Question

How do you determine if the equation $y = 0.25{\left( {1.03} \right)^5}$ represents exponential growth or decay?

Answer 1

Hint: Given an exponential expression in which we have to identify whether the expression represents growth or decay. First, we will compare the value of the coefficient of the expression whether it is greater than zero. Then, compare the value of the base of exponent whether it is between zero and one or greater than one.

Complete step-by-step solution:
We are given the exponential expression in the form $y = a{\left( b \right)^x}$. Then, compare the value of $a$ with zero by substituting $a = 0.25$
$ \Rightarrow 0.25 > 0$
Now, we will compare the value of $b$ with one by substituting $b = 1.03$
$1.03 > 1$
Here, the value of $a$ is greater than $0$ and $b$ is greater than $1$, which means the function represents the exponential growth.

Hence, the equation $y = 0.25{\left( {1.03} \right)^5}$ represents exponential growth.

Note: In the exponential function, $y = a{\left( b \right)^x}$ if the value of $a$ is greater than $0$, and the value of $b$ is greater than $1$, then the function is known as exponential growth. On the other hand, if the value of $a$ is greater than $0$, but the value of $b$ is less than $0$, then the function is known as exponential decay. The value of $b$ is known as the growth factor or decay factor of the expression.