Let's start with the original function y = e^x.
(a) Reflecting about the line y = 6:
To reflect the graph of y = e^x about the line y = 6, you can think of it as a vertical shift. First, shift the graph 6 units upward to get it centered at y = 6, and then reflect it.
Shift the graph upward by adding 6 to the original function:
y = e^x + 6
Now, we'll reflect this shifted graph about the line y = 6. To reflect about a horizontal line, you negate the y-coordinate. So, to reflect about y = 6, subtract 6 from y:
y = -(e^x) + 6
So, the equation of the graph that results from reflecting y = e^x about the line y = 6 is y = -(e^x) + 6.
(b) Reflecting about the line x = 3:
To reflect the graph of y = e^x about the line x = 3, you can think of it as a horizontal shift. First, shift the graph 3 units to the right to get it centered at x = 3, and then reflect it.
Shift the graph to the right by subtracting 3 from the x-coordinate:
y = e^(x - 3)
Now, we'll reflect this shifted graph about the line x = 3. To reflect about a vertical line, you negate the x-coordinate. So, to reflect about x = 3, negate (x - 3):
y = e^(-(x - 3))
Simplify:
y = e^(3 - x)
So, the equation of the graph that results from reflecting y = e^x about the line x = 3 is y = e^(3 - x).