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

To determine the equation of the transformed function $g(x)$, we need to identify the transformations applied to the basic parabola $y = x^2$.

Analysis of the Graph:

1. Vertex: The vertex of the graph is located at $(-3, -4)$.
2. Shape and Direction: The graph is a parabola that opens upwards, which indicates no reflection over the x-axis.
3. Horizontal Shift: The graph is shifted 3 units to the left.
4. Vertical Shift: The graph is shifted 4 units down.

Equation of the Transformation:

The standard form for a transformed quadratic function is:

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

where:

• $(h, k)$ is the vertex of the parabola,
• $a$ determines the vertical stretch or compression and the direction of the parabola.

Since the vertex is at $(-3, -4)$, we have $h = -3$ and $k = -4$. There appears to be no vertical stretch or compression, so $a = 1$.

Final Equation:

$g(x) = (x + 3)^2 - 4$

Would you like more details or have any questions?

Related Questions:

1. How would the equation change if the parabola were reflected over the x-axis?
2. What would the equation look like if the graph were vertically stretched by a factor of 2?
3. How can you determine the transformations if given only the equation?
4. What is the effect of changing the value of $a$ in the equation?
5. How do horizontal and vertical shifts affect the vertex form of a quadratic equation?

Tip:

Always identify the vertex and the direction of opening first when analyzing the transformation of a parabola.