# How do you write an exponential function of the form y = a b^x the graph of which passes through (-1, 4/5) and (2, 100)?

Jun 12, 2016

$y = 4 {\left(5\right)}^{x}$

#### Explanation:

Plug in $\left(x , y\right) = \left(- 1 , \frac{4}{5}\right)$ and $\left(x , y\right) = \left(2 , 100\right)$ to see the two equations that we're left with in terms of $a$ and $b$.

For $\left(- 1 , \frac{4}{5}\right)$, we get

$\frac{4}{5} = a {\left(b\right)}^{-} 1 \text{ "" } \boldsymbol{\left(1\right)}$

and for (2,100) we get

$100 = a {\left(b\right)}^{2} \text{ "" } \boldsymbol{\left(2\right)}$

Note that $\boldsymbol{\left(1\right)}$ can be rewritten as

$\frac{4}{5} = \frac{a}{b} \text{ "" } \boldsymbol{\left(3\right)}$

We can solve for $a$ or $b$. Here, I'll solve for $a$.

$\frac{4}{5} b = a$

We can use $a = \frac{4}{5} b$ to replace $a$ with $\frac{4}{5} b$ in $\boldsymbol{\left(2\right)}$.

$100 = a {\left(b\right)}^{2} \text{ "=>" } 100 = \frac{4}{5} b {\left(b\right)}^{2}$

We can now solve this for $b$.

$100 = \frac{4}{5} {b}^{3}$

Multiplying both sides by $\frac{5}{4}$:

$125 = {b}^{3}$

$b = 5$

Returning to $\boldsymbol{\left(3\right)}$ with our newfound value of $b$, we see that

$\frac{4}{5} = \frac{a}{b} \text{ "=>" "4/5=a/5" "=>" } a = 4$