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For the graph shown below, determine the quadratic equation in vertex form that represents the graph.

To determine the quadratic equation in vertex form from the given graph, we need to use the vertex form of a quadratic equation, which is:

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

Here, \((h, k)\) is the vertex of the parabola. From the graph, we can see the vertex is at the point \((2, -3)\).

Thus, the equation in vertex form will be:

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

Next, we need to find the value of \(a\). To do this, we can use another point that lies on the parabola. From the graph, we can see that the point \((1, -1)\) lies on the parabola. 

We substitute \((x, y) = (1, -1)\) into the equation and solve for \(a\):

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

Simplify the equation:

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

Solving for \(a\):

\[ a - 3 = -1 \]
\[ a = 2 \]

Thus, the quadratic equation in vertex form that represents the graph is:

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

Would you like more details or have any questions? Here are 5 questions you might consider next:

1. How do we convert this vertex form to the standard form of a quadratic equation?
2. How can we verify the correctness of this quadratic equation using other points on the graph?
3. What is the significance of the vertex in a quadratic equation?
4. How does the value of \(a\) affect the shape and direction of the parabola?
5. Can we determine the axis of symmetry from the vertex form of the equation?

**Tip:** When finding the quadratic equation from a graph, always identify the vertex and another point on the graph to determine the value of \(a\).

Learn how to determine the quadratic equation in vertex form from a graph with a detailed step-by-step solution. Understand the vertex form of a quadratic equation and how to find 'a' using points on the parabola. Suitable for high school students studying quadratic equations.

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