Answer:
Symmetric about the y-axis only
Step-by-step 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
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).
[tex]\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[/tex]
[tex]\bf \stackrel{\textit{testing with respect to the origin, using }-r~\hfill }{(-r)=4-4cos(\theta )\implies r=-4+4cos(\theta )}~\hfill \bigotimes[/tex]
