Lesson 3.6 (Continued) Graphing Exponential Functions 1 3.4.2: Graphing Exponential Functions.
PC FUNCTIONS Graphing Polar Functions
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Transcript of 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
Equation of a Lineθ = 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 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
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 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)
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}
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.
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
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
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.
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
Graph From Last Slide
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.
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
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.
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
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 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
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
Spiral Example: r = eθ/5 All Symmetry test Fail.