Question

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

Answer 1

Symmetry about y-axis

Explanation:

We are given that

`r=4-3sin\theta`

When the graph is symmetric about x- axis then the point (x,y) change in to (x,-y) and the function remain same.

The point
`(r,\theta,)` is replaced by
`( r,-\theta)`

Substitute the value then we get

`r=4-3 sin(-\theta)`

`r=4+3sin\theta` (
`sin(-\theta)=-sin\theta)`

The value of function changes .Hence , the function is not symmetric about x- axis.

When the function is symmetric about y-axis then the point
`(r,\theta)` change into point
`(r,\pi-\theta)` and function remain same

Substitute the value

`r=4-3sin(\pi-\theta)`

`r=4-3 sin\theta` (
`sin(\pi-\theta)=sin\theta`)

The value of function does not change when point change ,Hence, the function is symmetric about y- axis.

When the graph is symmetric about origin then the point
`(\theta,r)` change into point
`(r,\pi+\theta)` and the value of function remain same.

Substitute the value then we get

`r=4-3sin(\pi+\theta)`

`r=4+3sin\theta`

`r=4+3sin\theta`

Hence, the graph has symmetry about y-axis .

Answer 2

see picture, it is symmetrical about the Y axis