Question

Given the Initial Value and Rate of Change, write the exponential equation for each of the following. Write y in terms of x.

Answer 1

The form of the exponential function is

`y=a(1\pm r)^x`

a is the initial value

r is the rate in decimal

(+) for growth

(-) for decay

1.

The initial value is 153, then

`a=153`

The growth rate is 20%, then change it to decimal by dividing it by 100

`r=(20)/(100)=0.2`

Substitute them in the form above

`\begin{gathered} y=153(1+0.2)^x \\ y=153(1.2)^x \end{gathered}`

2.

The initial amount is 127

`a=127`

The rate of growth is 7%

`r=(7)/(100)=0.07`

The equation is

`\begin{gathered} y=127(1+0.07)^x \\ y=127(1.07)^x \end{gathered}`

3.

The initial value is 146

`a=146`

The growth rate is 5.5%

`r=(5.5)/(100)=(55)/(1000)=0.055`

The equation is

`\begin{gathered} y=146(1+0.055)^x \\ y=146(1.055)^x \end{gathered}`

4.

The initial value is 116

`a=116`

The growth rate is 117%

`r=(117)/(100)=1.17`

The equation is

`\begin{gathered} y=116(1+1.17)^x \\ y=116(2.17)^x \end{gathered}`

5.

The initial amount is 94

`a=94`

The decay rate is 13%

`r=(13)/(100)=0.13`

Since it is decay, then we will use (1 - r)

The equation is

`\begin{gathered} y=94(1-0.13)^x \\ y=94(0.87)^x \end{gathered}`

6.

The initial value is 142

`a=142`

The decay rate is 3%

`r=(3)/(100)=0.03`

The equation is

`\begin{gathered} y=142(1-0.03)^x \\ y=142(0.97)^x \end{gathered}`

7.

The initial value is 171

`a=171`

The decay rate is 0.3%

`r=(0.3)/(100)=(3)/(1000)=0.003`

The equation is

`\begin{gathered} y=171(1-0.003)^x \\ y=171(0.997)^x \end{gathered}`