We are asked to write an exponential function of the form
[tex]y=ab^x[/tex]
The function passes through the following two points
(3, 2) and (6, 16)
Let us substitute these two points into the above equation.
For point (3, 2):
[tex]\begin{gathered} 2=ab^3 \\ a=\frac{2}{b^3} \end{gathered}[/tex]
For point (6, 16):
[tex]\begin{gathered} 16=ab^6 \\ a=\frac{16}{b^6} \end{gathered}[/tex]
Equate the two equations to solve for b
[tex]\begin{gathered} \frac{2}{b^3}=\frac{16}{b^6} \\ 2\cdot b^6=16\cdot b^3 \\ \frac{2b^6}{2b^3}=\frac{16b^3}{2b^3} \\ b^3=8 \\ b=\sqrt[3]{8} \\ b=2 \end{gathered}[/tex]
So, the value of b is 2
Finally, the value of a is given by
[tex]a=\frac{2}{b^3}=\frac{2}{(2)^3}=\frac{2}{8}=\frac{1}{4}[/tex]
So, the value of a is 1/4
Therefore, the exponential function is
[tex]y=\frac{1}{4}\cdot2^x[/tex]
