Question

Write an exponential function (1 , 10.4) and (4 , 665.6)

Answer 1

`f(x)=2.6(4)^x`

Explanation:

`\boxed{\begin{minipage}{9 cm}\underline{General form of an Exponential Function}\\\\$f(x)=ab^x$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the initial value ($y$-intercept). \\ \phantom{ww}$\bullet$ $b$ is the base (growth/decay factor) in decimal form.\\\end{minipage}}`

Given points:

• (1, 10.4)
• (4, 665.6)

Substitute the given points into the formula to create two equations:

`\implies 10.4=ab`

`\implies 665.6=ab^4`

Divide the second equation by the first to eliminate a, then solve for b:

`\implies (665.6)/(10.4)=(ab^4)/(ab)`

`\implies 64=(b^4)/(b)`

`\implies 64=b^3`

`\implies b=\sqrt[3]{64}`

`\implies b=4`

Substitute the found value of b into the first equation and solve for a:

`\implies 10.4=4a`

`\implies a=2.6`

Therefore, the exponential function is:

`f(x)=2.6(4)^x`