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To find the equation of the quadratic function \( f(x) \), we need to use the general form of a quadratic equation:

\[
f(x) = a(x - h)^2 + k
\]

Where:
- \( (h, k) \) is the vertex of the parabola.
- \( a \) is a constant that determines the "width" and direction (upward or downward) of the parabola.

From the graph provided:
- The vertex of the parabola is \( (3, 2) \), so \( h = 3 \) and \( k = 2 \).
  
Thus, we start with:

\[
f(x) = a(x - 3)^2 + 2
\]

Next, we use the point \( (5, -6) \), which lies on the graph, to find the value of \( a \).

Substitute \( x = 5 \) and \( f(x) = -6 \) into the equation:

\[
-6 = a(5 - 3)^2 + 2
\]

Simplify:

\[
-6 = a(2)^2 + 2
\]
\[
-6 = 4a + 2
\]
\[
-6 - 2 = 4a
\]
\[
-8 = 4a
\]
\[
a = -2
\]

Thus, the equation of the quadratic function is:

\[
f(x) = -2(x - 3)^2 + 2
\]

### Final Answer:
\[
f(x) = -2(x - 3)^2 + 2
\]

Would you like more details on any step? 

Here are some related questions to expand your understanding:
1. What happens to the graph if the value of \( a \) becomes positive?
2. How does changing the vertex \( (h, k) \) affect the graph of the parabola?
3. How can we rewrite the quadratic equation in standard form \( ax^2 + bx + c \)?
4. Can you solve for the roots of the quadratic equation using the quadratic formula?
5. How does the value of \( a \) affect the width of the parabola?

**Tip:** Always use the vertex form of a quadratic equation when given the vertex and another point on the parabola.

Find the equation of the quadratic function f whose graph is shown below with vertex at (3, 2) and passing through the point (5, -6).

Learn how to find the equation of a quadratic function by using the vertex form. This problem shows how to use the vertex (3, 2) and another point (5, -6) on the graph to find the complete equation of the quadratic function.

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