What kind of transformation converts the graph of f(x)=10x2–2 into the graph of g(x)=
–
10x2+2?



The transformation that converts the graph of \( f(x) = 10x^2 - 2 \) into the graph of \( g(x) = -10x^2 + 2 \) involves two distinct transformations:

1. **Vertical reflection**: 
   - The coefficient of \( x^2 \) in \( f(x) \) is positive (10), while in \( g(x) \) it is negative (-10). This means the graph of \( f(x) \) is vertically reflected across the x-axis to obtain the graph of \( g(x) \).
   - Specifically, this changes the direction in which the parabola opens: the graph of \( f(x) \) opens upwards, while the graph of \( g(x) \) opens downwards.

2. **Vertical shift**: 
   - In \( f(x) \), the constant term is -2, whereas in \( g(x) \), the constant term is +2. This means the graph is shifted **4 units upwards**. The transformation is a vertical translation of the graph of \( f(x) \) by 4 units to get the graph of \( g(x) \).

### Summary of Transformations:
1. **Vertical reflection**: Reflect across the x-axis.
2. **Vertical translation**: Shift 4 units upwards.

Would you like more details on how to apply these transformations?

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Here are five related questions to explore further:
1. What are the general steps for reflecting a graph vertically?
2. How would you shift the graph of \( f(x) \) if the constant term changed by more than 4 units?
3. What is the effect of changing the coefficient of \( x^2 \) in a quadratic function?
4. How do horizontal transformations differ from vertical ones?
5. Can you combine multiple transformations in one step?

**Tip**: When reflecting a graph, remember that a vertical reflection changes only the sign of the output (y-values), not the x-values.

What kind of transformation converts the graph of f(x)=10x2–2 into the graph of g(x)=–10x2+2?

Understand how to convert the graph of f(x) = 10x^2 - 2 into the graph of g(x) = -10x^2 + 2 by applying a vertical reflection and vertical shift. This problem focuses on graph transformations in quadratic functions and is suitable for students in Grades 10-12.

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