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### Problem 8:
It discusses the transformations of parabolas and requires analysis of graphs and transformations.

1. **Part (a):** 
   - The graph of \( y = (x - 3)^2 + 2 \) is a vertical shift of the graph \( y = (x - 3)^2 \). It is shifted **up by 2 units**.
   
2. **Part (b):**
   - The graph of \( y = (x - 3)^2 - 2 \) is a vertical shift of \( y = (x - 3)^2 \), but shifted **down by 2 units**.

3. **Part (c):** 
   - The graph of \( y = (x + 3)^2 + 2 \) represents a horizontal shift **left by 3 units** and a vertical shift **up by 2 units** compared to \( y = x^2 \).

4. **Part (d):** 
   - This part asks to explain the connection between the vertex form of a parabola \( y = a(x - h)^2 + k \) and its vertex \((h, k)\). The vertex is at the point \((h, k)\), and the transformations applied to the basic parabola \( y = x^2 \) can be directly observed in the values of \( h \) and \( k \).

5. **Part (e):** 
   - The task is to sketch different parabolas based on transformations:
     1. \( y = (x - 2)^2 + 3 \): Shift right by 2 units and up by 3 units.
     2. \( y = (x + 2)^2 + 3 \): Shift left by 2 units and up by 3 units.
     3. \( y = (x - 4)^2 - 2 \): Shift right by 4 units and down by 2 units.
     4. \( y = (x + 1)^2 - 2 \): Shift left by 1 unit and down by 2 units.

### Problem 9:
It requires sketching parabolas that have undergone transformations, especially involving reflection:

- **Part (a):** \( y = (x + 4)^2 \): This is shifted left by 4 units.
- **Part (b):** \( y = -(x + 4)^2 \): Shift left by 4 units, but the parabola is reflected over the x-axis.
- **Part (c):** \( y = -(x + 4)^2 + 3 \): Shift left by 4 units, reflect over the x-axis, and shift up by 3 units.

### Problem 10:
This focuses on finding equations of parabolas that result from transformations applied to \( y = x^2 \):

- **Part (a):** Move up by 2 units: \( y = x^2 + 2 \).
- **Part (b):** Move down by 2 units: \( y = x^2 - 2 \).
- **Part (c):** Move 2 units to the right: \( y = (x - 2)^2 \).
- **Part (d):** Move 2 units to the left: \( y = (x + 2)^2 \).
- **Part (e):** Turn it upside down and move it up by 4 units: \( y = -x^2 + 4 \).
- **Part (f):** Turn it upside down and move it down by 2 units: \( y = -x^2 - 2 \).
- **Part (g):** Move it up by 2 units and reflect in the x-axis: \( y = -x^2 + 2 \).
- **Part (h):** Move it 2 units to the right and turn it upside down: \( y = -(x - 2)^2 \).
- **Part (i):** Turn it upside down and move it 3 units to the left and then down by 2 units: \( y = -(x + 3)^2 - 2 \).

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Would you like further details or explanations on any of these transformations or solutions?

Here are 5 related questions to expand on this topic:
1. What happens to a parabola's vertex if you apply both horizontal and vertical shifts?
2. How does reflecting a parabola over the x-axis change its equation?
3. How do we derive the vertex form of a parabola from its standard form?
4. What impact does a horizontal shift have on the axis of symmetry of a parabola?
5. Can we have transformations where both vertical and horizontal stretching/compression are involved? How does that affect the graph?

**Tip:** When transforming parabolas, always focus on the vertex form \( y = a(x - h)^2 + k \) to easily track shifts, stretches, and reflections.

The graphs of y = (x - 3)^2 and y = (x - 3)^2 + 2 are shown on the diagram. Analyze the transformations of parabolas and sketch the graphs accordingly.

This lesson covers the graphing and transformation of quadratic functions, focusing on the equation y = (x - 3)^2 and its variations. It explores vertical and horizontal shifts, reflections, and provides a step-by-step guide to sketching the graphs. Suitable for students in Grades 9-11.

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