The given graph is a transformation of the basic quadratic function $x^2$, and we need to write the equation of the function based on the transformation.

Observations:

1. Vertex: From the graph, the vertex of the parabola is at the point $(0, 3)$, indicating a vertical shift.
2. Direction: The parabola opens downwards, which means there is a reflection over the x-axis.
3. Vertical Stretch: The function is narrower than $x^2$, indicating a vertical stretch.

Transformation equation structure:

The general form of a vertically stretched, reflected, and shifted quadratic function is:

$g(x) = a(x - h)^2 + k$

Where:

• $(h, k)$ is the vertex of the parabola.
• $a$ determines the direction (if $a < 0$, it opens downward) and the stretch/compression.

From the graph:

• The vertex $(h, k)$ is $(0, 3)$, so the equation becomes: $g(x) = a(x - 0)^2 + 3 = a x^2 + 3$
• Since the parabola opens downward, $a < 0$.

Finding $a$:

Using another point on the graph to find $a$. From the graph, the point $(1, 2)$ lies on the parabola. Substituting $x = 1$ and $g(x) = 2$ into the equation:

$2 = a(1)^2 + 3$ $2 = a + 3$ $a = -1$

Thus, the equation of the transformed function is:

$g(x) = -x^2 + 3$

Would you like more details or explanations on any part of this solution?

Here are 5 related questions to expand on this:

1. What are the general forms of other types of quadratic transformations (horizontal shifts, compressions)?
2. How would you identify the transformations from the equation alone?
3. How does changing the sign of $a$ affect the parabola?
4. What happens to the graph if we move the vertex to a different location?
5. How do vertical stretches and compressions affect the steepness of the parabola?

Tip: When analyzing transformations, always locate the vertex first as it gives you critical information on how the graph has shifted.