Question

Find a formula for the exponential function passing through the points (-1,5/4) and (3,320).

Answer 1

`{\Large \begin{array}{llll} y = ab^x \end{array}} \\\\[-0.35em] ~\dotfill\\\\ \begin{cases} x=-1\\ y=(5)/(4) \end{cases}\implies \cfrac{5}{4}=ab^(-1)\implies \cfrac{5}{4}=\cfrac{a}{b} \\\\[-0.35em] ~\dotfill`

`\begin{cases} x=3\\ y = 320 \end{cases}\implies 320=ab^3\implies 320=ab^(4-1)\implies 320=ab^4 b^(-1) \\\\\\ 320=ab^4\cdot \cfrac{1}{b}\implies \stackrel{\textit{substituting from the previous equation}}{320=\cfrac{a}{b}b^4\implies 320=\cfrac{5}{4}b^4}\implies 320\cdot \cfrac{4}{5}=b^4 \\\\\\ 256=b^4\implies \sqrt[4]{256}=b\implies \boxed{4=b} \\\\\\ \cfrac{5}{4}=\cfrac{a}{b}\implies \cfrac{5}{4}=\cfrac{a}{4}\implies \boxed{5=a}~\hfill {\Large \begin{array}{llll} y =5(4)^x \end{array}}`