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Find the quadratic function whose graph is shown to the right. Write the function in the form f left parenthesis x right parenthesis equals a left parenthesis x minus h right parenthesis squared plus k.
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Part 1
​f(x)equals
  
enter your response here
​(Do not​ simplify.)

The graph provided shows a parabola, and we are tasked with finding the quadratic function in the form:

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

### From the graph:
- The vertex is at \((-3, 12)\), so \(h = -3\) and \(k = 12\).
- The point \((0, 3)\) lies on the parabola, which we can use to find the value of \(a\).

### Step 1: Write the general form
Using the vertex \((-3, 12)\), we substitute into the vertex form:

\[
f(x) = a(x + 3)^2 + 12
\]

### Step 2: Use the point \((0, 3)\) to find \(a\)
Now substitute \(x = 0\) and \(f(x) = 3\) into the equation to solve for \(a\):

\[
3 = a(0 + 3)^2 + 12
\]
\[
3 = a(9) + 12
\]
\[
3 - 12 = 9a
\]
\[
-9 = 9a
\]
\[
a = -1
\]

### Step 3: Final equation
Now substitute \(a = -1\) back into the equation:

\[
f(x) = -(x + 3)^2 + 12
\]

This is the equation of the quadratic function whose graph is shown.

Would you like any more details on this? Here are 5 related questions to explore further:

1. How do you find the vertex form of a quadratic equation in other cases?
2. What happens to the parabola if the value of \(a\) is positive?
3. How do you convert the vertex form to standard form of a quadratic function?
4. How do you determine the direction of opening of a parabola from the vertex form?
5. How can we use other points to verify or adjust the quadratic equation?

**Tip**: In vertex form, \(a\) controls the width and direction of the parabola, while \(h\) and \(k\) determine its location.

Find the quadratic function whose graph is shown to the right. Write the function in the form f(x) = a(x - h)^2 + k.

Discover how to find the quadratic function in the vertex form f(x) = a(x - h)^2 + k by using the vertex and a point on the parabola. This problem is ideal for students in Grades 9-10 and covers core algebraic concepts.

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