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Let's go through the problem step by step based on the image you've provided, which deals with the function \( y = e^{-x} \).

### Problem #13: Transformations of \( y = e^{-x} \)
We are asked to apply several transformations to the given function and find the new equations.

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#### (a) Reflecting about the line \( y = 6 \):

When reflecting about the line \( y = 6 \), we perform a vertical reflection around this line. The formula for reflecting \( y = f(x) \) about \( y = k \) is:

\[
y_{\text{new}} = 2k - f(x)
\]

Here, \( k = 6 \), and \( f(x) = e^{-x} \). Thus, the transformation gives:

\[
y_{\text{new}} = 2(6) - e^{-x} = 12 - e^{-x}
\]

You already entered this correctly.

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#### (b) Shifting 8 units to the right:

A horizontal shift of a function \( f(x) \) by \( h \) units to the right is given by the formula:

\[
y_{\text{new}} = f(x - h)
\]

In this case, \( h = 8 \). So the transformed equation becomes:

\[
y_{\text{new}} = e^{-(x - 8)} = e^{-(x - 8)}
\]

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#### (c) Reflecting about the line \( x = 3 \):

To reflect about the line \( x = 3 \), the transformation involves flipping the graph horizontally around this line. The formula for reflecting \( f(x) \) about \( x = h \) is:

\[
y_{\text{new}} = f(2h - x)
\]

Here, \( h = 3 \), so the transformation gives:

\[
y_{\text{new}} = e^{-(2(3) - x)} = e^{-(6 - x)} = e^{x - 6}
\]

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#### (d) Shifting up 7 units:

A vertical shift by \( k \) units upwards for the function \( f(x) \) is given by:

\[
y_{\text{new}} = f(x) + k
\]

In this case, \( k = 7 \), so the new equation becomes:

\[
y_{\text{new}} = e^{-x} + 7
\]

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### Summary of Answers:

- (a) \( 12 - e^{-x} \) (this was already correct)
- (b) \( e^{-(x - 8)} \)
- (c) \( e^{x - 6} \)
- (d) \( e^{-x} + 7 \)

Would you like me to explain any of these transformations in more detail? 

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Here are 5 related questions to consider:
1. What happens when you reflect about the x-axis instead of the y-axis?
2. How would a vertical stretch affect the function \( y = e^{-x} \)?
3. What is the result of shifting the function \( y = e^{-x} \) down by 5 units?
4. How would you reflect a function about the line \( y = 0 \)?
5. What happens to the domain and range of \( y = e^{-x} \) after a horizontal shift?

**Tip:** Always remember that horizontal transformations affect the input \( x \), while vertical transformations affect the entire output \( f(x) \).

Starting with the graph of y = e^(-x), find the equation of the function whose graph results from the following transformations: (a) reflecting about the line y = 6, (b) shifting 8 units to the right, (c) reflecting about the line x = 3, (d) shifting up 7 units.

Learn how to transform the exponential function y = e^(-x) with reflections and shifts. This guide covers reflecting about specific lines, horizontal and vertical shifts, and includes step-by-step explanations with formulas. Suitable for high school students in Grades 10-12.

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