Determine if the graph is symmetric about the x-axis, the y-axis, or the origin. r = 4 cos 3θ

Solution:

In a graph, symmetry can be found visually. Both equations are symmetric if both the images are mirror images of one another. It can be found analytically by testing the symmetry.

If f(r, θ) = f(-r,-θ), symmetric to the pole or the origin

If f(r, θ) = f(r,-θ), symmetric to the polar axis or the x-axis

If f(r, θ) = f(-r,θ), symmetric to the y-axis

(i) About the x - axis

f(r, θ) ⇒ r = 4 cos3θ

So f(r, -θ)⇒ r = 4 cos3(-θ)

= 4 cos3θ

Thus f(r, θ) = f(-r,θ)

So the graph is symmetric about the x-axis

(ii) About the y-axis

f(r,θ) ⇒ r= 4 cos3θ

So f(-r,θ) ⇒ -r = 4 cos3θ

⇒ r = -4 cos3θ

f(r,θ) ≠ f(-r,θ)

So the graph is not symmetric about the y-axis

(iii) About the origin

f(r,θ)⇒ r=4 cos3θ

So f(-r,-θ)⇒ -r = 4 cos3(-θ)

Here r = -4 cos3θ

f(r,θ) ≠ f(-r,-θ)

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 = 4 cos 3θ

Summary:

The graph r = 4 cos 3θ is symmetric about the x-axis.