What kind of transformation converts the graph of f(x)=
–
5(x–8)2–3 into the graph of g(x)=
–
5(x–1)2–6?



To understand the transformation from the graph of \( f(x) = -5(x - 8)^2 - 3 \) to the graph of \( g(x) = -5(x - 1)^2 - 6 \), we need to analyze the changes in both the horizontal and vertical shifts, as well as any changes in the shape or orientation of the graph.

1. **Horizontal Shift**:  
   - In \( f(x) = -5(x - 8)^2 - 3 \), the vertex of the parabola is at \( x = 8 \).
   - In \( g(x) = -5(x - 1)^2 - 6 \), the vertex of the parabola is at \( x = 1 \).
   - The horizontal shift is from \( x = 8 \) to \( x = 1 \), which is a shift **7 units to the right**.

2. **Vertical Shift**:  
   - In \( f(x) \), the vertical shift is \( -3 \), meaning the graph is shifted 3 units down.
   - In \( g(x) \), the vertical shift is \( -6 \), meaning the graph is shifted 6 units down.
   - The vertical shift is from \( -3 \) to \( -6 \), which is a shift **3 units downward**.

3. **Shape and Reflection**:  
   - The coefficients of \( x^2 \) in both functions are \( -5 \), meaning that the shape (narrowness) and the fact that the parabola opens downward are **unchanged** between the two functions.

### Summary of the transformation:
- **7 units to the right**
- **3 units downward**
- **No change in shape or orientation**

Would you like more details on any part of this transformation?

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Here are 5 related questions to deepen understanding:

1. How would the transformation change if the coefficient of \( x^2 \) was different in \( g(x) \)?
2. What would happen if we applied a vertical stretch instead of a shift?
3. How would you describe the transformation from \( f(x) \) to \( g(x) \) using function notation?
4. How does a shift in the vertex form affect the graph's symmetry?
5. Can you transform \( f(x) \) into \( g(x) \) using a single transformation?

**Tip**: Always look at the vertex form \( a(x-h)^2 + k \) to easily identify shifts and transformations!

What kind of transformation converts the graph of f(x) = -5(x - 8)^2 - 3 into the graph of g(x) = -5(x - 1)^2 - 6?

Learn how to transform the graph of f(x) = -5(x - 8)^2 - 3 to g(x) = -5(x - 1)^2 - 6 by analyzing horizontal and vertical shifts. This problem covers key concepts in graph transformations, including a 7-unit horizontal shift to the right and a 3-unit vertical shift downward. Suitable for students in Grades 9-11.

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