Question

what is the decay factor of the exponential function represented by the table?

1/3
2/3
2
6

Answer 1

if you look at the table, we know that when x = 0, f(x) = 6, thus

$\bf \qquad \textit{Amount for Exponential Decay}\\\\ f(x)=I(1 - r)^x\qquad \begin{cases} f(x)=\textit{accumulated amount}\\ I=\textit{initial amount}\\ r=rate\to r\%\to \frac{r}{100}\\ x=\textit{elapsed time}\\ ----------\\ x=0\\ f(x)=6 \end{cases} \\\\\\ 6=I(1-r)^0\implies 6=I\cdot 1\implies 6=I\qquad then~~\boxed{f(x)=6(1-r)^x}$

now... let's notice from the table, when x = 1, f(x) = 2, thus

$\bf \qquad \textit{Amount for Exponential Decay}\\\\ f(x)=6(1 - r)^x\qquad \begin{cases} f(x)=\textit{accumulated amount}\\ I=\textit{initial amount}\to &6\\ r=rate\to r\%\to \frac{r}{100}\\ x=\textit{elapsed time}\\ ----------\\ x=1\\ f(x)=2 \end{cases} \\\\\\ 2=6(1-r)^1\implies \cfrac{2}{6}=(1-r)^1\implies \cfrac{1}{3}=1-r \\\\\\ r=1-\cfrac{1}{3}\implies r=\cfrac{2}{3}\quad\quad\quad therefore~~~~~\boxed{f(x)=6\left(1-\frac{2}{3} \right)^x}$

Answer 2

Answer: Hello mate!

this function is written in a next way:

f(x) = A(1 - r)^x

where A is the initial value, r is the decay factor.

The first thing we need to do is look at f(0)

in the table, we can see that f(0) = 6 = A(1-r)^0 = A*1

now we know that A = 6

now we got that the function is f(x) = 6(1 - r)^x

in the table, you can see that wehn x = 1, f(1) = 2, with this we can obtain the value of r.

f(1) = 6(1-r)^1 = 6(1 - r) = 2

You can use any point of the table in this step (where x is not zero, because we already used that point), i used x = 1 because the math was easier this way

6 - 6r = 2

6r = 4

r = 4/6 = 2/3

and now we know that f(x) = 6(1 - 2/3)^x