Determine whether each function is an example of exponential growth or decay. Then find the y-intercept.
a) y = 2(4)^x
growth or decay
y-intercept: b) y = 3(.25)^x
growth or decay
y-intercept:
a^x is growth if a > 1
a^x is decay if a < 1
a^x passes through (0,1) for any value of a.
To determine whether a function represents exponential growth or decay, we need to look at the base of the exponential term.
a) The function is y = 2(4)^x. The base of the exponent is 4, which is greater than 1. When the base is greater than 1, it represents exponential growth. Therefore, the function represents exponential growth.
b) The function is y = 3(.25)^x. The base of the exponent is 0.25, which is between 0 and 1. When the base is between 0 and 1, it represents exponential decay. Therefore, the function represents exponential decay.
To find the y-intercept, we can substitute x = 0 into the respective functions and solve for y.
a) For y = 2(4)^x, substitute x = 0:
y = 2(4)^0
= 2(1)
= 2
Therefore, the y-intercept of the function is 2.
b) For y = 3(.25)^x, substitute x = 0:
y = 3(.25)^0
= 3(1)
= 3
Therefore, the y-intercept of the function is 3.