SOLUTION: An exponential function with a base of 4 has been compressed horizontally by a factor of 1/3 and reflected in the x-axis. Its asymptote is the line y = 3. Its y-intercept is (0, 2)

Algebra ->  Rational-functions -> SOLUTION: An exponential function with a base of 4 has been compressed horizontally by a factor of 1/3 and reflected in the x-axis. Its asymptote is the line y = 3. Its y-intercept is (0, 2)      Log On


   

Question 1138871: An exponential function with a base of 4 has been compressed horizontally by a factor of 1/3 and reflected in the x-axis. Its asymptote is the line y = 3. Its y-intercept is (0, 2). Write an equation of the function.
Answer by greenestamps(13209) About Me  (Show Source):
You can put this solution on YOUR website!


The basic exponential function with base 4 is y=4%5Ex (red):

graph%28400%2C400%2C-.5%2C.5%2C-5%2C5%2C4%5Ex%29

Compressed horizontally by a factor of 1/3 means the exponent has been multiplied by 3 to give y+=+4%5E%283x%29 (green):

graph%28400%2C400%2C-.5%2C.5%2C-5%2C5%2C4%5Ex%2C4%5E%283x%29%29

Reflected in the x-axis changes each y value to its opposite; that means the function has been multiplied by -1, to give y+=+-4%5E%283x%29 (blue):

graph%28400%2C400%2C-.5%2C.5%2C-5%2C5%2C4%5Ex%2C4%5E%283x%29%2C-4%5E%283x%29%29

With the asymptote at y=3, the whole graph has been moved up 3 units, moving the asymptote from y=0 to y=3, to give y+=+-4%5E%283x%29%2B3, or y+=+3-4%5E%283x%29 (purple):



The y-intercept is supposed to be (0,2), which it is, so our equation is good.

If the y-intercept had not been as desired, the equation would have involved a coefficient on the 4^(3x) term; and that would have moved the y-intercept, making the determination of the function more difficult.