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.

(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.

(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:

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

(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:

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

(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:

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

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?

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 $f(x)$.