Answer
$ f(x)=-\frac{3}{2} x+7$
Work Step by Step
The equation of the line passing through the two points $P_{1}\left(x_{1}, y_{1}\right), P_{2}\left(x_{2}, y_{2}\right)$ is
$$y-y_{1}=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\left(x-x_{1}\right)$$
We are given $P_{1}(4,1), P_{2}(8,-5)$ so substituting the values of $x_1, x_2, y_1, y_2$ into the equation above gives:
$y-1=\dfrac{-5-1}{8-4} \cdot (x-4)$
$y-1=\dfrac{-6}{4} \cdot (x-4)$
$y-1=-\frac{3}{2}(x-4)$
$y-1=-\frac{3}{2}x+(\frac{3}{2})4$
$y-1=-\frac{3}{2}x+6$
$y=-\frac{3}{2}x+6+1$
$y=-\frac{3}{2}x+7$
Therefore, the equation of the linear function is $f(x)=-\frac{3}{2}x+7$.
