To solve this problem, we need to find the values of $a$, $h$, and $k$ for a quadratic function in vertex form, given that the vertex is at $(4, -10)$ and the function passes through the point $(-2, 2)$.

Vertex Form of a Quadratic Function

The vertex form of a quadratic function is: $f(x) = a(x - h)^2 + k$ where:

• $(h, k)$ is the vertex of the parabola.
• $a$ is a coefficient that affects the "width" and direction of the parabola.

Step 1: Substitute the Vertex Coordinates

Since the vertex is at $(4, -10)$, we have: $h = 4 \quad \text{and} \quad k = -10$ Thus, the function becomes: $f(x) = a(x - 4)^2 - 10$

Step 2: Use the Point $(-2, 2)$ to Solve for $a$

Now, substitute $x = -2$ and $f(x) = 2$ into the equation to find $a$: $2 = a(-2 - 4)^2 - 10$ $2 = a(-6)^2 - 10$ $2 = 36a - 10$ Add 10 to both sides: $12 = 36a$ Now, divide by 36: $a = \frac{12}{36} = \frac{1}{3}$

Solution Summary

We have found the values:

• $a = \frac{1}{3}$
• $h = 4$
• $k = -10$

Thus, the function in vertex form is: $f(x) = \frac{1}{3}(x - 4)^2 - 10$

Would you like further details or explanations on any part of the solution?

Here are some related questions for further practice:

1. How do you convert a quadratic function from standard form to vertex form?
2. What is the effect of changing the value of $a$ on the graph of the quadratic function?
3. How can you find the vertex of a quadratic function given in standard form?
4. How do you determine the direction of opening of a parabola?
5. What are other forms (like factored form) of representing a quadratic function, and how do they compare?

Tip: When identifying the vertex from the vertex form equation $f(x) = a(x - h)^2 + k$, remember that $h$ and $k$ represent the coordinates of the vertex directly.