Parametric and Polar - Unit 9 HPreCalc

Given an x = t and y = t, how to sketch the curve given an interval

• If there’s restrictions, just pick t’s that satisfy the restriction, solve for corresponding x and y’s, then plot accordingly.

• The order (shown by arrows) is determined BY T.

Domain restrictions on a parameter

When its roots or sumn/0 on anything. Restrict t, to find the restriction of x, then find x of the whole thing and combine it into 1 domain

Theta as parameter to rectangular form

Given y = a trig function, then isolate the function and plug into trig identities. Ex: x = cos and y = sin, x² + y² = 1

When parameter is not defined for you

Js say x = t as one of your equations, then replace x with t in the given one, that’s 2 equations

When r is negative

opposite direction or plus 180 degrees

Coordinate conversion

x = rcos theta & y = rsin theta

Tan theta = y/x

r² = x² + y²

Multiple pairs of polar coordinates

• congruent angles, or angles + 180 with a negative radius

If y end up with 1 = tantheta

Theta = tan^-1 (1)

r = 3 is what type of graph?

Circle around the center/pole

Theta = pi is what type of graph?

Linear

R = 3 cos theta is what type of graph?

Circle centered on x axis

r = 3 sin theta is what type of graph?

Circle centered on y axis

r = a cos (n theta) (or sin) is what type of graph?

Rose curve

• n is odd, n number of petals

• N is even, 2n petals

Limacons

a + or - b cos(or sin) theta

• a = b → cardioid

• a > b → dimpled

• a < b → inner loop

• a > 2B → convex (looks like a circle)

r² = a² cos(2theta)

Lemniscate

• cos us horizontal

• Sin is slanted Q1 and Q3

Conics in polar equations (eccentricity)

p is distance from focus (pole) from directrix

Make the denominator if it’s 4 - 8sin, convert the a into a 1 by dividing by 4 everywhere, so e =2

Roots of 3 and 2

3 → 1.7

2 → 1.4