Question

An exponential function f(x) passes through the points (2, 360) and (3, 216). Write an equation for f(x).

Answer 1

`{\Large \begin{array}{llll} y=ab^x \end{array}} \\\\[-0.35em] ~\dotfill\\\\ \begin{cases} x=2\\ y=360 \end{cases}\implies 360=ab^2\implies 360=abb\implies \cfrac{360}{b}=ab \\\\[-0.35em] ~\dotfill\\\\ \begin{cases} x=3\\ y=216 \end{cases}\implies 216=ab^3\implies 216=abb^2\implies \stackrel{\textit{substituting from above}}{216=\left( \cfrac{360}{b} \right)b^2} \\\\\\ 216=360b\implies \cfrac{216}{360}=b\implies \boxed{\cfrac{3}{5}=b} \\\\[-0.35em] ~\dotfill`

`\stackrel{\textit{since we know that}}{360=ab^2}\implies 360=a\left( \cfrac{3}{5} \right)^2\implies 360=\cfrac{9a}{25} \\\\\\ \cfrac{25}{9}\cdot 360=a\implies \boxed{1000=a}~\hfill {\Large \begin{array}{llll} y=1000\left( (3)/(5) \right)^x \end{array}}`