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 = 140 cdot 2.9^{x} ) the function ( y = 140 cdot 2.9^{x} ) represents exponential dropdown. rewriting the base in terms of the rate of growth or decay results in the function ( y = 140 cdot (square)^{x} ). in this function, ( r = square ), which indicates that the value of ( y ) dropdown by dropdown% each time period.

Explanation:

Step1: Identify growth/decay

Exponential functions have the form $y = a \cdot b^x$. If $b>1$, it is growth. Here, $b=2.9>1$, so it is growth.

Step2: Rewrite base for growth rate

The growth form is $y = a(1+r)^x$. Set $1+r = 2.9$.

Step3: Solve for r

$r = 2.9 - 1 = 1.9$

Step4: Convert r to percentage

$1.9 \times 100 = 190\%$, so $y$ increases by 190% per period.

Answer:

The function $y = 140 \cdot 2.9^x$ represents exponential growth. Rewriting the base in terms of the rate of growth or decay results in the function $y = 140 \cdot (1 + 1.9)^x$. In this function, $r = 1.9$ which indicates that the value of $y$ increases by $190$% each time period.