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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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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: _____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________


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

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

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

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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? ______________________________________


Find the distance between each set of points:

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


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

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

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