Answer:
The multiplicative rate of change of the function is [tex]\dfrac{2}{3}[/tex]
Step-by-step explanation:
You are given the table
[tex]\begin{array}{cc}x&y\\ \\1&6\\2&4\\ \\3&\dfrac{8}{3}\\ \\4&\dfrac{16}{9}\end{array}[/tex]
An exponential function can be written as
[tex]y=a\cdot b^x,[/tex]
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:
[tex]6=a\cdot b^1\\ \\4=a\cdot b^2[/tex]
Divide the second equality by the first equality:
[tex]\dfrac{4}{6}=\dfrac{a\cdot b^2}{a\cdot b^1}\Rightarrow b=\dfrac{2}{3}[/tex]
Substitute it into the first equality:
[tex]6=a\cdot \dfrac{2}{3}\Rightarrow a=\dfrac{6\cdot 3}{2}=9[/tex]
So, the function expression is
[tex]y=9\cdot \left(\dfrac{2}{3}\right)^x[/tex]
Check whether remaining two values of x and y suit this expression:
[tex]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}[/tex]
So, the multiplicative rate of change of the function is [tex]\dfrac{2}{3}[/tex]
