Which of the following is the equation of the function shown?

g(x)=(x−2)2−2

g(x)=(x+2)2−2

g(x)=(x+3)2−1

g(x)=(x−3)2+1

To determine the correct equation of the function \( g(x) \) based on the options provided, let's analyze the general form of a quadratic function in vertex form:

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

Here, \( (h, k) \) represents the vertex of the parabola, and the parabola opens upwards if \( a > 0 \).

Given the options:

1. \( g(x) = (x - 2)^2 - 2 \)
2. \( g(x) = (x + 2)^2 - 2 \)
3. \( g(x) = (x + 3)^2 - 1 \)
4. \( g(x) = (x - 3)^2 + 1 \)

Each equation's vertex is as follows:

1. \( g(x) = (x - 2)^2 - 2 \) has a vertex at \( (2, -2) \).
2. \( g(x) = (x + 2)^2 - 2 \) has a vertex at \( (-2, -2) \).
3. \( g(x) = (x + 3)^2 - 1 \) has a vertex at \( (-3, -1) \).
4. \( g(x) = (x - 3)^2 + 1 \) has a vertex at \( (3, 1) \).

To determine which equation matches the function, we need the location of the vertex and the direction the parabola opens (upwards or downwards). 

1. **If the vertex is at \( (2, -2) \) and the parabola opens upwards, the correct equation is \( g(x) = (x - 2)^2 - 2 \).**
2. **If the vertex is at \( (-2, -2) \) and the parabola opens upwards, the correct equation is \( g(x) = (x + 2)^2 - 2 \).**
3. **If the vertex is at \( (-3, -1) \) and the parabola opens upwards, the correct equation is \( g(x) = (x + 3)^2 - 1 \).**
4. **If the vertex is at \( (3, 1) \) and the parabola opens upwards, the correct equation is \( g(x) = (x - 3)^2 + 1 \).**

Without the graph, you would compare the vertex and direction of the parabola with the options given to identify the correct function. If you have information about the graph's vertex and direction, you can match it directly.

Would you like more details on any specific option or general guidance on how to match equations with graphs?

Here are five related questions:
1. How do you determine the vertex of a quadratic function from its equation?
2. What are the effects of translating a parabola horizontally and vertically?
3. How can you identify the direction a parabola opens from its equation?
4. What is the importance of the vertex form of a quadratic function?
5. How does changing the coefficient \( a \) in the vertex form affect the shape of the parabola?

**Tip:** The vertex form of a quadratic equation is very useful for quickly identifying the vertex and direction of the parabola.

Learn how to identify the correct equation of a quadratic function given its vertex form and characteristics. This problem involves analyzing the vertex and direction of parabolas represented by equations g(x) = (x - 2)^2 - 2, (x + 2)^2 - 2, (x + 3)^2 - 1, (x - 3)^2 + 1. Suitable for high school students and those preparing for standardized tests.

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