To find the formula for the function $f(x)$ in the form $f(x) = a(x - h)^2 + k$, we need to determine the values of $a$, $h$, and $k$.

From the graph provided:

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

   • Observing the graph, it appears that the vertex is at $(2, -9)$, so $h = 2$ and $k = -9$.

2. Find a point on the graph to determine $a$: We can pick another point on the graph to solve for $a$.

   • A clear point on the graph is $(0, -1)$.

3. Substitute into the formula:

   • Substitute $(x, y) = (0, -1)$ and $(h, k) = (2, -9)$ into the vertex form to solve for $a$: $-1 = a(0 - 2)^2 - 9$ Simplify this equation: $-1 = a \cdot 4 - 9$ $8 = 4a$ $a = 2$

4. Write the function: Now that we have $a = 2$, $h = 2$, and $k = -9$, the equation of the function is: $f(x) = 2(x - 2)^2 - 9$

So, the function in vertex form is: $f(x) = 2(x - 2)^2 - 9$

Would you like more details on any part of this solution, or do you have further questions?

Here are 5 related questions for further practice:

1. How would the function change if the vertex was at $(3, -5)$ instead?
2. What would happen to the graph if $a$ was negative?
3. How can you rewrite $f(x) = 2(x - 2)^2 - 9$ in standard form?
4. How does the value of $a$ affect the "width" of the parabola?
5. What is the axis of symmetry for the function $f(x) = 2(x - 2)^2 - 9$?

Tip: To find the vertex quickly, always look for the lowest or highest point on a parabola (depending on whether it opens up or down).