Let's analyze the problems from the image.

For Problem 13, you're given a graph of a function $f(x)$ with specific points and asked to sketch the transformations of this function based on different formulas. I'll break down each transformation:

1. (a) $y = f(x) + 2$: This is a vertical shift upwards by 2 units. All points on the graph move 2 units up.

2. (b) $y = -f(x)$: This is a reflection of the graph across the x-axis. All the y-values of the points change sign.

3. (c) $y = f(x - 2)$: This is a horizontal shift to the right by 2 units. All points on the graph move 2 units to the right.

4. (d) $y = f(x + 3)$: This is a horizontal shift to the left by 3 units. All points on the graph move 3 units to the left.

5. (e) $y = 2f(x)$: This is a vertical stretch by a factor of 2. The distance of each point from the x-axis doubles.

6. (f) $y = f(-x)$: This is a reflection of the graph across the y-axis. The x-values change sign, reflecting the graph horizontally.

7. (g) $y = f\left(\frac{1}{2}x\right)$: This is a horizontal stretch by a factor of 2. The graph stretches out, making it wider.

For Problem 14, the function $f(x)$ is given with different transformation options:

1. (a) $y = f(x) - 1$: This is a vertical shift downwards by 1 unit. All points on the graph move 1 unit down.

2. (b) $y = f(x + 1)$: This is a horizontal shift to the left by 1 unit. All points on the graph move 1 unit to the left.

3. (c) $y = f(x - 1)$: This is a horizontal shift to the right by 1 unit. All points move 1 unit to the right.

4. (d) $y = -f(x - 2)$: This combines a horizontal shift to the right by 2 units and a reflection across the x-axis.

5. (e) $y = f(-x)$: This is a reflection of the graph across the y-axis. The x-values change sign.

6. (f) $y = \frac{1}{2}f(x)$: This is a vertical compression by a factor of $\frac{1}{2}$, meaning the y-values are halved, bringing the graph closer to the x-axis.

7. (g) $y = f(2x)$: This is a horizontal compression by a factor of 2, making the graph narrower.

Do you want further details or have any questions on how these transformations affect the graph?

Here are 5 related questions to explore:

1. How does a horizontal shift differ from a vertical shift in terms of affecting the graph?
2. What would happen if you applied both a horizontal and vertical shift to the function?
3. How do reflections across the x-axis and y-axis affect the graph's symmetry?
4. What is the effect of combining multiple transformations (e.g., stretch and shift)?
5. How does a horizontal stretch/compression differ from a vertical stretch/compression?

Tip: When sketching transformations, always start with basic shifts (vertical/horizontal) before applying stretches or reflections.