One Parameter Families

(i) $m$ arbitrary, $b$ fixed. Fixing the $y$-intercept at, say, $b=5$ and letting $m$ range over the reals gives the family $y=mx+5$ — every member shares the single point $(0,5)$ on the $y$-axis, and different values of $m$ (e.g. $-2,-1,\tfrac13,1,4$) fan out from that shared point at different slopes.

(ii) $b$ arbitrary, $m$ fixed. Fixing the slope at, say, $m=-2$ and letting $b$ range over the reals gives the family $y=-2x+b$ — every member is parallel to every other member (they all share slope $-2$), just shifted vertically by different amounts $b$ (e.g. $b=-3,-1,0,1,2,3,4$).

Two special named families, both of type (ii):

• Family of parallel lines. The family of lines parallel to $ax+by+c=0$ is $ax+by+\lambda=0$; different real values of $\lambda$ give different lines, all parallel to the given one.
• Family of perpendicular lines. The family of lines perpendicular to $ax+by+c=0$ is $bx-ay+\lambda=0$; different $\lambda$ give different lines, all perpendicular to the given one. …