Determine the quadratic function of the form 

f left parenthesis x right parenthesis equals a left parenthesis x minus h right parenthesis squared plus kf(x)=a(x−h)2+k 

whose graph is given on the right.

Question content area bottom left

Part 1

​f(x)equals=enter your response here

​(Do not​ simplify.)

...

Question content area right

Part 1

-55-20-15-10-5510xy(3,4)(0,-14)

To determine the quadratic function \( f(x) = a(x - h)^2 + k \) from the given points on the graph, we'll use the vertex form of a quadratic function and the points provided.

### Given Information:

- **Vertex**: \((h, k) = (3, 4)\)
- **Point on the graph**: \((x, f(x)) = (0, -14)\)

### Steps to Find the Quadratic Function:

1. **Substitute the vertex into the vertex form**:
   \[
   f(x) = a(x - 3)^2 + 4
   \]

2. **Use the point \((0, -14)\) to find \(a\)**:
   Substitute \(x = 0\) and \(f(x) = -14\) into the equation:
   \[
   -14 = a(0 - 3)^2 + 4
   \]

3. **Simplify the equation**:
   \[
   -14 = a(-3)^2 + 4
   \]
   \[
   -14 = 9a + 4
   \]

4. **Solve for \(a\)**:
   \[
   -14 - 4 = 9a
   \]
   \[
   -18 = 9a
   \]
   \[
   a = -2
   \]

5. **Write the quadratic function**:
   \[
   f(x) = -2(x - 3)^2 + 4
   \]

### Conclusion:

The quadratic function in vertex form that fits the given graph is:
\[
f(x) = -2(x - 3)^2 + 4
\]

### Would you like more details or have any questions?

### Related Questions:

1. How do you find the vertex of a quadratic function given in standard form?
2. How do you determine the axis of symmetry for a quadratic function in vertex form?
3. What is the effect of the coefficient \(a\) on the width of a parabola?
4. How can you convert a quadratic function from vertex form to standard form?
5. How can you use the vertex form to determine the maximum or minimum value of a quadratic function?

### Tip:

The vertex form of a quadratic function \( f(x) = a(x - h)^2 + k \) makes it easy to identify the vertex \((h, k)\) and determine whether the parabola opens upward or downward based on the sign of \(a\).

Learn how to determine the quadratic function \( f(x) = a(x - h)^2 + k \) from given points on its graph, including (3, 4) and (0, -14). This problem involves using the vertex form of a quadratic equation and solving for the coefficients. Suitable for high school students.

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