# How do you find the exponential model y=ae^(bx) that goes through the points (-3,2)(1,164)?

Feb 16, 2017

Exponential model is $y = 54.5 {e}^{1.10168 x}$

#### Explanation:

As $y = a {e}^{b x}$ goes through $\left(- 3 , 2\right)$,

we have $2 = a {e}^{- 3 b}$ ......................(1)

and as $y = a {e}^{b x}$ also goes through $\left(1 , 164\right)$,

we have $164 = a {e}^{b}$ ......................(2)

Dividing (2) by (1), we get

${e}^{b} / {e}^{- 3 b} = \frac{164}{2} = 82$ or ${e}^{4 b} = 82$ and $4 b = \ln 82 = 4.40672$

and $b = \frac{4.40672}{4} = 1.10168$

Putting this value of $b$ in (2), we get

$a {e}^{1.10188} = 164$ or $3.0092 a = 164$

and $a = \frac{164}{3.0092} = 54.5$

and hence exponential model is $y = 54.5 {e}^{1.10168 x}$