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Question

Is there an intersection at the origin of the graph: $r=2+3cos\theta$

I have realized that if I was to graph the equation into a graphing calculator and zoom at the pole, there were no intersection at the point but then I thought this was strange so I did some calculations

I know that if r=0 then that is an intersection at the pole so I set it to 0 and realize that it is possible to get a value $\theta$ which satisfies r being zero,

Why does the calculator show otherwise?

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    I don't think one can answer the question without knowing something about your specific calculator and how you used it... There is definitely an intersection and Wolfram Alpha (https://www.wolframalpha.com/input/?i=polar+plot+r+%3D+2+%2B+3+cos(t)) shows you that. – 2017-02-27
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    Can we see waht the calculator graph looks like? – 2017-02-27
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    it is desmos.com – 2017-02-27

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If we plot the graph:

enter image description here

We see there is in fact an intersection at $r=0$. The graphing calculator you used probably did a cartesian plot.