https://file.aiquickdraw.com/mathbot/file/17262721566235dml8bug.jpg

In the image you provided, the task involves graphing transformations of the parent function \( f(x) = x^2 \) and identifying the type of transformation. Let me walk through each of the parts visible in the image:

### Part a. \( g(x) = x^2 + 2 \)
- **Graph**: The graph of \( g(x) = x^2 + 2 \) is a vertical shift of the parent function \( f(x) = x^2 \) upward by 2 units.
- **Transformation**: Vertical translation up by 2 units.

### Part b. \( h(x) = (x + 2)^2 \)
- **Graph**: The graph of \( h(x) = (x + 2)^2 \) is a horizontal shift of the parent function \( f(x) = x^2 \) to the left by 2 units.
- **Transformation**: Horizontal translation to the left by 2 units.

### Part c. \( k(x) = 2x^2 \)
- **Graph**: The graph of \( k(x) = 2x^2 \) stretches the graph of \( f(x) = x^2 \) vertically by a factor of 2, making the parabola narrower.
- **Transformation**: Vertical stretch by a factor of 2.

### Part d. \( p(x) = -x^2 \)
- **Graph**: The graph of \( p(x) = -x^2 \) reflects the parent function \( f(x) = x^2 \) across the x-axis.
- **Transformation**: Reflection over the x-axis.

If you need a more detailed breakdown of any part or further explanations, feel free to ask!

### Follow-up Questions:
1. What would happen if \( g(x) = x^2 - 3 \)? How would the graph shift?
2. How do horizontal shifts differ from vertical shifts in transformation?
3. What would be the transformation if \( k(x) = \frac{1}{2}x^2 \)?
4. What kind of transformation occurs if the function is \( h(x) = (x - 4)^2 \)?
5. Can you explain why the graph reflects over the x-axis in \( p(x) = -x^2 \)?

### Tip:
When dealing with horizontal shifts, the transformations inside the parentheses (like \( (x + 2) \)) move the graph in the opposite direction to the sign.

Graph each of the following transformations to the parent function f(x) = x^2. Plot at least 5 points (if possible). Then, identify the type of transformation that occurs.

Explore graph transformations of the quadratic function f(x) = x^2. Understand vertical shifts, horizontal shifts, reflections, and vertical stretches through detailed graph plots and transformations. Suitable for students in Grades 8-10.

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