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To find the equation of the parabola shown, we'll use the vertex form of a quadratic function:

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

where:
- \( (h, k) \) is the vertex of the parabola,
- \( a \) determines the direction and width of the parabola.

### Step 1: Determine the Vertex
From the graph:
- The vertex appears to be at \( (-3, -1) \), so \( h = -3 \) and \( k = -1 \).

### Step 2: Substitute the Vertex into the Equation
Substitute \( h = -3 \) and \( k = -1 \):

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

### Step 3: Find the Value of \( a \)
To determine \( a \), we need another point on the parabola. Observing the graph, it appears that the point \( (-4, 0) \) lies on the curve.

Substitute \( x = -4 \) and \( y = 0 \) into the equation:

\[
0 = a(-4 + 3)^2 - 1
\]
\[
0 = a(1)^2 - 1
\]
\[
0 = a - 1
\]
\[
a = 1
\]

### Final Equation
With \( a = 1 \), the equation of the parabola is:

\[
y = (x + 3)^2 - 1
\]

Would you like more details on any specific step?

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### Related Questions
1. How can we confirm the values of \( h \) and \( k \) for a vertex form equation?
2. What happens if \( a \) is negative in the vertex form?
3. How would you transform this parabola if you wanted it to open downward?
4. How can you find the axis of symmetry from the vertex form equation?
5. What is the role of the value of \( a \) in changing the parabola's shape?

### Tip
Remember, the sign of \( h \) in the equation \( (x - h) \) is opposite to the \( x \)-value of the vertex.

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

Learn how to find the equation of a parabola using the vertex form y = a(x - h)^2 + k. This example involves identifying the vertex and another point on the graph to solve for 'a', providing a step-by-step solution. Suitable for students in Grades 9-11.

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