# How do you write a power model for the function that passes through the points ( 2, 3) and (6, 12)?

Dec 17, 2017

Assuming this is a straight line

$y = \frac{4}{9} x - \frac{3}{2}$

#### Explanation:

Using the standard form $y = m x + c$ where $m$ is the gradient

$m = \left(\text{change in y")/("change in x") color(white)("d}\right)$ reading left to right on the x-axis

Set the left most point as ${P}_{1} \to \left({x}_{1} , {y}_{1}\right) = \left(2 , 3\right)$
Set the right most point as ${P}_{2} \to \left({x}_{2} , {y}_{2}\right) = \left(6 , 12\right)$

$m = \left(\text{change in y")/("change in x}\right) \to \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}} = \frac{12 - 3}{6 - 2} = \frac{9}{4}$

As $m > 0$ then the gradient (slope) is upwards reading left to right
It goes up 9 for every 4 along.

To determine the value of $c$ substitute a known point. I choose ${P}_{1}$

$3 = \frac{9}{4} \times 2 + c$

$c = 3 - \frac{9}{2} \textcolor{w h i t e}{\text{ddd") ->color(white)("ddd") 6/2-9/2color(white)("d")= color(white)("d}} - \frac{3}{2}$

$y = \frac{4}{9} x - \frac{3}{2}$