# How do you write an exponential equation that passes through (0, -2) and (2, -50)?

##### 1 Answer
Feb 13, 2016

You set $f \left(x\right) = a \cdot {e}^{b x}$ and find the $a$ and $b$ constants via the 2 passing points.

$f \left(x\right) = - 2 \cdot {e}^{\ln \frac{25}{2} \cdot x}$
or
$f \left(x\right) = - 2 \cdot {e}^{1.60944 \cdot x}$

#### Explanation:

Let the exponential function be:

$f \left(x\right) = a \cdot {e}^{b x}$

where $a$ and $b$ are constants to be found. From the two points that the function is passing we know that:

Point (0,-2)

$f \left(0\right) = - 2$

$a \cdot {e}^{b \cdot 0} = - 2$

$a \cdot 1 = - 2$

$a = - 2$

Point (2,-50)

$f \left(2\right) = - 50$

$- 2 \cdot {e}^{b \cdot 2} = - 50$

${e}^{2 b} = \frac{- 50}{-} 2$

${e}^{2 b} = 25$

$\ln {e}^{2 b} = \ln 25$

$2 b = \ln 25$

$b = \ln \frac{25}{2} = 1.60944$

Function

So now that the constants are known:

$f \left(x\right) = - 2 \cdot {e}^{\ln \frac{25}{2} \cdot x}$
or
$f \left(x\right) = - 2 \cdot {e}^{1.60944 \cdot x}$