# How do you find an exponential function given the points are (-1,8) and (1,2)?

Jan 9, 2016

$y = 4 {\left(\frac{1}{2}\right)}^{x}$

#### Explanation:

An exponential function is in the general form

$y = a {\left(b\right)}^{x}$

We know the points $\left(- 1 , 8\right)$ and $\left(1 , 2\right)$, so the following are true:

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

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

Multiply both sides of the first equation by $b$ to find that

$8 b = a$

Plug this into the second equation and solve for $b$:

$2 = \left(8 b\right) b$

$2 = 8 {b}^{2}$

${b}^{2} = \frac{1}{4}$

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

Two equations seem to be possible here. Plug both values of $b$ into the either equation to find $a$. I'll use the second equation for simpler algebra.

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

$2 = a \left(\frac{1}{2}\right)$

$a = 4$

Giving us the equation: color(green)(y=4(1/2)^x

If $b = - \frac{1}{2}$:

$2 = a \left(- \frac{1}{2}\right)$

$a = - 4$

Giving us the equation: $y = - 4 {\left(- \frac{1}{2}\right)}^{x}$

However! In an exponential function, $b > 0$, otherwise many issues arise when trying to graph the function.

The only valid function is

color(green)(y=4(1/2)^x