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: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]