https://file.aiquickdraw.com/mathbot/file/1726676062600tbu1273l.jpg

The image presents a graph of a quadratic function \( f(x) \) and asks to provide the corresponding function equation. From the graph, we can determine several key points to find the quadratic equation in the form:

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

where \( (h, k) \) is the vertex of the parabola, and \( a \) represents the shape and direction of the parabola.

### Step 1: Identify the Vertex
From the graph, the vertex appears to be at \( (3, -1) \). This tells us:
\[
h = 3 \quad \text{and} \quad k = -1
\]
Thus, the equation so far is:
\[
f(x) = a(x - 3)^2 - 1
\]

### Step 2: Find \( a \)
To find the value of \( a \), we need another point on the parabola. From the graph, we can see that \( f(0) = 4 \) (the y-coordinate when \( x = 0 \)). Substituting this point into the equation:

\[
4 = a(0 - 3)^2 - 1
\]
\[
4 = 9a - 1
\]
\[
5 = 9a
\]
\[
a = \frac{5}{9}
\]

### Step 3: Final Equation
Thus, the quadratic function is:
\[
f(x) = \frac{5}{9}(x - 3)^2 - 1
\]

Would you like further clarification on any step? 

Here are some related questions:
1. How do you find the vertex of a quadratic function from its equation?
2. What role does the coefficient \( a \) play in the shape of a parabola?
3. How can you find the x-intercepts of a quadratic function?
4. What is the general form of a quadratic equation and how does it relate to the vertex form?
5. How does completing the square help in converting a quadratic function into vertex form?

**Tip**: When identifying the vertex from a graph, always check both the x- and y-coordinates carefully for accuracy.

Geben Sie zum abgebildeten Graphen eine entsprechende Funktionsgleichung an.

Learn how to find the equation of a quadratic function from a graph. The graph has its vertex at (3, -1) and another point at (0, 4), allowing us to calculate the function in vertex form. This detailed step-by-step guide covers quadratic equations and vertex form, suitable for Grades 9-11.

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