# An exponential function passes through the points (-3, 8) and (0, 1). What is the equation of the function?

Mar 2, 2017

The function has equation $y = {\left(\frac{1}{2}\right)}^{x}$.

#### Explanation:

It may look like you only have $1$ point, but in fact you have $2$.

Every exponential function has its y-intercept at $\left(0 , 1\right)$, because ${n}^{0} = 1$ for every real value of $n$.

We conclude the equation can be written in the form

$y = a {b}^{x}$

We can write a system of equations in two variables to solve for parameters $a$ and $b$.

$\left\{\begin{matrix}8 = a {b}^{-} 3 \\ 1 = a {b}^{0}\end{matrix}\right.$

We can see by inspection, using the property ${n}^{0} = 1$, that $a = 1$.

We can now readily solve for $b$.

$8 = 1 \left({b}^{-} 3\right)$

$8 = {b}^{-} 3$

${2}^{3} = {b}^{-} 3$

${2}^{3} = \frac{1}{b} ^ 3$

${b}^{3} = \frac{1}{2} ^ 3$

${\left({b}^{3}\right)}^{\frac{1}{3}} = {\left(\frac{1}{2} ^ 3\right)}^{\frac{1}{3}}$

$b = \frac{1}{2}$

The function therefore has equation $y = {\left(\frac{1}{2}\right)}^{x}$

Practice Exercises

1. Find the equation of the exponential function given:

a) It passes through the points $\left(2 , 12\right)$ and $\left(- 1 , \frac{3}{2}\right)$.

b) The following graph:

Solutions
1a) $y = 3 {\left(2\right)}^{x}$
1b) $y = 9 {\left(\frac{1}{3}\right)}^{x}$

Hopefully this helps!