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Starting with the graph of y = ex, find the equation of the graph that results from the following changes. (a) reflecting about the line y = 6 y = (b) reflecting about the line x = 3 y =

User Dmansfield
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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).

User Danny Harding
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