To find the equation of the parabola passing through the points $(-1, 0)$, $(3, -12)$, and $(4, 0)$, let's follow the format $y = a(x - p)(x - q)$.

Given:

1. The parabola passes through $(-1, 0)$, meaning one root is $x = -1$.
2. It also passes through $(4, 0)$, so the other root is $x = 4$.

This gives us the equation in factored form: $y = a(x + 1)(x - 4)$ where $a$ is a constant that we need to determine.

Step 1: Substitute the Point $(3, -12)$ to Find $a$

Since the parabola passes through the point $(3, -12)$, we substitute $x = 3$ and $y = -12$ into the equation: $-12 = a(3 + 1)(3 - 4)$ $-12 = a \cdot 4 \cdot (-1)$ $-12 = -4a$ $a = 3$

Step 2: Substitute $a$ Back Into the Equation

Now that we have $a = 3$, we can substitute it back into the equation: $y = 3(x + 1)(x - 4)$

Final Answer

The equation of the parabola is: $y = 3(x + 1)(x - 4)$

Would you like further details or have any questions?

Here are some related questions:

1. How do you find the vertex of a parabola given its equation?
2. What is the significance of the value of $a$ in a parabola's equation?
3. How can you convert this equation into standard quadratic form?
4. How do you find the axis of symmetry of a parabola given its equation?
5. What methods can you use to verify the accuracy of the parabola equation?

Tip: When working with parabolas, knowing the roots and an additional point can help you quickly determine the equation in factored form.