The problem involves modifying the graph of $y = f(x)$ to create the graph of $y = 2f(x) - 1$. Here's a breakdown of the transformations:

1. Vertical Scaling (Stretching): The coefficient $2$ scales the graph of $f(x)$ vertically. This means that every $y$-value in the original graph of $f(x)$ is multiplied by $2$. For example:

   • If $f(x) = 1$, then $2f(x) = 2$.
   • If $f(x) = -1$, then $2f(x) = -2$.

2. Vertical Translation: The $-1$ translates the graph downward by 1 unit. This means that after scaling the graph vertically, you subtract 1 from each resulting $y$-value.

Steps to Sketch $y = 2f(x) - 1$:

1. For each key point on $y = f(x)$, perform the following calculations:
   • Multiply the $y$-coordinate by 2.
   • Subtract 1 from the result.
2. Plot these transformed points on the same axes.
3. Connect the points smoothly, maintaining the shape of the graph after the transformations.

Example Key Points:

Let’s take a few points from the graph of $y = f(x)$:

• At $x = 0$, $f(0) = 1$:
  $y = 2(1) - 1 = 1$
• At $x = 90$, $f(90) = 0$:
  $y = 2(0) - 1 = -1$
• At $x = 180$, $f(180) = -1$:
  $y = 2(-1) - 1 = -3$
• Similarly, apply these transformations to all key points on the curve.

Would you like a more detailed solution or a step-by-step sketching guide for this? Let me know! Below are some additional questions and tips:

5 Related Questions:

1. What does the transformation $2f(x) - 1$ represent geometrically on a graph?
2. How would the graph change if the transformation was $-2f(x) + 1$ instead?
3. How does vertical scaling affect the amplitude of periodic graphs like sine or cosine?
4. Can horizontal transformations (e.g., $f(2x)$) be combined with vertical ones like $2f(x) - 1$?
5. How do we calculate the new $x$- and $y$-intercepts after a vertical transformation?

1 Tip:

Always apply transformations step by step: first scaling, then translations. This ensures clarity and accuracy!