What kind of transformation converts the graph of f(x)=

–

9(x–10)2–6 into the graph of g(x)=

–

9(x–8)2–6?

To determine the transformation that converts the graph of \( f(x) = -9(x - 10)^2 - 6 \) to the graph of \( g(x) = -9(x - 8)^2 - 6 \), we will analyze the functions' forms and identify any changes.

### Analyzing the Functions

1. **Function \( f(x) = -9(x - 10)^2 - 6 \):**
   - **Vertex Form:** The function is in the vertex form \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex.
   - **Vertex:** The vertex of this parabola is at \((10, -6)\).
   - **Direction:** The parabola opens downwards because the coefficient of \((x - 10)^2\) is negative.
   - **Vertical Stretch:** The factor \(-9\) indicates a vertical stretch by a factor of 9 and a reflection over the x-axis.

2. **Function \( g(x) = -9(x - 8)^2 - 6 \):**
   - **Vertex Form:** This is also in vertex form, \( g(x) = a(x - h)^2 + k \).
   - **Vertex:** The vertex of this parabola is at \((8, -6)\).
   - **Direction:** The parabola opens downwards with the same coefficient of \(-9\), indicating the same vertical stretch and reflection over the x-axis as in \( f(x) \).

### Transformation Analysis

Comparing the two functions:

- **Horizontal Shift:** The vertex of \( f(x) \) is at \((10, -6)\), and the vertex of \( g(x) \) is at \((8, -6)\). The transformation that occurs here is a shift of 2 units to the right to go from \( (10, -6) \) to \( (8, -6) \).

- **Vertical Shift:** There is no vertical shift as the y-coordinate of the vertex (\(-6\)) remains the same for both functions.

### Conclusion

The transformation that converts the graph of \( f(x) = -9(x - 10)^2 - 6 \) to \( g(x) = -9(x - 8)^2 - 6 \) is a **horizontal shift 2 units to the right**.

Would you like more details or have any other questions?

Here are five related questions to expand your understanding:

1. How does a horizontal shift affect the vertex form of a quadratic function?
2. What would the graph of \( f(x) = -9(x - 10)^2 - 6 \) look like if it was shifted 5 units up?
3. How does changing the coefficient of \( (x - h)^2 \) affect the graph of a quadratic function?
4. Can you explain the effects of combining horizontal and vertical shifts on a quadratic graph?
5. How would you determine the transformation if \( g(x) \) had a different vertical stretch factor than \( f(x) \)?

**Tip:** Remember that horizontal shifts are reflected in changes to the \( h \) value of the vertex form \( (x - h) \), while vertical shifts change the \( k \) value.

Explore the transformation process converting the graph of f(x) = -9(x - 10)^2 - 6 into g(x) = -9(x - 8)^2 - 6. Understand the effects of horizontal shifts in quadratic functions with detailed analysis and examples. Suitable for high school students studying quadratic equations and transformations.

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