The graph you have uploaded contains two parts:

1. The first part shows the graph of $f(x) = |x|$, which is a V-shaped graph with a vertex at the origin (0,0).
2. The second part asks you to sketch the graph of $y = f(3 - x)$, which is a transformation of the graph of $f(x) = |x|$.

Let's break down how to approach the transformation for the graph of $y = f(3 - x)$:

Step-by-Step Process:

• Step 1 (Understanding $f(x) = |x|$): The original graph of $f(x) = |x|$ is a V-shaped graph with a vertex at (0,0). It opens upwards and has two linear pieces: one for $x \geq 0$ (where $f(x) = x$) and another for $x < 0$ (where $f(x) = -x$).

• Step 2 (Horizontal Shift): The function $f(3 - x)$ is a horizontal transformation of $f(x) = |x|$. Specifically, the graph of $f(3 - x)$ is a horizontal reflection followed by a horizontal shift. First, reflect the graph about the y-axis (because of the negative sign in $3 - x$), then shift it 3 units to the right.

Key Points for $y = f(3 - x)$:

• The vertex of the original graph $f(x) = |x|$ was at (0,0). After the transformation, the new vertex will shift to $x = 3$, so the vertex of the new graph is at $(3,0)$.
• The V-shape remains, but the graph will be reflected horizontally, meaning the left part of the V (which originally went leftwards from the vertex) will now go right, and the right part will go left.

Now, sketch the graph as follows:

1. The vertex is at $(3,0)$.
2. The left part of the graph will decrease as $x$ increases, following a line with slope $-1$ (from $x = 3$ to the right).
3. The right part of the graph will increase with a slope of $1$ as you move leftwards from $x = 3$.

Would you like further clarification, or do you have any questions on this?

Related Questions:

1. What does a horizontal shift by 3 units mean in general for any function $f(x)$?
2. How does the graph change if you add or subtract constants to the function $f(x)$?
3. What is the difference between vertical and horizontal transformations?
4. How would you reflect the graph of $f(x) = |x|$ across the x-axis?
5. How can you algebraically verify that $y = f(3 - x)$ is a reflection and shift of $f(x) = |x|$?

Tip:

When transforming graphs, always handle reflections first (if there is a negative sign involved) before applying shifts.