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To describe the transformations applied to \( y = x^2 \) to obtain the function \( y = -2(x - 1)^2 + 23 \), we break it down step by step:

1. **Horizontal Translation**: 
   - The expression inside the parentheses, \( (x - 1) \), indicates a shift to the **right by 1 unit**. This is because \( x - h \) represents a horizontal shift of \( h \) units to the right.

2. **Vertical Stretch**:
   - The coefficient of 2 outside the parentheses, \( -2 \), indicates a **vertical stretch** by a factor of 2. Normally, this would stretch the parabola, making it narrower.

3. **Reflection**:
   - The negative sign in front of \( -2 \) indicates a **reflection over the x-axis**. This flips the parabola upside down.

4. **Vertical Translation**:
   - The constant \( +23 \) outside the squared term indicates a **vertical shift upwards by 23 units**.

### Summary of Transformations:
- Shift right by 1 unit.
- Vertically stretched by a factor of 2.
- Reflected over the x-axis.
- Shifted up by 23 units.

Would you like further details or have any other questions?

Here are 5 related questions to expand your understanding:
1. How would the transformations change if the equation were \( y = 2(x + 3)^2 - 5 \)?
2. What does the coefficient of the squared term do to the shape of the parabola?
3. How does a horizontal shift affect the vertex of a parabola?
4. Can you describe how reflections over the y-axis differ from reflections over the x-axis?
5. How would the graph of \( y = (x - 1)^2 \) look without the reflection and vertical shift?

**Tip:** Always start by identifying changes inside parentheses as they indicate horizontal transformations, while constants outside indicate vertical shifts or reflections.

Describe the transformations that were applied to y = x^2 to obtain the function y = -2(x - 1)^2 + 23.

Learn how to describe the transformations applied to y = x^2 to obtain the quadratic function y = -2(x - 1)^2 + 23. This problem explores shifts, stretches, reflections, and transformations in algebra. Suitable for students in Grades 9-10.

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