Answer
(a) $$y=mx+2$$
(b) $$y=-x+2$$
(c)
red: $y=2x+2$
green: $y=-x+2$
blue: $y=-5x+2$
Work Step by Step
(a) The equation for the family of lines with $y$-intercept $b$ is$$y=mx+b,$$so the equation for the family of lines with $y$-intercept $b=2$ is$$y=mx+2.$$
(b) If a line has angle of inclination $\theta $, then the slope of the line is$$m= \tan \theta ,$$so for the member of the family with angle of inclination $\theta = 135 ^{\circ }$ we have$$m= \tan 135 ^{\circ } = - \tan 45 ^{\circ } = -1 \quad \Rightarrow \quad y= -x+2.$$
