3.4 Graphing functions using polar coordinates

• What does a polar coordinate represent in terms of angle and distance?

  Angle and distance
• Steps to plot a point in polar coordinates

  1️⃣ Determine the angle (θ) in radians or degrees

  2️⃣ Determine the distance (r) from the origin

  3️⃣ Plot the point using the (θ, r) pair
• Match the coordinate system with its advantage:

  Polar ↔️ Better for circular functions

  Cartesian ↔️ Simpler for linear functions
• To convert from Cartesian to polar coordinates, the angle (θ) is calculated using the formula $\theta =$$\tan^{ - 1}\left(\frac{y}{x}\right)$, where x cannot be zero.
• What are the polar coordinates of the Cartesian point (1, 1)?

  ($\sqrt{2}$, $\frac{\pi}{4}$)
• What are the Cartesian coordinates for Example 2?

  (1, $\sqrt{3}$)
• What is the formula to calculate the x-coordinate from polar coordinates?

  $x =$$r \cos(\theta)$
• When converting (1, 1) to polar coordinates, the angle $\theta$ is $\frac{\pi}{4}$
• What defines the extent of the polar plot?

  Maximum r Values
• What is the approximate y-coordinate when converting the polar coordinate (π/3, 5) to Cartesian coordinates?

  4.33
• Steps to graph polar equations

  1️⃣ Identify symmetry about the x-axis, y-axis, or origin

  2️⃣ Determine maximum r values

  3️⃣ Find the period of the function

  4️⃣ Locate intercepts

  5️⃣ Determine the domain of θ
• The interval over which the function repeats its pattern is called the period
• Steps for graphing polar equations

  1️⃣ Identify symmetry

  2️⃣ Find maximum r values

  3️⃣ Determine the period

  4️⃣ Locate intercepts

  5️⃣ Define the domain
• What role do intercepts play in sketching polar plots?

  Provide reference points
• The graph of a polar function is symmetric about the x-axis if replacing (r, θ) with (r, -θ) yields the same equation.


  True
• What condition must a polar function satisfy to have symmetry about the origin?

  r(θ) = r(θ + π)
• For what type of functions are polar coordinates better suited?

  Circular or periodic
• When converting from Cartesian to polar coordinates, the angle θ is found using $\theta =$$\tan^{ - 1}(\frac{y}{x})$, but must be adjusted for the quadrant
• The highest r value achieved by a polar function defines the extent
• The highest r value achieved by the function defines the extent of the polar plot.
• The domain of a polar function is the valid range of θ values for which the function is defined.
• Match the feature of a polar graph with its definition:

  Symmetry ↔️ Mirroring about x-axis, y-axis, or origin

  Maximum r Values ↔️ Highest r achieved by the function

  Periods ↔️ Interval over which the function repeats
• The maximum r value defines the extent or size of the polar plot.
• Polar functions can exhibit symmetry about the x-axis, y-axis, or the origin.
• The polar equation $r =$$4 \cos \theta$ satisfies x-axis symmetry.


  True
• Using technology to graph polar equations allows for visualizing complex shapes and properties quickly.


  True
• Polar coordinates use an (x, y) pair to represent a point.

  False
• Polar coordinates are better suited for circular and periodic functions than Cartesian coordinates.


  True
• What are the formulas used to convert between rectangular and polar coordinates?

  $x =$$r \cos \theta$ and $y =$$r \sin \theta$
• What are the Cartesian coordinates of the polar point (π/3, 5)?

  (2.5, 4.33)
• What are the Cartesian coordinates of the polar point (2, π/3)?

  (1, $\sqrt{3}$)
• Converting between Cartesian and polar coordinates requires using formulas that relate the coordinates.


  True
• Polar coordinates are better suited for circular or periodic functions.


  True
• What are the polar coordinates when converting (1, 1) from Cartesian coordinates?

  (\sqrt{2}, \frac{\pi}{4})</latex>
• The relationship between polar and Cartesian coordinates is given by $x =$$r \cos \theta$ and $y =$$r \sin \theta$.


  True
• Symmetry in polar functions simplifies their analysis and graphing.


  True
• Polar functions can exhibit symmetry about the x-axis, y-axis, or origin
• What do the points where the graph crosses the x or y-axis represent?

  Intercepts
• What types of symmetry can polar functions exhibit?

  X-axis, y-axis, origin
• The valid range of θ values for which the function is defined is called the domain