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To write the equation for the rational function, let's break down the given information and construct the equation step by step.

### 1. **Vertical Asymptotes**:  
The function has vertical asymptotes at \(x = -4\) and \(x = 4\). This means the denominator of the rational function must have factors that become zero at these points. So, the denominator will contain \((x + 4)\) and \((x - 4)\).

### 2. **X-Intercepts**:  
The x-intercepts are at \(x = 3\) and \(x = -3\). The function crosses the x-axis at these points, so the numerator will have factors \((x - 3)\) and \((x + 3)\).

### 3. **Y-Intercept**:  
The y-intercept is given at \(y = 9\) when \(x = 0\). To ensure this, we will include a constant factor \(k\) that we will solve for.

### 4. **General Form of the Rational Function**:
From the above points, the general form of the rational function is:

\[
y = k \cdot \frac{(x - 3)(x + 3)}{(x + 4)(x - 4)}
\]

Now, we use the y-intercept to determine the constant \(k\). The y-intercept occurs when \(x = 0\), and \(y = 9\):

\[
9 = k \cdot \frac{(0 - 3)(0 + 3)}{(0 + 4)(0 - 4)}
\]

Simplifying:

\[
9 = k \cdot \frac{(-3)(3)}{(4)(-4)} = k \cdot \frac{-9}{-16} = k \cdot \frac{9}{16}
\]

Solving for \(k\):

\[
k = 9 \cdot \frac{16}{9} = 16
\]

### 5. **Final Equation**:
Substitute \(k = 16\) into the equation:

\[
y = 16 \cdot \frac{(x - 3)(x + 3)}{(x + 4)(x - 4)}
\]

This is the equation of the rational function in factored form.

### Do you want further details or have any questions about this?

Here are 5 related questions you can explore:
1. How do vertical asymptotes affect the behavior of a rational function near certain x-values?
2. What is the role of x-intercepts in shaping the graph of a rational function?
3. How do you calculate the constant \(k\) in rational functions?
4. What happens if the degree of the numerator and denominator are equal in a rational function?
5. How would the function change if there were no vertical asymptotes?

**Tip:** Always check the intercepts and asymptotes of your rational function after solving to ensure accuracy!

Write an equation for a rational function with: vertical asymptotes at x = -4 and x = 4, x intercepts at (3,0) and (-3,0), y intercept at (0,9).

Explore the rational function with vertical asymptotes at x = -4 and x = 4, x intercepts at (3,0) and (-3,0), and a y intercept at (0,9). This detailed solution explains how to construct the function and calculate the constant factor, covering key concepts such as asymptotes and intercepts.

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