Answer
$ y+8=\displaystyle \frac{5}{4}(x-4) \qquad$ ... point-slope form
$y=\displaystyle \frac{5}{4}x-13 \qquad$ ... slope-intercept form
$f(x)=\displaystyle \frac{5}{4}x-13$ $\qquad$ ... function notation
Work Step by Step
$(x_{1},y_{1})=(4,-8),\ \; (x_{2},y_{2})=(8,-3)$
$m=\displaystyle \frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{-3-(-8)}{8-4}=\frac{5}{4}$.
So with $m=\displaystyle \frac{5}{4}$ and $(x_{1},y_{1})=(4,-8)$, we write the point-slope form
$y-y_{1}=m(x-x_{1})$
$ y+8=\displaystyle \frac{5}{4}(x-4) \qquad$ ... point-slope form
Simplify to slope-intercept form, $ y=mx+b$
... distribute
$ y+8=\displaystyle \frac{5}{4}x-5 \qquad$ ...add $(-8)$
$ y=\displaystyle \frac{5}{4}x-13 \qquad$ ... is the slope-intercept form
For function notation, replace $y$ with $f(x)$.
