https://file.aiquickdraw.com/mathbot/file/1728876810617ka0gqbuz.jpg

Let's analyze the graph to find the equation in the form \( y = a(x - h)^2 + k \).

### Step 1: Identify the **vertex** (h, k)
From the graph:
- The vertex is at the point \( (1, 2) \).

Thus, \( h = 1 \) and \( k = 2 \).

So far, the equation looks like:
\[
y = a(x - 1)^2 + 2
\]

### Step 2: Identify another point on the graph
A visible point on the parabola is \( (0, 3) \). We can substitute \( x = 0 \) and \( y = 3 \) into the equation to solve for \( a \).

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

### Step 3: Write the final equation
Now that we know \( a = 1 \), the equation of the parabola is:
\[
y = (x - 1)^2 + 2
\]

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Let me know if you need further details or explanations!

Here are five questions you might explore next:
1. What happens to the graph if the value of \(a\) is negative?
2. How do you convert a quadratic equation from vertex form to standard form?
3. How can you determine the axis of symmetry of a parabola from its equation?
4. How does the value of \(k\) affect the vertical shift of a parabola?
5. How do we find the focus and directrix of a parabola?

**Tip:** When \( a = 1 \), the parabola has the standard "width". If \( |a| > 1 \), the parabola becomes narrower; if \( |a| < 1 \), it becomes wider.

Find an equation for the graph shown below. Hint: use the form y = a(x - h)^2 + k.

Learn how to determine the quadratic equation in vertex form (y = a(x - h)^2 + k) from a graph. This example involves identifying the vertex and solving for the value of 'a'. Ideal for students in Grades 8-10.

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