Determine if the graph is symmetric about the x-axis, the y-axis, or the origin. r = 8 cos3θ

Solution:

Symmetric about the x-axis

If (r, θ) lies on the graph, then (r, -θ) or (-r, π - θ) lies on the graph.

Symmetric about the y-axis

If (r, θ) lies on the graph, then (r, -θ) or (-r, -θ) lies on the graph.

Symmetric about the origin

If (r, θ) lies on the graph, then (r, -θ) or (-r, π + θ) lies on the graph.

r = 8 cos3θ is the given polar equation

Now check the equation by (r, -θ)

r = 8 cos(-3θ) = 8 cos(3θ) = r

So the graph is symmetric about the x-axis.

Now check the equation by (-r, -θ)

- r = 8 cos(-3θ) = 8 cos(3θ) ⇒r ≠ - r

So the graph is not symmetric about the y-axis.

Now check the equation by (-r, θ)

- r = 8 cos(3θ) ⇒ r ≠ - r

So the graph is not symmetric about the origin.

Therefore, the graph is symmetric about the x-axis.

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Determine if the graph is symmetric about the x-axis, the y-axis, or the origin. r = 8 cos3θ

Summary:

The graph is symmetric about the x-axis.