Let's walk through the questions visible in the image:

1. The graph of $y = f(x) - 5$ is obtained by doing what to the graph of $y = f(x)$?

• Explanation: Subtracting a constant from a function’s output (i.e., $-5$) moves the graph vertically down by 5 units.
• Answer: "Shifting the graph down 5 units."

2. The graph of $y = f(x + 5)$ is obtained by doing what to the graph of $y = f(x)$?

• Explanation: Adding 5 inside the function shifts the graph horizontally to the left by 5 units. This might seem counterintuitive, but positive values inside the function result in a leftward shift.
• Answer: "Shifting the graph left 5 units."

3. The graph of $y = -3f(x)$ is obtained by doing what to the graph of $y = f(x)$?

• Explanation:

  1. The multiplication by -1 reflects the graph across the x-axis.
  2. The multiplication by 3 stretches the graph vertically by a factor of 3.

• Answer: "Reflecting the graph across the x-axis and vertically stretching it by a factor of 3."

4. The graph of $y = f(x/2)$ is obtained by doing what to the graph of $y = f(x)$?

• Explanation: Dividing the input by 2 compresses the x-values, which results in a horizontal stretch of the graph by a factor of 2.
• Answer: "Horizontally stretching the graph by a factor of 2."

Do you want me to provide more detailed explanations for these transformations?

Related questions:

1. What happens if we multiply the function output by a fraction (e.g., $\frac{1}{2}f(x)$)?
2. How does adding a constant inside versus outside the function affect the graph?
3. What is the result of reflecting the graph over the y-axis instead of the x-axis?
4. How does $f(-x)$ change the graph compared to $-f(x)$?
5. What is the general rule for stretching and compressing graphs vertically and horizontally?

Tip: When modifying a function's input (inside the parentheses), the effect is on the horizontal axis. When modifying the output (outside the parentheses), the effect is on the vertical axis.