Question #9415f

Oct 10, 2016

$f \left(x\right) = - 40 x + 1700$

Explanation:

If we treat the data as a set of points ordered pairs $\left(t , V\right)$, then we are given the data as $\left(0 , 1700\right) , \left(4 , 1540\right) , \left(8 , 1380\right)$. The first thing we need find a linear function is its slope $m$.

Given two points $\left({x}_{1} , {y}_{1}\right)$ and $\left({x}_{2} , {y}_{2}\right)$ on the graph $y = f \left(x\right)$ of a linear equation, we can find the slope as the ratio of the change in $y$ to the change in $x$.

$m = \frac{\triangle y}{\triangle x} = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$

Picking the first two points from the data set, we get the slope of our desired function as

$m = \frac{1540 - 1700}{4 - 0} = - \frac{160}{4} = - 40$

Now that we have the slope, we can choose a point and use the point-slope form of a linear equation, $y - {y}_{1} = m \left(x - {x}_{1}\right)$, to find the equation of the graph $y = f \left(x\right)$.

Taking the first point $\left(0 , 1700\right)$ and the slope $- 40$, we get

$y - 1700 = - 40 \left(x - 0\right)$

$\implies y - 1700 = - 40 x$

$\implies y = - 40 x + 1700$

Thus, our linear function is $f \left(x\right) = - 40 x + 1700$