Let's break down how each graph can be obtained from the graph of $y = f(x)$:

a) $y = -f(x)$

• This transformation reflects the graph of $y = f(x)$ across the x-axis. All the y-values of the function are multiplied by -1, flipping the graph upside down.

b) $y = f(x + 2) - 3$

• The $(x + 2)$ results in a horizontal shift to the left by 2 units.
• The $-3$ at the end results in a vertical shift down by 3 units.

c) $y = -3f(x + 7)$

• The $(x + 7)$ results in a horizontal shift to the left by 7 units.
• The multiplication by -3 causes two effects:
  1. A vertical stretch by a factor of 3 (since the absolute value is greater than 1).
  2. A reflection across the x-axis due to the negative sign.

Each part of the transformation adjusts the base graph $f(x)$ by shifting, stretching, or reflecting it.

Do you want more detailed explanations for any specific part?

Here are some related questions to explore:

1. What is the general formula for vertical and horizontal shifts in a function?
2. How do vertical stretches affect the steepness of a graph?
3. How do you identify reflections in graphs based on transformations?
4. What would happen if the function was stretched by a factor of 1/2 instead of 3?
5. How would the graph change if the $f(x)$ was replaced by $f(-x)$?

Tip: A vertical shift affects the y-values directly, while a horizontal shift affects the x-values in the opposite direction of the sign.