Question

given the following exponential function, identify whether the change represents growth or decay, and determine the percentage rate of increase or decrease.

$y = 47(1.09)^x$

Explanation:

Step1: Identify growth or decay

For an exponential function of the form $y = a(b)^x$, if $b>1$, it represents growth. Here $b = 1.09>1$, so it is growth.

Step2: Calculate percentage rate

The general form for growth is $y=a(1 + r)^x$, where $r$ is the rate of growth. Comparing $y = 47(1.09)^x$ with $y=a(1 + r)^x$, we have $1 + r=1.09$. Solving for $r$, we get $r=1.09 - 1=0.09$. To convert $r$ to a percentage, we multiply by 100. So the percentage rate is $0.09\times100 = 9\%$.

Answer:

Growth, 9%