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

Answer:

2/3

Step-by-step explanation:

Answer 2

Answer:

The multiplicative rate of change of the function is $\dfrac{2}{3}$

Step-by-step explanation:

You are given the table

$\begin{array}{cc}x&y\\ \\1&6\\2&4\\ \\3&\dfrac{8}{3}\\ \\4&\dfrac{16}{9}\end{array}$

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:

$\dfrac{4}{6}=\dfrac{a\cdot b^2}{a\cdot b^1}\Rightarrow b=\dfrac{2}{3}$

Substitute it into the first equality:

$6=a\cdot \dfrac{2}{3}\Rightarrow a=\dfrac{6\cdot 3}{2}=9$

So, the function expression is

$y=9\cdot \left(\dfrac{2}{3}\right)^x$

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

$9\cdot \left(\dfrac{2}{3}\right)^3=9\cdot \dfrac{8}{27}=\dfrac{8}{3}\\ \\9\cdot \left(\dfrac{2}{3}\right)^4=9\cdot \dfrac{16}{81}=\dfrac{16}{9}$

So, the multiplicative rate of change of the function is $\dfrac{2}{3}$