Symmetry in Graphs of Polar Equations On to Sec. 6.5a!!!

Recall our three types of symmetry… 1. The x-axis (polar axis) as a line of symmetry x y

Recall our three types of symmetry… 2. The y-axis (the line ) as a line of symmetry x y

Recall our three types of symmetry… 3. The origin (the pole) as a point of symmetry x y

Symmetry Tests for Polar Graphs The graph of a polar equation has the indicated symmetry if either replacement produces an equivalent polar equation. To Test for Symmetry Replace By 1. about the x-axis 2. about the y-axis 3. about the origin

Guided Practice Use symmetry tests to determine if the graph if symmetric about the x-axis, y-axis, or origin. Confirm your answers graphically. Symmetric about the y-axis (check the graph first???)

Guided Practice Use symmetry tests to determine if the graph if symmetric about the x-axis, y-axis, or origin. Confirm your answers graphically. Symmetric about the x-axis

Guided Practice Use symmetry tests to determine if the graph if symmetric about the x-axis, y-axis, or origin. Confirm your answers graphically. Symmetric about the y-axis

Guided Practice Use symmetry tests to determine if the graph if symmetric about the x-axis, y-axis, or origin. Confirm your answers graphically. Symmetric about the y-axis

Guided Practice Use symmetry tests to determine if the graph if symmetric about the x-axis, y-axis, or origin. Confirm your answers graphically. Symmetric about the x-axis

Analyzing Graphs of Polar Equations

But first… p.548: # –3 Now, do you have the graph?

We can “analyze” the graphs of polar equations in much the same way as we have for rectangular equations… As before, we check: Domain RangeSymmetry ContinuityBoundedness Asymptotes Maximum r-value (this is the new one…)

Let’s focus on that last one for a bit… Find the maximum r-value of Maximum r-value = 4  Check algebraically and graphically…

Let’s focus on that last one for a bit… Identify the points on the graph of for that give maximum r-values

Guided Practice Analyze the graph of the given polar curve. Domain: Range: Symmetry: About the x-axis, y-axis, and origin Continuity: Continuous Boundedness: Bounded Asymptotes: None Maximum r-value: 8 The graph???

Guided Practice Analyze the graph of the given polar curve. Domain: Range: Symmetry: About the origin Continuity: Continuous Boundedness: Unbounded Asymptotes: None Maximum r-value: None The graph???

One Last Definition The graphs of and where n is an integer greater than 1, are r rr rose curves If n is odd there are n petals; if n is even there are 2n petals

Analyze the graph: The graph??? How many petals??? Domain: Range: Symmetry: About the x-axis, y-axis, and origin Continuity: Continuous Boundedness: Bounded Asymptotes: None Maximum r-value: 3