https://file.aiquickdraw.com/mathbot/file/17255828014533n7sdojw.jpg

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.

Sketch the graph of y = f(3 - x) based on the given graph of f(x) = |x|.

Learn how to transform the graph of f(x) = |x| to y = f(3 - x). This problem covers key algebraic concepts like graph reflection and horizontal shifting. Suitable for students in Grades 8-10.

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