Answered

Determine if the graph is symmetric about the x-axis, the y-axis, or the origin.


r = 2 + 3 sin θ


Choices:

No symmetry

y-axis only

x-axis only

Origin only

Verified Answer :

caylus

Hello,

Answer B: y-axis only
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View image caylus

Answer:

y-axis only.

Step-by-step explanation:

The given equation is

\(r=2+3sin \theta\)

Notice that this equation is in polar form.

If \((r, \theta)\) can be replace with \((r, \pi - \theta)\) or \((-r, -\theta)\), then the graph is symmetric to the line \(\theta = \frac{\pi}{2}\), which is a vertical line.

Let's evaluate the given equation.

\(r=2+3sin \theta=2+3sin(\pi -\theta)= 2-3sin(- \theta)\)

But, \(sin(-\theta)=-sin \theta\)

So,  \(r= 2-3sin(- \theta)=2-(-3sin\theta)=2+3sin\theta=r\)

Notice that the change produces the same equation.

Therefore, the given polar expression is symmetric to \(\theta = \frac{\pi}{2}\), which is the y-axis only in the coordinate system.

So, the right answer is y-axis only.

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