Pc Functions Graphing Polar Functions

Polar Equations and Graphs Equation of a Circle r=a a is any constant Example: Graph the Circle r = 3 Center at (0, 0) and Radius = 3

Equation of a Line θ=a θ is any angle. Example: Graph the Line

θ = π/4

No “r” ∴ No End Points

π/4

Horizontal Line “a” units from the Pole r sin θ = a Vertical Line “a” units from the Pole r cos θ = a Example: Identify & Graph:

r = 4sin θ

Use Calculator in Polar Mode to Graph

Straight Line Graph Properties in a Polar Equation 1. Horizontal Line r sin θ = a a = units above the Pole if a > 0 |a| units below the Pole if a < 0 Example: r = sin θ = 2

2. Vertical Line r cos θ = a a units to the right of the pole if a > 0 |a| units to the left of the pole if a < 0 Example: r cos θ = - 3

Equations of a Circle Let “a” be a positive real number, then: 1.

Equation r = 2a sin θ

Radius a

Center in Rect. Coord. (0, a)

2.

r = - 2a sin θ

a

(0, -a)

3.

r = 2a cos θ

a

(a, 0)

4.

r = - 2a cos θ

a

(-a, 0)

Example: r = 4 sin θ ∴ r = 4 sin θ = 2 (2) sin θ ∴ radius = 2 Center = (0, 2) Graph on the Calculator: 1. Mode NormalDegree Pol. EnterQuit 2. Y=r1 = 4sinθGraph {Note: looks better if you use ZoomZSquare}

Example: r = - 2 cos θ ∴ r = -2 cos θ = -2 (1) cos θ ∴ radius = 1 Center = (-1, 0) Graph on the Calculator

Symmetry 1. Points Symmetric with respect to the Polar Axis Test: Replace “θ” with “-θ”. If you get an equivalent equation then it is symmetric with respect to the Polar Axis. Example: r = 5 cos θ r = 5 cos(-θ) = 5 cos θ

Symmetry 2. Points Symmetric with respect to the line θ = π/2 Test: Replace “θ” with “π -θ”. If you get an equivalent equation then it is symmetric with respect to the line θ = π/2.

Example: r = -2 sin θ r = - 2 sin θ = -2 sin(π-θ) = - 2 sin θ

3. Points Symmetric with respect to the Pole. Test: Replace “r” with “-r”. If you get an equivalent equation then it is symmetric with respect to the Pole.

Example: r2 = 3 sin θ r2 = 3 sin θ

(-r)2 = 3 sin θ r2 = 3 sin θ

Cardioids r = a(1 + cos θ) r = a(1 – cos θ)

r = a(1 + sin θ) r = a(1 – sin θ)

where a > 0. The graph of a cardioid passes through the Pole.

Example: Graph the equation: r = 1 – sin θ Check for Symmetry: (a). Polar Axis r = 1 – sin(-θ) = 1 + sin θ {Fails} (b). Line θ = π/2 r = 1 – sin(π – θ) = 1 – [(sin π)(cos θ) – (cos π)(sin θ)] = 1 – [0·cos θ – (-1)sin θ] = 1 – sin θ ∴ Is Symmetric to Line (c). Pole – r = 1 – sin θ  r = -1 + sin θ {Fails} To Graph Use calculator: modedegreepolY=r1 = 1 – sin θ Graph ”zoom in” if necessaryZsquare Graph on paper using table. Warning: “θ max” setting Must be set at 360º Graph on next Slide

Graph From Last Slide

Limaçon “Without” Inner Loop r = a + b cos θ r = a – b cos θ

r = a + b sinθ r = a – b sin θ

a > 0, b > 0 and “a > b” It does NOT pass through the pole.

Example: Graph the equation: r = 3 + 2 cos θ Check for Symmetry: (a). Polar Axis r = 3 + 2 cos(-θ) = 3 + 2 cos θ ∴ Is Symmetric to Polar Axis (b). Line θ = π/2 r = 3 + 2 cos(π – θ) = 3 + 2 [(cos π)(cos θ) + (sin π)(sin θ)] = 3 + 2[-1·cos θ + (0)sin θ] = 3 – 2 cos θ {Fails} (c). Pole – r = 3 + 2 cos θr = -3 – 2 sin θ {Fails} To Graph Use calculator: modedegreepolY=r1 = 3 + 2 cos θ Graph ”zoom in” if necessaryZsquare Graph on paper using table. Warning: “θ max” setting Must be set at 360º Graph on next Slide

Limaçon “With” Inner Loop r = a + b cos θ r = a + b sinθ r = a – b cos θ r = a – b sin θ a > 0, b > 0 and “a < b” It “does” pass through the pole.

Example: Graph the equation: r = 1 + 2 cos θ Check for Symmetry: (a). Polar Axis r = 1 + 2 cos(-θ) = 1 + 2 cos θ ∴ Is Symmetric to Polar Axis (b). Line θ = π/2 r = 1 + 2 cos(π – θ) = 1 + 2 [(cos π)(cos θ) + (sin π)(sin θ)] = 1 + 2[-1·cos θ + (0)sin θ] = 1 – 2 cos θ {Fails} (c). Pole – r = 1 + 2 cos θ r = -1 – 2 sin θ {Fails} To Graph: {With Calculator} Graph on next Slide

Rose r = a cos (nθ)

r = a sin(nθ)

If n ≠ 0 is even, the rose has 2n petals. If n ≠ 1 is odd, the rose has n petals.

Example: Graph the equation: r = 2 cos 2θ Check for Symmetry: (a). Polar Axis r = 2 cos 2(-θ) = 2 cos 2θ ∴ Is Symmetric to Polar Axis (b). Line θ = π/2 r = 2 cos 2(π – θ) = 2 cos (2π - 2θ) = 2 cos 2θ ∴ Is Symmetric to Line (c). Pole Since the graph is symmetric with respect to both the polar axis and the line θ = π/2 it must be symmetric to the pole. ∴ Is Symmetric to Line To Graph: {With Calculator} Graph on next Slide

Lemniscate r2 = a2 sin (nθ) r2 = a2 cos (nθ) where a ≠ 0 & have graphs propeller shaped. Example: Graph the equation: r2= 4sin 2θ Check for Symmetry: (a). Polar Axis r2= 4sin 2(-θ) = - 4sin 2θ ∴ Fails (b). Line θ = π/2 r2= 4sin 2(π – θ) = 4sin (2π - 2θ) = 4sin (-2θ) = -4sin 2θ ∴ Fails (c). Pole r2 = 4sin 2θ (-r2) = 4sin 2θ  r2= 4sin 2θ ∴ Is Symmetric to Line To Graph: {With Calculator} Graph on next Slide

Spiral

Example: r = eθ/5 All Symmetry test Fail.