Question

The table represents an exponential function. A 2-column table has 4 rows. The first column is labeled x with entries 1, 2, 3, 4. The second column is labeled y with entries 6, 4, eight-thirds, sixteen-ninths. What is the multiplicative rate of change of the function?

Answer 1

B. The multiplicative rate of change of the function is 2/3

Explanation:

Answer 2

The multiplicative rate of change of the function is
`(2)/(3)`

Explanation:

You are given the table

`\begin{array}{cc}x&y\\ \\1&6\\2&4\\ \\3&(8)/(3)\\ \\4&(16)/(9)\end{array}`

An exponential function can be written as

`y=a\cdot b^x,`

where b is the multiplicative rate of change of the function.

Find a and b. Substitute first two corresponding values of x and y into the function expression:

`6=a\cdot b^1\\ \\4=a\cdot b^2`

Divide the second equality by the first equality:

`(4)/(6)=(a\cdot b^2)/(a\cdot b^1)\Rightarrow b=(2)/(3)`

Substitute it into the first equality:

`6=a\cdot (2)/(3)\Rightarrow a=(6\cdot 3)/(2)=9`

So, the function expression is

`y=9\cdot \left((2)/(3)\right)^x`

Check whether remaining two values of x and y suit this expression:

`9\cdot \left((2)/(3)\right)^3=9\cdot (8)/(27)=(8)/(3)\\ \\9\cdot \left((2)/(3)\right)^4=9\cdot (16)/(81)=(16)/(9)`

So, the multiplicative rate of change of the function is
`(2)/(3)`