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Part 1

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 kf(x)=a(x−h)2+k.

To determine the quadratic function from the given graph, we can use the vertex form of a quadratic equation:

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

Where:
- $(h, k)$ is the vertex of the parabola.
- $a$ is a constant that determines the shape and direction of the parabola.

### Step 1: Identify the vertex from the graph
From the graph, the vertex appears to be at the point $(-2, 9)$. Therefore, we have:
\[
h = -2, \quad k = 9
\]

### Step 2: Use another point on the parabola
Another point on the graph is $(0, 5)$. This point will help us find the value of $a$.

### Step 3: Plug the point into the equation
Using the vertex form and substituting $f(0) = 5$, we get:
\[
5 = a(0 - (-2))^2 + 9
\]
Simplify:
\[
5 = a(2)^2 + 9
\]
\[
5 = 4a + 9
\]
Solve for $a$:
\[
5 - 9 = 4a
\]
\[
-4 = 4a
\]
\[
a = -1
\]

### Step 4: Write the quadratic equation
Now that we know $a = -1$, $h = -2$, and $k = 9$, we can write the equation as:
\[
f(x) = -(x + 2)^2 + 9
\]

Thus, the quadratic function is:
\[
f(x) = -(x + 2)^2 + 9
\]

Would you like more details, or have any further questions on this? Here are some related questions for further exploration:

1. How do we confirm that the vertex is the maximum point of the parabola?
2. Can we rewrite the equation in standard form $ax^2 + bx + c$?
3. What is the axis of symmetry for this quadratic function?
4. How does changing the value of $a$ affect the graph of a parabola?
5. What are the x-intercepts of the quadratic function?

**Tip:** The sign of $a$ in the vertex form tells you whether the parabola opens upward or downward. A negative $a$ means it opens downward.

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

Learn how to find a quadratic function from its graph in the form f(x) = a(x - h)^2 + k. This step-by-step solution explains how to use the vertex and another point to determine the equation, perfect for high school students learning about quadratic functions.

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