# How do you write an exponential equation that passes through (0,3) and (2,6)?

Jul 11, 2015

$f \left(x\right) = 3 {\left(\sqrt{2}\right)}^{x}$

#### Explanation:

Suppose the required function is $f \left(x\right) = a \cdot {b}^{x}$ with $b > 0$

We are told $f \left(0\right) = 3$ and $f \left(2\right) = 6$

So $3 = f \left(0\right) = a \cdot {b}^{0} = a \cdot 1 = a$

And $2 = \frac{6}{3} = f \frac{2}{f} \left(0\right) = \frac{a \cdot {b}^{2}}{a \cdot {b}^{0}} = {b}^{2}$

So, since $b > 0$, we must have $b = \sqrt{2}$

Putting these together, $f \left(x\right) = 3 {\left(\sqrt{2}\right)}^{x}$