Question

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

Answer 1

Hello,

Answer B: y-axis only
==================

Answer 2

y-axis only.

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 = (\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 = (\pi)/(2)`, which is the y-axis only in the coordinate system.

So, the right answer is y-axis only.