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:

$g(x) = -(x + 2)^2 + 4$

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.