Question

Please help thank you

Answer 1

the tale-tell fellow is the number inside the parentheses.

if that number, the so-called "growth or decay factor", is less than 1, then is Decay, if it's more than 1, is Growth.

`\bf f(x)=0.001(1.77)^x\qquad \leftarrow \qquad \textit{1.77 is greater than 1, Growth} \\\\[-0.35em] ~\dotfill\\\\ f(x)=2(1.5)^{(x)/(2)}\qquad \leftarrow \qquad \textit{1.5 is greater than 1, Growth} \\\\[-0.35em] ~\dotfill\\\\ f(x)=5(0.5)^(-x)\implies f(x)=5\left( \cfrac{05}{10} \right)^(-x)\implies f(x)=5\left( \cfrac{1}{2} \right)^(-x) \\\\\\ f(x)=5\left( \cfrac{2}{1} \right)^(x)\implies f(x)=5(2)^x\qquad \leftarrow \qquad \textit{Growth} \\\\[-0.35em] ~\dotfill`

`\bf f(t)=5e^(-t)\implies f(t)=5\left( \cfrac{e}{1} \right)^(-t)\implies f(t)=5\left( \cfrac{1}{e} \right)^t \\\\\\ \cfrac{1}{e}\qquad \leftarrow \qquad \textit{that's a fraction less than 1, Decay}`

now, let's take a peek at the second set.

`\bf f(x)=3(1.7)^(x-2)\qquad \leftarrow \qquad \begin{array}{llll} \textit{the x-2 is simply a horizontal shift}\\\\ \textit{1.7 is more than 1, Growth} \end{array} \\\\[-0.35em] ~\dotfill\\\\ f(x)=3(1.7)^(-2x)\implies f(x)=3\left(\cfrac{17}{10}\right)^(-2x)\implies f(x)=3\left(\cfrac{10}{17}\right)^(2x) \\\\\\ \textit{that fraction is less than 1, Decay} \\\\[-0.35em] ~\dotfill`

`\bf f(x)=3^5\left( \cfrac{1}{3} \right)^x\qquad \leftarrow \qquad \textit{that fraction is less than 1, Decay} \\\\[-0.35em] ~\dotfill\\\\ f(x)=3^5(2)^(-x)\implies f(x)=3^5\left( \cfrac{2}{1} \right)^(-x)\implies f(x)=3^5\left( \cfrac{1}{2} \right)^x \\\\\\ \textit{that fraction in the parentheses is less than 1, Decay}`