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!