If the function y=e^−2x is vertically compressed by a factor of 3, reflected across the y-axis, and then shifted down 2 units, what is the resulting function? Write your answer in the form y=ce^ax+b
AAAaannndd the bot gets it wrong yet again!
original : y=e^(-2x)
vertical compression: y = 1/3 e^(-2x)
reflected across y-axis: y = 1/3 e^(-2(-x)) = 1/3 e^(2x)
shifted down: y = 1/3 e^(2x) - 2
To vertically compress the function by a factor of 3, we replace x with 3x. To reflect it across the y-axis, we replace x with -3x. Finally, to shift it down 2 units, we subtract 2 from the y-value. So the resulting function is:
y = ce^ax + b = c(e^(-2*(-3x))/3) - 2
Simplifying the exponent:
y = c(e^(6x)/3) - 2
Multiplying c/3 by the e^(6x):
y = (c/3)e^(6x) - 2
Therefore, the resulting function is y = (c/3)e^(6x) - 2, in the form y=ce^ax+b.