The graph provided is a transformation of the standard quadratic function $f(x) = x^2$.

Observations:

1. Vertex Shift: The vertex of the parabola is located at $(2, -1)$, indicating a horizontal shift to the right by 2 units and a vertical shift downward by 1 unit.
2. Reflection: The parabola opens downward, which suggests that there is a vertical reflection (indicating a negative coefficient for $x^2$).
3. No Stretching/Compressing: The shape of the parabola seems standard, with no visible stretching or compressing.

Transformation Equation:

Starting with the basic quadratic form $f(x) = a(x-h)^2 + k$, where:

• $h$ is the horizontal shift (2 units to the right),
• $k$ is the vertical shift (down by 1 unit),
• $a$ controls the reflection and vertical stretching (in this case, $a = -1$ due to the downward opening).

Thus, the equation becomes:

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

This is the function of the graph.

Would you like more details, or do you have any questions?

Related Questions:

1. How do we determine the reflection in quadratic transformations?
2. What happens if the value of $a$ in a parabola changes?
3. Can a quadratic function have more than one transformation applied?
4. How do we find the axis of symmetry for a transformed parabola?
5. What are the steps to find the vertex of any quadratic function?

Tip: The vertex form of a quadratic function is handy for identifying transformations like shifts, reflections, and stretching/compressing.