5.4 Graphs of Polar Equations

Objectives Use point plotting to graph polar equations. Use symmetry to graph polar equations.

Using Polar Grids to Graph A polar equation is an equation whose variables are r and θ. The graph of a polar equation is the set of all points whose polar coordinates satisfy the equation. We use polar grids like the one in the figure to graph polar equations.

Circles in Polar Coordinates The graph of r = a cos θ and r = a sin θ, a > 0, are circle.

Example 1: Graphing an Equation Using the Point-Plotting Method (1 of 4) We construct a partial table of coordinates for using multiples of Then we plot the points and join them in a smooth curve.

Example 1: Graphing an Equation Using the Point-Plotting Method (2 of 4)

Example 1: Graphing an Equation Using the Point-Plotting Method (3 of 4)

Example 1: Graphing an Equation Using the Point-Plotting Method (4 of 4) We can verify that the graph is a circle by changing from polar to rectangular form. The graph is a circle with center at (0, 2) and r = 2.

Tests for Symmetry in Polar Coordinates (1 of 3) Symmetry with Respect to the Polar Axis (x - Axis) Replace θ with −θ. If an equivalent equation results, the graph is symmetric with respect to the polar axis.

Tests for Symmetry in Polar Coordinates (2 of 3) Symmetric with Respect to the Line Replace (r, θ) with (−r, −θ). If an equivalent equation results, the graph is symmetric with respect to

Tests for Symmetry in Polar Coordinates (3 of 3) Symmetric with Respect to the Pole (Origin) Replace r with −r. If an equivalent equation results, the graph is symmetric with respect to the pole.

Example 2: Graphing a Polar Equation Using Symmetry (1 of 6) Check for symmetry and then graph the polar equation: Symmetry with respect to the polar axis (x-axis): on the graph of the function. The polar equation does not change when θ is replaced with −θ, so the graph is symmetric with respect to the polar axis.

Example 2: Graphing a Polar Equation Using Symmetry (2 of 6) Check for symmetry and then graph the polar equation: Symmetry with respect to the line (y-axis): on the graph of the function. The polar equation changes when θ is replaced with −θ. The graph is not symmetric with respect to the line

Example 2: Graphing a Polar Equation Using Symmetry (3 of 6) Check for symmetry and then graph the polar equation: Symmetry with respect to the polar (origin): on the graph of the function. The polar equation changes when r is replaced with −r. The graph is not symmetric with respect to the pole.

Example 2: Graphing a Polar Equation Using Symmetry (4 of 6) Check for symmetry and then graph the polar equation: Complete a table of values for the function:

Example 2: Graphing a Polar Equation Using Symmetry (5 of 6)

Example 2: Graphing a Polar Equation Using Symmetry (6 of 6) The graph of is an example of a limaçon.

Limaçons The graph of are called limaçon. The ratio determines a limaçon’s shape.

Example 3: Graphing a Polar Equation (1 of 3) Graph the polar equation: We first check for symmetry:

Example 3: Graphing a Polar Equation (2 of 3) Graph the polar equation: Complete a table of values for the function:

Example 3: Graphing a Polar Equation (3 of 3) Graph the polar equation: The graph of is an example of a rose curve.

Rose Curves The graph of are called rose curves. If n is even, the rose has 2n petals. If n is odd, the rose has n, petals.

Example 4: Graphing a Polar Equation (1 of 3) Graph the polar equation: We first check for symmetry:

Example 4: Graphing a Polar Equation (2 of 3) Graph the polar equation: Complete a table of values for the function:

Example 4: Graphing a Polar Equation (3 of 3) Graph the polar equation: The graph of the polar equation is an example of a lemniscate.

Lemniscates The group of are called lemniscates.