https://file.aiquickdraw.com/mathbot/file/1723047784473iyu0nudu.jpg

To determine which equation will produce the graph shown in the image, let's analyze the graph in detail. 

The graph depicts a parabola that opens upwards with its vertex at the point \((2, -2)\).

The standard form of a parabola with vertex \((h, k)\) is:

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

Given the vertex is at \((2, -2)\), we can substitute \(h = 2\) and \(k = -2\) into the equation:

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

We need to determine the value of \(a\). If the point \((2, -2)\) is the vertex, we can use another point on the graph to find \(a\). Let's consider a point on the graph such as \((0, 2)\). Substituting \((0, 2)\) into the equation, we get:

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

Therefore, the equation of the parabola is:

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

To confirm, we check other points on the graph. For instance, at \(x = 1\):

\[ y = (1 - 2)^2 - 2 = 1 - 2 = -1 \]

This matches the graph.

Thus, the equation that produces the graph shown is:

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

Do you want any details or have any questions?

Here are 8 related questions to expand your understanding:

1. What is the general form of a quadratic equation?
2. How do you find the vertex of a parabola given its equation in standard form?
3. What does the coefficient \(a\) in a quadratic equation indicate about the graph's shape?
4. How can you determine if a parabola opens upwards or downwards?
5. How do you find the axis of symmetry for a quadratic function?
6. How can you convert a quadratic equation from vertex form to standard form?
7. What is the importance of the vertex in graphing a parabola?
8. How do you find the x-intercepts of a quadratic function?

**Tip:** When graphing a quadratic function, always start by identifying the vertex and the direction the parabola opens to get a clear idea of its shape.

Explore how to find the equation of a parabola when given its vertex and another point on the graph. This problem covers quadratic equations and the vertex form of parabolas, providing step-by-step analysis and examples. Suitable for high school students.

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