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To find the equation of the quadratic function \( f(x) \) from the graph provided, we can use the vertex form of a quadratic equation, which is:

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

where \((h, k)\) is the vertex of the parabola.

From the graph:
- The vertex is at \((1, 4)\), so \(h = 1\) and \(k = 4\).

Now the equation becomes:

\[
f(x) = a(x - 1)^2 + 4
\]

To find the value of \(a\), we can use another point on the parabola. From the graph, another point is \((3, 12)\).

Substitute \((x, y) = (3, 12)\) into the equation:

\[
12 = a(3 - 1)^2 + 4
\]

Simplify and solve for \(a\):

\[
12 = a(2)^2 + 4
\]
\[
12 = 4a + 4
\]
\[
8 = 4a
\]
\[
a = 2
\]

Now, substitute \(a = 2\) back into the equation:

\[
f(x) = 2(x - 1)^2 + 4
\]

Thus, the equation of the quadratic function is:

\[
f(x) = 2(x - 1)^2 + 4
\]

Would you like more details or have any questions?

Here are some related questions you might find interesting:
1. How do you convert the vertex form of a quadratic equation to the standard form?
2. What is the importance of the vertex in the graph of a quadratic function?
3. How can we determine the axis of symmetry for a quadratic function from its equation?
4. What are the properties of the graph of a quadratic function when the leading coefficient \(a\) is negative?
5. How can we find the x-intercepts of a quadratic function?

**Tip:** Always verify the vertex and another point from the graph to ensure your equation accurately represents the given parabola.

Find the equation of the quadratic function f(x) whose graph is shown below.

Learn how to find the equation of a quadratic function using its vertex and another point on the graph. This step-by-step guide covers the vertex form of a quadratic equation, ideal for students in Grades 9-11.

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