Let's analyze each function transformation step by step:

a. $y = f(x) - 1$

• Transformation: This is a vertical shift downward by 1 unit.
  • Every point on the graph of $f(x)$ is shifted down by 1 unit.

Result: The new graph moves vertically down.

b. $y = \frac{1}{2}(f(x) + 2)$

• First: $f(x) + 2$ indicates a vertical shift upward by 2 units.
  • This moves the graph of $f(x)$ 2 units up.
• Then: $\frac{1}{2}$ scales the graph vertically by a factor of 1/2 (compression).
  • This brings all the points closer to the x-axis.

Result: The graph shifts 2 units up and is vertically compressed.

c. $y = -2(f(x) - 5)$

• First: $f(x) - 5$ shifts the graph downward by 5 units.
• Then: $-2$ applies two transformations:
  1. The negative sign reflects the graph across the x-axis.
  2. The factor 2 stretches the graph vertically by a factor of 2.

Result: The graph is shifted down by 5 units, reflected across the x-axis, and stretched vertically.

Would you like further clarification or a detailed graph for any of these transformations?

Related Questions:

1. What is the difference between vertical stretching and compression?
2. How does adding a constant inside vs. outside the function change the transformations?
3. What happens if the reflection is applied without any stretching?
4. How would the transformations change if the coefficient in part (b) were negative?
5. How can horizontal shifts and reflections be applied to the function?

Tip: A general rule is that constants outside the function affect vertical changes, while constants inside affect horizontal shifts.