# How do you find the equation of the exponential function y=a(b)^x which goes through the points(2, 18) and (6,91.125). What is the intercepts of this function?

Feb 20, 2017

$y = 8 {\left(\frac{5}{4}\right)}^{x}$
intercept: (0;8)

#### Explanation:

It's
$a \cdot {b}^{2} = 18$ and $a \cdot {b}^{6} = 91.125$ and $b > 0$

then

$a = \frac{18}{b} ^ 2$ and $\frac{18}{b} ^ 2 \cdot {b}^{6} = 91.125$

$a = \frac{18}{b} ^ 2$ and $18 \cdot {b}^{4} = 91.125$

$a = \frac{18}{b} ^ 2$ and ${b}^{4} = \frac{91.125}{18} = 5.0625$

$a = \frac{18}{b} ^ 2$ and ${b}^{2} = 2.25$

$a = \frac{18}{2.25}$ and $b = 1.25$

$a = 8$ and $b = \frac{5}{4}$

Then $y = 8 {\left(\frac{5}{4}\right)}^{x}$

If $x = 0$, then $y = 8 \cdot {\left(\frac{5}{4}\right)}^{0} = 8 \cdot 1 = 8$

It's the one intercept of the function since $y = 0 \forall x$
graph{8*(5/4)^x [-10, 5, -1, 15]}