# If the first point is (-2, 45/4) and the second point is (1, 10/3), how do I find the formula for exponential growth?

May 30, 2017

Any exponential growth/decay will fit the general form:

$y = A {e}^{\lambda x} \text{ [1]}$

Use the two points to write two equations and then solve for $\lambda$ and A.

#### Explanation:

Substitute the point $\left(- 2 , \frac{45}{4}\right)$ into equation [1]:

$\frac{45}{4} = A {e}^{- 2 \lambda} \text{ [2]}$

Substitute the point $\left(1 , \frac{10}{3}\right)$ into equation [1]:

$\frac{10}{3} = A {e}^{\lambda} \text{ [3]}$

We can make A disappear by dividing equation [2] by equation [3]:

$\frac{\frac{45}{4}}{\frac{10}{3}} = \frac{A {e}^{- 2 \lambda}}{A {e}^{\lambda}}$

$\left(\frac{45}{4}\right) \left(\frac{3}{10}\right) = {e}^{- 3 \lambda}$

$\frac{135}{40} = \frac{27}{8} = {e}^{- 3 \lambda}$

$\ln \left(\frac{27}{8}\right) = - 3 \lambda$

$- \frac{1}{3} \ln \left(\frac{27}{8}\right) = \lambda$

$\lambda = \ln \left(\frac{2}{3}\right)$

Substitute the value of $\lambda$ into equation [1]:

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

The exponential function and the natural logarithm are inverses, therefore, they cancel:

$y = A {\left(\frac{2}{3}\right)}^{x} \text{ [4]}$

Use the point, $\left(1 , \frac{10}{3}\right)$, to find the value of A:

$\frac{10}{3} = A \left(\frac{2}{3}\right)$

$A = \frac{10}{3} \left(\frac{3}{2}\right)$

$A = 5$

Substitute into equation [4]:

$y = 5 {\left(\frac{2}{3}\right)}^{x} \text{ [5]}$