Question

determine whether the function represents exponential growth or decay. write the base in terms of the rate of growth or decay, identify ( r ), and interpret the rate of growth or decay.
( y = 500 cdot 4^x )

the function ( y = 500 cdot 4^x ) represents exponential __ rewriting the base in terms of the rate of growth or decay results in the function ( y = 500 cdot square^x ). in this function, ( r = square ) which indicates that the value of ( y ) by __% each time period.

Explanation:

Step1: Identify growth/decay

Exponential form: $y = a \cdot b^x$. If $b>1$, it's growth. Here $b=4>1$, so growth.

Step2: Rewrite base as $1+r$

We need $4 = 1+r$, so $r = 4-1 = 3$. The function becomes $y=500 \cdot (1+3)^x$.

Step3: Interpret the rate

$r=3$ means a $300\%$ increase per period.

Answer:

The function $y = 500 \cdot 4^x$ represents exponential growth. Rewriting the base in terms of the rate of growth or decay results in the function $y = 500 \cdot \boldsymbol{(1+3)}^x$. In this function, $r = \boldsymbol{3}$ which indicates that the value of $y$ increases by $\boldsymbol{300}\%$ each time period.