SOLUTION: The exponential function y=4^x has been compressed vertically by a factor of 1/3, stretched horizontally by a factor of 5, and reflected in the x-axis. Its asymptote is the line y=

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: The exponential function y=4^x has been compressed vertically by a factor of 1/3, stretched horizontally by a factor of 5, and reflected in the x-axis. Its asymptote is the line y=      Log On


   

Question 924774: The exponential function y=4^x has been compressed vertically by a factor of 1/3, stretched horizontally by a factor of 5, and reflected in the x-axis. Its asymptote is the line y=3. Its y-intercept is 2 2/3. Write the equation of the transformed function.
Answer by Edwin McCravy(20086) About Me  (Show Source):
You can put this solution on YOUR website!
y%22%22=%22%224%5Ex
That has this red graph:

Now we'll compress the red graph vertically by 1/3 multiplying the right
side by 1/3. That's the green graph. Notice that it is lower by 1/3 than
the red graph. It may not look so at the top, but remember that the red
graph is cut off at the top and actual goes three times as high as the
green graph:
y%22%22=%22%22expr%281%2F3%294%5Ex

Now we'll stretch the green graph horizontally by a factor of 5 by replacing
x in the right sides by x/5. That's the blue graph below. Notice that it
sticks 5 times as far to the right as the green graph. Its equation is:


Next we refect the blue graph in the x-axis by multiplying the right side
by -1. That's the purple graph below. It equation is:


Next we make the horizontal asymptote of the purple graph become y=3 instead
of the x-axis, y=0. The graph is getting too cluttered, so I'll erase all the
graphs except the purple one. The purple graph shifted upward so that the
horizontal asymptote is y=3 is obtained by adding 3 to the right side of the
equation. That graph is the greenish-grey one. The red dotted line is the
new horizontal asymptote y=3. The equation of the greenish-grey graph is:


Finally it is to have the y-intercept of 2%262%2F3.
Let's see what the y-intercept of the greenish-grey graph is.
We do that by substituting 0 for x:



matrix%282%2C3%2C%22%22%2C%22%22%2C%22%22%2Cy%2C%22%22=%22%22%2C-1%2F3%2B3+%29
matrix%282%2C3%2C%22%22%2C%22%22%2C%22%22%2Cy%2C%22%22=%22%22%2C2%262%2F3+%29
Wow! It already has that y-intercept! So we don't need to
stretch, compress, reflect or shift it. The greenish-grey graph is
the graph of the equation we were looking for.
The answer is
Edwin