Question

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

r = 4 - 4 cos θ

Answer 1

Symmetric about the y-axis only

Explanation:

Graph the function using a calculator

alternatively,

Sketch the graph via the following steps

-sketch cosθ

- reflect about the x-axis to get (- cosθ)

- multiply vertex values by 4 to get -4cosθ

- shift the graph in the positive y direction by 4 units to get 4 - 4cosθ

you will end up with the attached graph:

By observation, we can see that the graph is

1) NOT symmetric about the x-axis

2) Symmetric about the y-axis

3) Not symmetric bout the origin

Answer 2

to run a symmetry test on a polar, we can simply do a few ( r , θ ) replacements, if the equation resulting from the replacements, resembles the original, then it has that symmetry type, now, let's recall the symmetry trigonometry identity, cos(-θ) = cos(θ).

`\bf r=4-4cos(\theta ) \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{testing with respect to the y-axis, using }-r,-\theta ~\hfill }{(-r)=4-4cos(-\theta )\implies -r=4-4cos(\theta )}\implies r=-4+4cos(\theta )~\hfill \bigotimes \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{testing with respect to the x-axis, using }-\theta ~\hfill }{r=4-4cos(-\theta )\implies r=4-4cos(\theta )}~\hfill \stackrel{\textit{symmetry with respect}}{\textit{to x-axis}~\hfill }\textit{\Large\checkmark} \\\\[-0.35em] ~\dotfill`

`\bf \stackrel{\textit{testing with respect to the origin, using }-r~\hfill }{(-r)=4-4cos(\theta )\implies r=-4+4cos(\theta )}~\hfill \bigotimes`