https://file.aiquickdraw.com/mathbot/file/1725865821089zdqbt29k.jpg

To solve this problem, we need to match each equation with the corresponding graph, using the transformations of the base function \( y = f(x) \). Here are the steps to do that:

1. **Equation (a): \( y = f(x - 4) \)**
   - This represents a horizontal shift to the right by 4 units.
   - Look for the graph where the function \( f(x) \) is shifted to the right.

2. **Equation (b): \( y = f(x) + 3 \)**
   - This represents a vertical shift upwards by 3 units.
   - Look for the graph where the function is moved upward by 3 units.

3. **Equation (c): \( y = 2f(x + 6) \)**
   - This represents a horizontal shift to the left by 6 units and a vertical stretch by a factor of 2.
   - Look for the graph that is stretched and shifted to the left.

4. **Equation (d): \( y = -f(2x) \)**
   - This represents a reflection over the x-axis and a horizontal compression by a factor of 2.
   - Look for the graph that is flipped upside down and horizontally compressed.

### Now, let's analyze the graphs:

1. **Graph (1)**:
   - This graph appears to be horizontally shifted to the right by 4 units.
   - This corresponds to equation **(a): \( y = f(x - 4) \)**.

2. **Graph (2)**:
   - This graph is stretched vertically and shifted to the left by 6 units.
   - This corresponds to equation **(c): \( y = 2f(x + 6) \)**.

3. **Graph (3)**:
   - This graph is shifted upwards by 3 units.
   - This corresponds to equation **(b): \( y = f(x) + 3 \)**.

4. **Graph (4)**:
   - This graph is reflected over the x-axis and horizontally compressed.
   - This corresponds to equation **(d): \( y = -f(2x) \)**.

### Final Matching:

- **(a)** \( y = f(x - 4) \): Graph **1**
- **(b)** \( y = f(x) + 3 \): Graph **3**
- **(c)** \( y = 2f(x + 6) \): Graph **2**
- **(d)** \( y = -f(2x) \): Graph **4**

Would you like more details or have any questions?

Here are 5 related questions you might explore next:
1. How do horizontal shifts affect the graph of a function?
2. What is the effect of vertical stretching on a graph?
3. How does a reflection over the x-axis change the graph of a function?
4. Can you determine the transformation if a function is compressed horizontally?
5. What is the combined effect of multiple transformations on a graph?

**Tip:** Always check the order of transformations, as some transformations (like shifts) are affected by others (like reflections or stretches).

Match the equation with its graph: (a) y = f(x - 4), (b) y = f(x) + 3, (c) y = 2f(x + 6), (d) y = -f(2x)

Learn how to match equations with their graphs by understanding key function transformations, such as horizontal and vertical shifts, vertical stretching, and reflections. This problem covers graph transformations and is suitable for high school students in Grades 9-12.

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