Question

How do you determine whether each function represents exponential growth or decay $y=3{{\left( \dfrac{5}{2} \right)}^{x}}$ ?

Answer 1

Hint:
In this problem we have found the exponential growth or decay of the given expression that is $y=3{{\left( \dfrac{5}{2} \right)}^{x}}$. First, we want to check whether the given expression is exponential form or not. The general form of the exponential function is $y=a{{b}^{x}}$. Now we will compare the given equation with the standard equation and we will write the values of $a$, $b$. Now from the values of the $a$ and $b$ we will decide whether the given equation is exponential decay or growth. If the value of $a$ is positive and the value $b$ is greater than $1$ then we can call it as exponential growth and if the value of $a$ is positive and $b$ is less than $1$ then we can call it as exponential decay.

FORMULA USE:
exponential function form of $y=a{{b}^{x}}$.

Complete step by step solution:
Given that, $y=3{{\left( \dfrac{5}{2} \right)}^{x}}$.
To whether the given equation is in exponential form or not. We are going to check whether the given equation is the form of $y=a{{b}^{x}}$. By observing the given equation, we can say that the given equation is an exponential function and the values of $a$, $b$ are
$a=3$ , $b=\dfrac{5}{2}$
In the given equation we have the value of $a$ as the positive. Considering the value of $b$ . We have $b=\dfrac{5}{2}$ which is greater than $1$. So, we can say that the given exponential equation has growth.

Note:
We can also plot the graph and see the growth of the given equation. The graph of the given equation is