# How do you determine the formula for the graph of an exponential function given (0,3) and (2,6)?

Apr 5, 2018

$y = 3 {e}^{\ln \left(\sqrt{2}\right) x}$

#### Explanation:

One form of an exponential function is:

$y = {y}_{0} {e}^{k x} \text{ [1]}$

Substitute the point $\left(0 , 3\right)$ into equation [1]:

$3 = {y}_{0} {e}^{k 0}$

$3 = {y}_{0} \left(1\right)$

${y}_{0} = 3 \text{ [2]}$

Substitute equation [2] into equation [1]:

$y = 3 {e}^{k x} \text{ [1.1]}$

Substitute the point $\left(2 , 6\right)$ into equation [1.1]:

$6 = 3 {e}^{k \left(2\right)}$

$2 = {e}^{2 k}$

$\ln \left(2\right) = 2 k$

$k = \frac{1}{2} \ln \left(2\right)$

$k = \ln \left(\sqrt{2}\right) \text{ [3]}$

Substitute equation [3] into equation [1.1]:

$y = 3 {e}^{\ln \left(\sqrt{2}\right) x}$