# Question 35861

Nov 2, 2016

$y = \left(- \frac{1}{20}\right) x \pm \frac{7}{20}$

#### Explanation:

The Average rate function is the straight line drawn between the points at each end of the interval.

So first we need to find our points,

$y = \frac{1}{x - 2}$

$y \left(- 3\right) = \frac{1}{\left(- 3\right) - 2}$

$y \left(- 3\right) = \frac{1}{-} 5$

giving us $\left(- 3 , - \frac{1}{5}\right)$

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

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

giving us $\left(- 2 , \frac{1}{-} 4\right)$

So the gradient is equal to $\text{rise"/"run}$

rise$= - \frac{1}{4} - \left(- \frac{1}{5}\right)$

rise$= - \frac{1}{20}$

run$= - 2 - \left(- 3\right)$

run$= 1$

So $\text{rise"/"run} = \frac{- \frac{1}{20}}{1} = - \frac{1}{20}$

This is our average rate of change between the two points.

to find the function we use,

$y = m x + c$

$y = \left(- \frac{1}{20}\right) x + c$

And subbing in one of our points, $\left(- 3 , - \frac{1}{5}\right)$

$\left(- \frac{1}{5}\right) = \left(- \frac{1}{20}\right) \left(- 3\right) + c$

$c = - \frac{7}{20}$

leaving us with the average rate function between $\left[- 3 , - 2\right]$ as,

$y = \left(- \frac{1}{20}\right) x \pm \frac{7}{20}$

Oct 15, 2017

Average rate of change $= - \left(\frac{1}{20}\right)$
$x + 20 y + 7 = 0$

#### Explanation:

$y = \frac{1}{x - 2}$
Given $a = - 3 , b = - 2$
$f \left(a\right) = \frac{1}{- 3 - 2} = - \left(\frac{1}{5}\right)$
$f \left(b\right) = \frac{1}{- 2 - 2} = - \left(\frac{1}{4}\right)$

Average rate of change =(f(b)-f(a))/(b-a)=(-(1/4)+(1/5))/((-2)-(-3)#
$= - \frac{\frac{1}{20}}{1} = - \left(\frac{1}{20}\right)$

To find the equation :
$a = x = - 3 , y = f \left(a\right) = - \left(\frac{1}{5}\right) , s l o p e = - \left(\frac{1}{20}\right)$

Equation is $y = m x + c$
$- \left(\frac{1}{5}\right) = - \left(\frac{1}{20}\right) \left(- 3\right) + c$
$c = - \left(\frac{1}{5}\right) - \left(\frac{3}{20}\right) = - \left(\frac{7}{20}\right)$

$y = - \left(\frac{1}{20}\right) x - \left(\frac{7}{20}\right)$
$x + 20 y + 7 = 0$