Let's go over each problem based on the image you've uploaded:

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: $y = -(x + 3)^2 - 2$.

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.