What kind of transformation converts the graph of f(x)=
–
4(x–7)2–4 into the graph of g(x)=
–
4(x+2)2–4?

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.

Understand how the graph of f(x)=-4(x-7)^2-4 transforms into g(x)=-4(x+2)^2-4 through horizontal and vertical shifts. Explore the vertex form of quadratic functions and learn about transformations in algebra.

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