https://file.aiquickdraw.com/mathbot/file/1727065052445j5vr2jgh.jpg

The graph shown is a transformation of the basic quadratic function \( f(x) = x^2 \).

### Observations:
1. **Vertical Reflection**: The parabola opens downwards, indicating a vertical reflection, meaning there is a negative coefficient for \(x^2\).
2. **Vertical Stretch**: The vertex is located at \((-2, 4)\), which is a transformation from the origin in the base function \( f(x) = x^2 \). The function is shifted vertically and horizontally.
3. **Horizontal Shift**: The vertex is moved from \( (0, 0) \) to \( (-2, 4) \), indicating a horizontal shift of \( +2 \) and a vertical shift of \( +4 \).

### Formula Construction:
The transformation is of the form:

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

Where:
- \( h = -2 \) (horizontal shift),
- \( k = 4 \) (vertical shift),
- \( a \) (vertical stretch/reflection).

Since the parabola opens downward, \( a \) is negative, and we observe the parabola passes through \( (0, 0) \). This gives us a way to determine \( a \).

Plugging \( (0, 0) \) into the general formula:

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

Solving for \( a \):

\[
0 = 4a + 4
\]
\[
-4 = 4a
\]
\[
a = -1
\]

Thus, the function is:

\[
g(x) = -(x + 2)^2 + 4
\]

### Final Answer:
\[
g(x) = -(x + 2)^2 + 4
\]

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Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:
1. What is the effect of the vertical stretch factor \( a \) on the shape of the parabola?
2. How would the equation change if the parabola opened upwards instead?
3. What happens if the vertex of the parabola was moved to \( (1, -3) \)?
4. How can you determine the horizontal and vertical shifts by just looking at the graph?
5. How does the transformation affect the axis of symmetry?

**Tip:** Always use the vertex of a parabola and another point on the graph to determine the transformation parameters accurately.

The graph above is a transformation of the function x^2. Give the function in the graph above.

Learn how to transform the quadratic function x^2 to find g(x), incorporating horizontal and vertical shifts, as well as reflections. This problem involves recognizing shifts to the vertex and determining the transformation's formula, including vertical reflections. Suitable for Grades 9-12.

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