Families of functions

Most curves you meet belong to a family with a shared shape and a few dials (parameters) that stretch, shift or flip it:

• Line — y = a·x + b (slope a, height b)
• Parabola — y = a·x² + b·x + c (a U or ∩)
• Sine wave — y = a·sin(b·x) (amplitude |a|, angular frequency b; ordinary frequency |b|/(2π))

Change a parameter and the whole family member reshapes — but the character stays. For nonzero quadratic coefficient, a parabola is a smooth bowl. A sine with nonzero amplitude and frequency is a wave; setting a coefficient to zero can produce a degenerate case. Learning the families lets you predict a graph before you plot a single point.