https://file.aiquickdraw.com/mathbot/file/1730501098917d00zh0t1.jpg

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?

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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).

Use the graph of the quadratic function f to write its formula as f(x) = a(x - h)^2 + k.

Learn how to write the formula of a quadratic function in vertex form f(x) = a(x - h)^2 + k using a graph. This problem covers key algebraic concepts and requires identifying the vertex and calculating the value of 'a' based on points from the graph. Suitable for students in Grades 9-10.

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