https://file.aiquickdraw.com/mathbot/file/1727802743778ltoejzix.jpg

The graph shown is a downward-facing parabola, indicating that the function is a quadratic function of the form:

\[
f(x) = ax^2 + bx + c
\]

From the graph, we can deduce the following characteristics:
1. **Vertex**: The vertex is located at \( (4, 16) \). This implies the function is in the form \( f(x) = a(x - h)^2 + k \), where \( (h, k) = (4, 16) \).
   
2. **Intercepts**: The graph intersects the x-axis at \( x = 0 \) and \( x = 8 \), making 0 and 8 the roots of the function. Therefore, the factored form can be written as:

\[
f(x) = a(x)(x - 8)
\]

To determine the value of \( a \), we use the information that the vertex is at \( (4, 16) \). Substituting \( x = 4 \) and \( f(4) = 16 \) into the factored form:

\[
16 = a(4)(4 - 8)
\]

\[
16 = a(4)(-4) = -16a
\]

Solving for \( a \):

\[
a = -1
\]

Thus, the equation of the function is:

\[
f(x) = -(x)(x - 8) = -(x^2 - 8x) = -x^2 + 8x
\]

The function that matches the graph is:

\[
f(x) = -x^2 + 8x
\]

Would you like further details or have any questions?

Here are 5 related questions to expand on this:

1. How do you find the axis of symmetry for a parabola?
2. What are the steps to convert a quadratic function from standard form to vertex form?
3. How can you determine the range of a quadratic function given its vertex?
4. What happens to the graph of a parabola if the coefficient of \( x^2 \) is positive?
5. Can you use the quadratic formula to find the roots of any quadratic equation?

**Tip**: In a quadratic function, the vertex represents either the maximum or minimum point, depending on the direction the parabola opens.

Write a function in any form that would match the graph shown below.

Learn how to find a quadratic function from a graph with vertex (4, 16) and intercepts at 0 and 8. This problem focuses on algebraic concepts and forms of quadratic functions, including vertex form and factored form. Suitable for high school students in Grades 9-12.

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