# How do you write an exponential function whose graph passes through (0,-0.3) and (5,-9.6)?

Nov 3, 2016

#### Explanation:

An exponential function is:

$y = C {e}^{\alpha \left(x\right)}$

We can find the value of C, using the point $\left(0 , - 0.3\right)$

$- 0.3 = C {e}^{\alpha \left(0\right)}$

${e}^{\alpha \left(0\right)} = 1$ so we merely flip the equation and drop the exponential:

$C = - 0.3$

We can use the point $\left(5 , - 9.6\right)$ to find the value of $\alpha$:

$- 9.6 = \left(- 0.3\right) {e}^{\alpha \left(5\right)}$

Divide both side by $- 0.3$

$\frac{- 9.6}{- 0.3} = {e}^{\alpha \left(5\right)}$

$32 = {e}^{\alpha \left(5\right)}$

Take the natural logarithm of both sides, to make the exponential disappear:

$\ln \left(32\right) = \alpha \left(5\right)$

Divide both sides by 5:

$\alpha = \ln \frac{32}{5}$

The above is the same as $\ln \left(\sqrt[5]{32}\right) = \ln \left(2\right)$

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

The exponential function is:

$y = \left(- 0.3\right) {e}^{\ln \left(2\right) x}$