PC FUNCTIONS Graphing Polar Functions

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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

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Notes by Dr. David Archerteacher of Calculus at Andress High

Transcript of PC FUNCTIONS Graphing Polar Functions

Page 1: PC FUNCTIONS Graphing Polar Functions

Polar Equations and Graphs

Equation of a Circler = a a is any constant

Example: Graph the Circle r = 3

Center at (0, 0) and Radius = 3

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Equation of a Lineθ = a θ is any angle.

Example: Graph the Line θ = π/4

No “r” No End Points

π/4

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Horizontal Line “a” units from the Poler 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

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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

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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

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Equations of a CircleLet “a” be a positive real number, then:

Equation Radius Center in Rect. Coord. 1. r = 2a sin θ a (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)

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Example: r = 4 sin θ

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

r = 4 sin θ = 2 (2) sin θ

radius = 2 Center = (0, 2)

Graph on the Calculator: 1. Mode NormalDegree Pol. EnterQuit2. Y=r1 = 4sinθGraph

{Note: looks better if you use ZoomZSquare}

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Symmetry

1. Points Symmetric with respect to the Polar Axis

Example: r = 5 cos θ r = 5 cos(-θ) = 5 cos θ

Test: Replace “θ” with “-θ”.

If you get an equivalent equation then it is symmetric with respect to the Polar Axis.

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2. Points Symmetric with respect to theline θ = π/2

Symmetry

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

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3. Points Symmetric with respect to the Pole.

Test: Replace “r” with “-r”.If you get an equivalent equation then it issymmetric with respect to the Pole.

Example: r2 = 3 sin θ r2 = 3 sin θ (-r)2 = 3 sin θ

r2 = 3 sin θ

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Cardioidsr = a(1 + cos θ) r = a(1 + sin θ)r = a(1 – cos θ) r = a(1 – sin θ)

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

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Example: Graph the equation: r = 1 – sin θCheck for Symmetry:(a). Polar Axis r = 1 – sin(-θ)(b). Line θ = π/2

= 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}

= 1 + sin θ {Fails}

To Graph Use calculator:

r = 1 – sin(π – θ)

modedegreepolY=r1 = 1 – sin θ Graph”zoom in” if necessaryZsquare

Graph on paper using table. Warning: “θ max” setting Must be set at 360º

Graph on next Slide

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Graph From Last Slide

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Limaçon “Without” Inner Loop

r = a + b cos θ r = a + b sinθr = a – b cos θ r = a – b sin θ a > 0, b > 0 and “a > b”

It does NOT pass through the pole.

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Example: Graph the equation: r = 3 + 2 cos θCheck 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 θr = -3 – 2 sin θ {Fails}To Graph Use calculator:

modedegreepolY=r1 = 3 + 2 cos θGraph”zoom in” if necessaryZsquare

Graph on paper using table. Warning: “θ max” setting Must be set at 360º

Graph on next Slide

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Limaçon “With” Inner Loopr = a + b cos θ r = a + b sinθr = a – b cos θ r = a – b sin θ

a > 0, b > 0 and “a < b” It “does” pass through the pole.

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Example: Graph the equation: r = 1 + 2 cos θCheck 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 θr = -1 – 2 sin θ {Fails}

To Graph: {With Calculator}

Graph on next Slide

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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.

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Example: Graph the equation: r = 2 cos 2θCheck 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

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Lemniscater2 = 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 LineTo Graph: {With Calculator}

Graph on next Slide

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