https://file.aiquickdraw.com/mathbot/file/1727450228288yapwga0d.jpg

The graph appears to show a quadratic function with a vertex at (2, 5) and passing through the point (3, 2). The general form of a quadratic equation is:

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

where \((h, k)\) is the vertex of the parabola. Based on the graph, the vertex is \((2, 5)\), so we substitute \(h = 2\) and \(k = 5\):

\[
f(x) = a(x - 2)^2 + 5
\]

Now, we use the point \((3, 2)\) to solve for \(a\). Substituting \(x = 3\) and \(f(x) = 2\) into the equation:

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

Simplifying:

\[
2 = a(1)^2 + 5
\]
\[
2 = a + 5
\]
\[
a = 2 - 5 = -3
\]

Thus, the equation of the quadratic function is:

\[
f(x) = -3(x - 2)^2 + 5
\]

Let me know if you need further details or clarifications!

Here are some related questions for practice:
1. How would the equation change if the vertex moved to (1, -3)?
2. What is the role of the coefficient \(a\) in determining the parabola's direction?
3. How can you derive the equation of a parabola given its roots and vertex?
4. How does the vertex form differ from the standard form \(ax^2 + bx + c\)?
5. How can the axis of symmetry of a parabola be determined from its equation?

**Tip:** The sign of \(a\) determines whether the parabola opens upwards or downwards.

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

Learn how to find the equation of a quadratic function using the vertex form. In this problem, the vertex is at (2, 5) and the function passes through the point (3, 2). Follow a step-by-step solution to solve for the equation of the quadratic function.

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