Let's analyze the original graph $f(x)$ shown in the image and use it to sketch the transformations based on the equations provided:

Original Function $f(x)$:

From the graph:

• The line increases from $(-1, 0)$ to $(0, 2)$.
• It is constant at $y = 2$ from $x = 0$ to $x = 2$.

Now, we will apply the transformations based on each given equation.

(a) $y = -f(-x)$:

• The transformation $f(-x)$ reflects the graph across the $y$-axis.
• The negative sign outside $f(-x)$ reflects the graph vertically (over the $x$-axis).

Steps:

1. Reflect the graph horizontally over the $y$-axis.
2. Reflect the resulting graph vertically over the $x$-axis.

(b) $y = f(2 - x)$:

• The term $f(2 - x)$ reflects the graph horizontally over the line $x = 1$ (because of the $2 - x$).

Steps:

1. First, reflect the graph over the $y$-axis.
2. Then shift the reflected graph two units to the right.

(c) $y = 1 - f(2 - x)$:

• The $f(2 - x)$ part reflects the graph across $x = 1$ (same as in part (b)).
• Then the transformation $1 - f(x)$ reflects the graph vertically and shifts it up by 1 unit.

Steps:

1. Reflect the graph horizontally (across $x = 1$).
2. Reflect the graph vertically (multiply $f(x)$ by $-1$).
3. Shift the entire graph up by 1 unit.

(d) $y = \frac{1}{2} f(2x)$:

• The $f(2x)$ compresses the graph horizontally by a factor of $\frac{1}{2}$.
• The $\frac{1}{2} f(x)$ scales the graph vertically by a factor of $\frac{1}{2}$.

Steps:

1. Compress the graph horizontally by a factor of 2.
2. Scale the graph vertically by a factor of $\frac{1}{2}$.

These are the general transformations you can apply to sketch the new graphs. Let me know if you'd like me to calculate specific points for each transformation, or clarify any particular step.

5 Related Questions:

1. How do vertical and horizontal reflections differ in graph transformations?
2. What does the transformation $y = f(x - h) + k$ represent for the graph of a function?
3. How does the factor inside the function $f(bx)$ affect the width of the graph?
4. Can you explain how translations differ from reflections in terms of graph transformations?
5. How does scaling the graph by $a$ (i.e., $y = af(x)$) change its appearance?

Tip:

When transforming graphs, always consider one transformation at a time. Reflect, shift, or stretch/compress in separate steps to avoid confusion.