The graph shown in the image is a transformation of the basic quadratic function $x^2$. By analyzing the graph, we can determine the specific transformations.

The standard form of a quadratic function is:

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

Where:

• $a$ controls the vertical stretch or compression (and reflection),
• $h$ is the horizontal shift, and
• $k$ is the vertical shift.

Observations:

1. Vertex: The vertex of the parabola appears to be at $(-1, 2)$. This means the graph has been shifted 1 unit to the left and 2 units up.
2. Reflection: The parabola opens downward, which indicates that there is a reflection across the x-axis. This is represented by $a < 0$.
3. No Stretch or Compression: The shape of the parabola seems consistent with the standard parabola $x^2$, so $a = -1$.

Function:

Using these transformations, the equation of the function can be written as:

$g(x) = -(x + 1)^2 + 2$

This accounts for the reflection (negative sign), the shift left by 1 unit, and the shift up by 2 units.

Would you like further details or explanations on any part of this? Here are some related questions you might want to explore:

1. What does the parameter $a$ represent in transformations of quadratic functions?
2. How can we find the vertex of a parabola given its equation?
3. How does the graph change if we change the value of $a$ in $a(x - h)^2 + k$?
4. What are the effects of horizontal and vertical shifts on the graph of a quadratic function?
5. How can you determine if a parabola opens upwards or downwards just by looking at the equation?

Tip: Always pay attention to the vertex and direction of the parabola when working with quadratic transformations. These are key features that help you quickly identify the transformations applied.