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Transformations of Polar Graphs

Alteration of graph based on changing constants and/or function of a polar equation.

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Polar Coordinates and Graphs

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Vocabulary

Complete the chart.
Word Definition
Polar Coordinates ________________________________________________________
______________ a graph of two heart shaped loops reflected across the "x" axis
______________ a graph with a sinusoidal curve looping around the origin
Transformation ________________________________________________________

Polar Coordinates

How many radians are in a circle? _____________

How do you convert from degrees to radians? ____________________________

In your own words, explain how to graph polar points on a graph: _____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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Name the coordinates of the points on this graph: 

A: ____________

B: ____________

C: ____________

Graph the following on a calculator:

  1.  \begin{align*}r = 1 + 3 sin \theta\end{align*}r=1+3sinθ
  2. \begin{align*}r = 1 + 2 cos \theta\end{align*}r=1+2cosθ
  3. \begin{align*}\left(1.5, \frac{2\pi}{3}\right)\end{align*}(1.5,2π3)

Give three alternate sets of coordinates for the given point within the range \begin{align*}-360^\circ \leq\theta\leq 360^\circ\end{align*}360θ360 .

  1. \begin{align*}(3, 120^\circ )\end{align*}(3,120)
  2. \begin{align*}(1, 240^\circ )\end{align*}(1,240)
  3. \begin{align*}(4, 345^\circ )\end{align*}(4,345)

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Distance Between Two Polar Coordinates

What is the Law of Cosines? ______________________________________

How do we have to transform the Law of Cosines to make it a distance formula for polar coordinates? _____________________________________________________

Therefore, what is the Polar Distance Formula? ______________________________________

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Find the distance between each set of points:

  1. \begin{align*}(-3, 260^\circ )\end{align*}(3,260) and \begin{align*}(2, 90^\circ )\end{align*}(2,90)
  2. \begin{align*}(4, -45^\circ )\end{align*}(4,45) and \begin{align*}(6, 150^\circ )\end{align*}(6,150)
  3. \begin{align*}(-5, -60^\circ )\end{align*}(5,60) and \begin{align*}(1, 250^\circ )\end{align*}(1,250)
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Transformations

Give an equation for a line on a polar graph: _______________________________

Give an equation for a circle on a polar graph: _______________________________

Graph the following polar equations on the same polar grid and compare the graphs.

\begin{align*}r & = 5 + 5 \sin \theta && r = 5 - 5 \sin \theta \\ r & = 5(1 + \sin \theta) && r = 5(1 - \sin \theta) \end{align*}rr=5+5sinθ=5(1+sinθ)r=55sinθr=5(1sinθ)

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What are the graphs of \begin{align*}r = a \pm b \sin \theta\end{align*}r=a±bsinθ or \begin{align*}r = a \pm b \cos \theta\end{align*}r=a±bcosθ called? _________________________

Graph each function using your calculator and sketch on your paper.

  1. \begin{align*}r=2 -3\cos(\theta )\end{align*}r=23cos(θ)
  2. \begin{align*}r=1+2\sin(\theta )\end{align*}r=1+2sin(θ)
  3. \begin{align*}r=-2+5\cos(\theta )\end{align*}r=2+5cos(θ)
  4. How do sine and cosine graphs differ?
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