 2)}
\]

Substitute \(x = 0\) into the equation:
\[
f(0) = A \frac{(0 + 5)(0 + 3)}{(0 - 5)(0 + 2)} = A \frac{5 \cdot 3}{-5 \cdot 2} = A \frac{15}{-10} = -\frac{3}{2}A
\]
Since the y-intercept is 7, we set \(f(0) = 7\):
\[
-\frac{3}{2}A = 7
\]
Solve for \(A\):
\[
A = -\frac{14}{3}
\]

### 4. Final Equation:
Now that we know the constant \(A = -\frac{14}{3}\), the equation of the rational function is:
\[
f(x) = -\frac{14}{3} \frac{(x + 5)(x + 3)}{(x - 5)(x + 2)}
\]

This is the rational function that satisfies all the given conditions.

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Would you like further details on this, or do you have any questions?

Here are 5 related questions for further exploration:

1. How does changing the constant \(A\) affect the graph of the rational function?
2. How would the function change if the y-intercept were different?
3. What happens if a rational function has no vertical asymptotes?
4. How do you find horizontal asymptotes of a rational function?
5. What is the significance of a hole in the graph of a rational function?

**Tip:** Always verify the behavior at intercepts and asymptotes when constructing or analyzing rational functions.

Write an equation for a rational function with: Vertical asymptotes at x = 5 and x = -2, x-intercepts at x = -5 and x = -3, y-intercept at 7.

Learn how to construct a rational function with vertical asymptotes at x = 5 and x = -2, x-intercepts at x = -5 and x = -3, and a y-intercept at 7. This step-by-step guide covers the process of determining the function using algebraic methods.

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