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.