https://file.aiquickdraw.com/mathbot/file/1726019417633wykbxdh4.jpg

Let's analyze the transformations of the graphs and match them to the correct equations:

### General Transformation Rules:
1. **Vertical Stretch/Compression:**
   - If \( y = 2f(x) \), this represents a vertical stretch by a factor of 2.
   - If \( y = \frac{1}{2}f(x) \), this represents a vertical compression.
   
2. **Horizontal Stretch/Compression:**
   - If \( y = f(2x) \), this represents a horizontal compression by a factor of 2.
   - If \( y = f(\frac{1}{2}x) \), this represents a horizontal stretch.
   
3. **Horizontal Shift:**
   - If \( y = f(x + 2) \), this shifts the graph left by 2 units.
   
4. **Vertical Shift:**
   - If \( y = f(x) + 2 \), this shifts the graph up by 2 units.

### Matching the Graphs:

- **Graph 14:** 
  The graph is vertically stretched. So, the corresponding equation should be \( y = 2f(x) \) (Answer A).

- **Graph 15:**
  The graph is compressed horizontally, which indicates the equation should be \( y = f(2x) \) (Answer B).

- **Graph 16:**
  The graph is shifted left by 2 units, which matches the equation \( y = f(x + 2) \) (Answer C).

- **Graph 17:**
  The graph is shifted vertically up by 2 units, corresponding to the equation \( y = f(x) + 2 \) (Answer D).

### Summary:
- **14:** \( y = 2f(x) \) (A)
- **15:** \( y = f(2x) \) (B)
- **16:** \( y = f(x + 2) \) (C)
- **17:** \( y = f(x) + 2 \) (D)

Would you like further details on how transformations affect the graphs?

Here are 5 related questions for practice:
1. What is the effect of the transformation \( y = f(x - 3) \) on the graph of \( f(x) \)?
2. How would the graph of \( y = -f(x) \) differ from \( f(x) \)?
3. If \( y = \frac{1}{3}f(x) \), how does this transformation affect the original graph?
4. How do you graph \( y = f(x - 1) + 4 \)?
5. What happens if you apply both horizontal and vertical transformations simultaneously?

**Tip:** Always analyze whether transformations affect the graph vertically or horizontally based on whether they are inside or outside the function.

Match the graph of the transformation of f with the correct equation shown.

This problem focuses on graph transformations of functions and matching them to the corresponding equations. It covers vertical and horizontal stretching, shifting, and compression. Suitable for students in Grades 9-12.

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