To understand the transformation from the graph of $f(x) = -4(x - 7)^2 - 4$ to the graph of $g(x) = -4(x + 2)^2 - 4$, let's analyze the equations step by step.

Analyzing the Functions

1. Function $f(x) = -4(x - 7)^2 - 4$:

   • Vertex Form: This is already in vertex form, $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.
   • Vertex: $(7, -4)$
   • Direction: The parabola opens downwards since the coefficient of $(x - 7)^2$ is negative.
   • Vertical Stretch: The factor $-4$ indicates a vertical stretch by a factor of 4, and the negative sign reflects it across the x-axis.

2. Function $g(x) = -4(x + 2)^2 - 4$:

   • Vertex Form: This is also in vertex form, $g(x) = a(x - h)^2 + k$.
   • Vertex: $(-2, -4)$
   • Direction: The parabola also opens downwards with the same coefficient of $-4$, indicating the same vertical stretch and reflection across the x-axis as in $f(x)$.

Transformation Analysis

The change from $f(x)$ to $g(x)$ involves the following transformations:

• Horizontal Shift: The vertex of $f(x)$ is at $(7, -4)$, and the vertex of $g(x)$ is at $(-2, -4)$. To go from $(7, -4)$ to $(-2, -4)$, we move 9 units to the left on the x-axis.
• Vertical Shift: There is no vertical shift because the $k$-value (which represents the y-coordinate of the vertex) remains at $-4$ for both functions.

Conclusion

The transformation that converts the graph of $f(x) = -4(x - 7)^2 - 4$ to $g(x) = -4(x + 2)^2 - 4$ is a horizontal shift 9 units to the left.

Would you like more details or have any other questions?

Here are five related questions to expand your understanding:

1. What is a vertical shift in a graph transformation?
2. How would you transform $f(x) = x^2$ to shift it 5 units to the right and 3 units down?
3. What does a reflection over the x-axis do to the graph of a function?
4. Can you explain what a vertical stretch or compression means in terms of graph transformation?
5. How would the graph of $f(x) = -4(x - 7)^2 - 4$ change if the coefficient $-4$ were changed to $4$?

Tip: Understanding transformations helps in graphing functions quickly without plotting individual points.