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The given problem asks for a description of the transformation of the function \( f(x) = x^2 \) to the new function \( f(x) = -2x^2 + 7 \).

### Step-by-step breakdown:

1. **Vertical Stretch and Compression**: The coefficient of \( x^2 \) is \( -2 \). This means:
   - **Reflection over the x-axis**: The negative sign in front of the \( x^2 \) reflects the parabola over the x-axis.
   - **Vertical Stretch**: The factor of 2 causes a vertical stretch, making the parabola narrower.

2. **Vertical Shift**: The constant term \( +7 \) shifts the graph vertically.
   - **Vertical Shift Up**: Since the constant is positive, the parabola is shifted 7 units upwards.

### Summary of transformations:
- Reflection over the x-axis.
- Parabola is narrower due to vertical stretch.
- Vertical shift up.

### Correct Choices:
- Reflection over the x-axis.
- Parabola is narrower due to vertical stretch.
- Vertical shift up.

Would you like any more details or clarifications?

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Here are 5 related questions to explore further:

1. What would happen if the coefficient of \( x^2 \) was a fraction?
2. How would the graph change if the constant term was negative?
3. Can you describe the transformation for \( f(x) = -3x^2 - 5 \)?
4. What is the effect of horizontal shifts on parabolas?
5. How does a negative coefficient inside the squared term affect the graph?

**Tip**: To identify vertical shifts, always look at the constant term (without the \( x^2 \) part) in the equation.

Choose all descriptions of the transformation of f(x) = x^2 to f(x) = -2x^2 + 7.

This problem involves transforming the basic quadratic function f(x) = x^2 to f(x) = -2x^2 + 7. The solution explains vertical reflection, vertical stretching, and vertical shifts. Ideal for students in Grades 8-10 studying quadratic transformations.

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