The goal of this question is to find the quadratic function in vertex form, $f(x) = a(x-h)^2 + k$, based on the given graph.

From the graph:

• The vertex of the parabola is at $(h, k) = (0, 10)$.
• Another point on the parabola is $(2, 4)$.

Step-by-step Solution:

1. Vertex form of a quadratic function: $f(x) = a(x - h)^2 + k$ Here, $h = 0$ and $k = 10$. So, the equation becomes: $f(x) = a(x - 0)^2 + 10 = a x^2 + 10$

2. Using the point $(2, 4)$ to find $a$: We can plug in $x = 2$ and $f(x) = 4$ into the equation to solve for $a$: $4 = a(2)^2 + 10$ Simplifying: $4 = 4a + 10$ $4a = 4 - 10$ $4a = -6$ $a = -\frac{3}{2}$

3. Final equation: The quadratic function is: $f(x) = -\frac{3}{2}x^2 + 10$

Let me know if you would like more details or if you have any questions!

Here are 5 related questions:

1. How do you convert a quadratic equation from standard form to vertex form?
2. What are the key characteristics of the vertex form of a quadratic function?
3. How can you determine if the parabola opens upwards or downwards?
4. What is the significance of the vertex in the context of a quadratic function?
5. How can you find the axis of symmetry of a parabola?

Tip: When given a graph, always use points and the vertex to derive the equation accurately.