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To find the equation of the quadratic function \( f(x) \) whose graph is shown, we will use the general form of a quadratic equation:

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

Given points from the graph:

1. \( (1, 6) \)
2. \( (2, 3) \)

We need one more detail to fully determine the function. By observing the graph, it seems that the vertex of the parabola is at \( (1, 6) \), indicating this as the highest point (since the parabola opens downwards). 

Thus, we can write the quadratic equation in its **vertex form**:

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

Where \( (h, k) \) is the vertex. From the graph, \( h = 1 \) and \( k = 6 \), so the equation becomes:

\[
f(x) = a(x - 1)^2 + 6
\]

Now, use the point \( (2, 3) \) to find \( a \). Substituting \( x = 2 \) and \( f(2) = 3 \):

\[
3 = a(2 - 1)^2 + 6
\]

Simplifying:

\[
3 = a(1)^2 + 6
\]

\[
3 = a + 6
\]

\[
a = 3 - 6 = -3
\]

Thus, the equation of the quadratic function is:

\[
f(x) = -3(x - 1)^2 + 6
\]

Would you like further details or clarification on this? Here are some related questions:

1. How can we convert this equation to standard form \( ax^2 + bx + c \)?
2. What does the value of \( a \) tell us about the shape of the parabola?
3. Can we use any other points on the graph to verify our equation?
4. How can we find the axis of symmetry of the parabola?
5. What is the significance of the vertex in a quadratic equation?

**Tip**: The vertex form of a quadratic equation is especially useful when the vertex of the parabola is given or easily identifiable from the graph!

Find the equation of the quadratic function f whose graph is shown below. The points provided on the graph are (1, 6) and (2, 3).

This page provides a step-by-step solution for finding the equation of a quadratic function based on its graph. The graph provides the vertex at (1, 6) and another point (2, 3). Learn how to derive the quadratic equation in both vertex form and standard form.

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