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The question provided in the image asks about determining the x-intercepts of different transformations of a function \( f(x) \), where the original function \( y = f(x) \) has x-intercepts at \( x = -2 \) and \( x = 4 \). Let's go through each part:

### (a) What are the x-intercepts of the graph of \( y = f(x + 6) \)?
- A horizontal shift to the left by 6 units occurs when \( f(x) \) is replaced by \( f(x + 6) \).
- The original x-intercepts are \( -2 \) and \( 4 \). Shifting them left by 6 gives:
  \[
  -2 - 6 = -8, \quad 4 - 6 = -2
  \]
  So, the x-intercepts are \( -8 \) and \( -2 \).

### (b) What are the x-intercepts of the graph of \( y = f(x - 7) \)?
- A horizontal shift to the right by 7 units occurs when \( f(x) \) is replaced by \( f(x - 7) \).
- Shifting the original x-intercepts \( -2 \) and \( 4 \) to the right by 7 gives:
  \[
  -2 + 7 = 5, \quad 4 + 7 = 11
  \]
  So, the x-intercepts are \( 5 \) and \( 11 \).

### (c) What are the x-intercepts of the graph of \( y = 3f(x) \)?
- Multiplying the function by 3 (vertical stretch) does not affect the x-intercepts, because x-intercepts occur where \( y = 0 \), and multiplying by 3 still keeps \( y = 0 \).
- Therefore, the x-intercepts remain \( -2 \) and \( 4 \).

### (d) What are the x-intercepts of the graph of \( y = f(-x) \)?
- Reflecting across the y-axis occurs when \( f(x) \) is replaced by \( f(-x) \).
- The x-intercepts change signs because the function is reflected across the y-axis. The original intercepts \( -2 \) and \( 4 \) become:
  \[
  -(-2) = 2, \quad -(4) = -4
  \]
  So, the x-intercepts are \( 2 \) and \( -4 \).

Let me know if you need further clarification or have more questions. Here are five related questions for further understanding:

1. What happens to the x-intercepts when the graph is shifted vertically?
2. How does a reflection across the x-axis affect the x-intercepts?
3. What effect does multiplying a function by a constant have on its x-intercepts?
4. How do you determine the y-intercepts after a horizontal shift?
5. Can an absolute value transformation affect the x-intercepts?

**Tip**: Horizontal shifts are applied by adjusting the \( x \)-values, while vertical transformations do not change the x-intercepts unless the function crosses the x-axis differently.

Suppose that the x-intercepts of the graph of y = f(x) are -2 and 4. What are the x-intercepts for various transformations of the function?

Learn how to determine the x-intercepts of transformed functions, including horizontal shifts, vertical stretches, and reflections. Starting with the x-intercepts of y = f(x) at -2 and 4, find the intercepts for transformations like y = f(x + 6), y = f(x - 7), y = 3f(x), and y = f(-x).

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