	By identifying key points on the graph, algebraically develop the equation of the parabola in the form y=a(x-h)^2+k

To develop the equation of a parabola in the form \( y = a(x - h)^2 + k \), we need the following key pieces of information from the graph:

1. **Vertex**: The vertex of the parabola gives the values of \( h \) and \( k \). If the vertex is located at \( (h, k) \), then those values are directly substituted into the equation.
2. **A second point**: To find the value of \( a \), we substitute another known point on the parabola (not the vertex) into the equation along with the vertex values and solve for \( a \).

Here’s how we can algebraically develop the equation step-by-step:

### Step 1: Identify the vertex \( (h, k) \)
The vertex form of the parabola is \( y = a(x - h)^2 + k \). From the graph, identify the coordinates of the vertex. Let’s assume the vertex is \( (h, k) \).

### Step 2: Use the second point to find \( a \)
Choose another point \( (x_1, y_1) \) from the graph that lies on the parabola. Substitute this point along with the vertex \( (h, k) \) into the equation to solve for \( a \).

The general form becomes:
\[
y_1 = a(x_1 - h)^2 + k
\]
Solve for \( a \) by isolating it on one side.

### Example:
Suppose the vertex is \( (2, 3) \), and another point on the graph is \( (4, 7) \).

The equation becomes:
\[
7 = a(4 - 2)^2 + 3
\]

First, simplify the expression:
\[
7 = a(2)^2 + 3
\]
\[
7 = 4a + 3
\]

Now, solve for \( a \):
\[
7 - 3 = 4a
\]
\[
4 = 4a
\]
\[
a = 1
\]

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

### General Process Recap:
1. Identify the vertex \( (h, k) \).
2. Substitute the vertex into the vertex form of the equation \( y = a(x - h)^2 + k \).
3. Use a second point on the parabola to solve for \( a \).
4. Substitute \( a \), \( h \), and \( k \) into the equation.

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Would you like more details or have any questions? Here are 5 related questions you might explore:

1. How do you find the vertex of a parabola from its standard equation?
2. Can you derive the equation of a parabola from its intercepts?
3. How does changing the value of \( a \) affect the shape of the parabola?
4. How can you convert a standard form equation to vertex form?
5. What is the significance of the focus and directrix in the equation of a parabola?

**Tip**: The vertex \( (h, k) \) gives the minimum or maximum point of the parabola, depending on the sign of \( a \).

By identifying key points on the graph, algebraically develop the equation of the parabola in the form y=a(x-h)^2+k

Learn how to algebraically develop the equation of a parabola using its vertex form y=a(x-h)^2+k. This detailed step-by-step solution explores algebraic concepts, such as identifying the vertex and solving for the constant a using another point on the graph. Suitable for students in Grades 9-12.

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