To determine if the given function [tex]\( y = 0.2^{6x} \)[/tex] is a growth exponential or a decay exponential, we need to analyze the base of the exponential function.
1. Identify the base of the exponent:
The given function is [tex]\( y = 0.2^{6x} \)[/tex]. Here, the base of the exponent is 0.2.
2. Determine the range of the base:
- If the base [tex]\( a \)[/tex] is between 0 and 1 (i.e., [tex]\( 0 < a < 1 \)[/tex]), the function represents exponential decay.
- If the base [tex]\( a \)[/tex] is greater than 1 (i.e., [tex]\( a > 1 \)[/tex]), the function represents exponential growth.
3. Evaluate the base for [tex]\( y = 0.2^{6x} \)[/tex]:
- In this case, the base [tex]\( a \)[/tex] is 0.2.
- We check if [tex]\( 0 < 0.2 < 1 \)[/tex], which is true.
Given that the base 0.2 is between 0 and 1, we can conclude that the function [tex]\( y = 0.2^{6x} \)[/tex] is an exponential decay function, not growth.
Thus, the function is a decay exponential.
