Parametric functions, one parameter (article) | Khan Academy

Tags

Top Highlights

• ompletes one loop as � tt goes from 0 00 to 2 � 2π2, pi. Looking at the loopy curve, it seems we shift exactly 1 11 to the right after a single loop.

• always think about drawing it.

• imagine sketching it by trying to draw a circle counterclockwise while someone pushes your hand to the right at a steady velocity.

• start with parametric function for a circle:

• the point ( 1 , 0 ) (1,0)left parenthesis, 1, comma, 0, right parenthesis,

• (−2,0)left parenthesis, minus, 2, comma, 0, right parenthesis,

• xx value by − 3 −3minus, 3.

• −3

• with respect to time, unrelated to the motions

• steady increase

• cstart color #bc2612, c, end color #bc2612 times � tt to the � xx-component of the function.

• � cstart color #bc2612, c, end color #bc2612 times � tt to the � xx-component of the function.

• To figure out what the constant should be, we need to know how far right one has moved after completing one loop

• s we shift exactly 1 11 to the right after a single loop.

• 2πc=12, pi, start color #bc2612, c, end color #bc2612, equals, 1, and hence � = 1 2 � c= 2π 1 ​ start color #bc2612, c, equals, start fraction, 1, divided by, 2, pi, end fraction, end color #bc2612.

• we need to place bounds on � tt.

• our chosen function � ( � ) f(t)f, left parenthesis, t, right parenthesis completes one loop as � tt increases by 2 � 2π

• starting

• and tracing a circle

• 1 1