Question

FIND THE FORMULA FOR THE EXPONENTIAL GRAPHThe figure shows the graph of an exponential function. The dots on the graph are points with integer coordinates.The function graphed is P(t)Hint: The function may be written as P(t) = Pa^t

Answer 1

Based on the graph,

At,

f(2) = 1 (1)

f(9) = 6 (2)

`\text{ f(x) = ab}^{\text{x}}`

At f(2) = 1,

`\text{ 1 = ab}^2`
`\text{ a = }\frac{\text{ 1}}{b^2}`

At f(9) = 6 ,

`\text{ 6 = ab}^9`
`\text{ a = }\frac{\text{ 6}}{b^9}`

Equate them to find b, a = a,

`\text{ }\frac{\text{ 1}}{b^2}\text{ = }\frac{\text{ 6}}{b^9}\text{ }\rightarrow\text{ }(b^9)/(b^2)\text{ = }(6)/(1)`
`\text{ b}^7\text{ = 6}`
`\text{ b = }\sqrt[7]{6}`

Let's find a at f(2) = 1 and the seventh root of 6,

`\text{ f(x) = ab}^{\text{x}}`
`\text{ 1 = a(}\sqrt[7]{6})^2`
`\text{ a = }\frac{\text{ 1}}{\sqrt[7]{36}}`

Let's now complete the equation.

`\text{ f(x) = ab}^{\text{x}}`

`\text{ f(x) =}\frac{1}{\sqrt[7]{36}}\text{ x (}\sqrt[7]{6})^x=\frac{\text{(}\sqrt[7]{6})^x}{\sqrt[7]{36}}`
`\text{ f(x) }=\frac{\text{(}\sqrt[7]{6})^x}{\sqrt[7]{36}}`