To find the equation of a parabola that passes through the point $(-6, 9)$ and has $x$-intercepts at $-2$ and $3$, we can use the fact that the parabola's equation in factored form is:

$y = a(x - x_1)(x - x_2)$

where $x_1$ and $x_2$ are the $x$-intercepts. Since the $x$-intercepts are $-2$ and $3$, we can substitute them into the equation:

$y = a(x + 2)(x - 3)$

Now, we need to determine the value of $a$ using the point $(-6, 9)$, which lies on the parabola. Substitute $x = -6$ and $y = 9$ into the equation:

$9 = a(-6 + 2)(-6 - 3)$

Simplify within the parentheses:

$9 = a(-4)(-9)$

$9 = 36a$

Now, solve for $a$:

$a = \frac{9}{36} = \frac{1}{4}$

Thus, the equation of the parabola is:

$y = \frac{1}{4}(x + 2)(x - 3)$

We can expand this if you want it in standard form:

1. Distribute $(x + 2)(x - 3)$: $y = \frac{1}{4}(x^2 - 3x + 2x - 6)$ $y = \frac{1}{4}(x^2 - x - 6)$

2. Distribute $\frac{1}{4}$: $y = \frac{1}{4}x^2 - \frac{1}{4}x - \frac{3}{2}$

So, the equation in standard form is:

$y = \frac{1}{4}x^2 - \frac{1}{4}x - \frac{3}{2}$

Would you like further details or have any questions?

Related Questions

1. How would the equation change if the point given was $(-6, -9)$ instead?
2. Can we determine the vertex of this parabola from the equation?
3. How can we find the axis of symmetry for this parabola?
4. What are the steps to convert this equation into vertex form?
5. How does changing the value of $a$ affect the shape of the parabola?

Tip: Factored form is helpful for quickly identifying the roots of a parabola and setting up an equation based on those roots.