# How do you determine the formula for the graph of an exponential function given (0,2) and (3,1)?

Sep 25, 2016

I found: $y \left(x\right) = 2 {e}^{- 0.231 x}$

#### Explanation:

$y \left(x\right) = A {e}^{k x}$
where $A$ and $k$ are two constants we need to evaluate.
We use the coordinates of the points into the general expression to find the two constants:

First : $x = 0 \mathmr{and} y = 2$ getting:
$y \left(0\right) = A {e}^{0 k}$ that must be equal to $2$
or:
$A {e}^{0 k} = 2$
$A {e}^{0} = 2$
$A \cdot 1 = 2$
so that $A = 2$

Second : $x = 3 \mathmr{and} y = 1$ getting:
$y \left(3\right) = A {e}^{3 k}$ that must be equal to $1$;
we know from the previous step that $A = 2$ so:
$y \left(3\right) = 2 {e}^{3 k} = 1$
and so:
$2 {e}^{3 k} = 1$
${e}^{3 k} = \frac{1}{2}$
take the natural log of both sides:
$\ln \left({e}^{3 k}\right) = \ln \left(\frac{1}{2}\right)$
rearranging:
$3 k = \ln \left(\frac{1}{2}\right)$
and:
$k = - 0.231$

Finally our function will be:

$y \left(x\right) = 2 {e}^{- 0.231 x}$

Graphically:
graph{2e^(-0.231x) [-10, 10, -5, 5]}