To draw the graph of $y = 2f(x) - 1$ based on the given graph $y = f(x)$, let’s analyze the transformations step by step:

1. Vertical Stretch by a Factor of 2:

   • The function $2f(x)$ stretches the graph vertically. This means every $y$-coordinate of $f(x)$ is multiplied by 2.

2. Vertical Translation Down by 1 Unit:

   • The $-1$ subtracts 1 from the result of $2f(x)$, moving every point of the stretched graph down by 1 unit.

Steps to Sketch the New Graph:

1. Identify key points on $y = f(x)$ (e.g., peaks, troughs, and intercepts).
2. Multiply each $y$-coordinate by 2 (apply the vertical stretch).
3. Subtract 1 from the new $y$-coordinates (apply the downward translation).
4. Plot the transformed points and connect them smoothly, preserving the shape of the original curve.

Example with Key Points:

If we pick points from $f(x)$ (as seen in the graph):

• At $x = 0$, $f(0) = 1$:
  • $2f(0) - 1 = 2(1) - 1 = 1$
• At $x = 60$, $f(60) = 0$:
  • $2f(60) - 1 = 2(0) - 1 = -1$
• At $x = 120$, $f(120) = -1$:
  • $2f(120) - 1 = 2(-1) - 1 = -3$

Would you like a detailed plot or further assistance with sketching this graph?
Here are 5 questions to deepen your understanding:

1. What effect does multiplying $f(x)$ by 2 have on the graph’s symmetry?
2. How does subtracting 1 influence the graph’s range?
3. Can you describe the transformations using function composition?
4. What happens to the period of the graph if it’s periodic?
5. How would the graph look if instead of subtracting 1, we added 1?

Tip: Always perform vertical transformations in sequence: stretch/compress first, then translate.