How am I incorrectly solving this polar equations graph?

Question

Graph: $r = 1 + 2\cos(\theta)$

I set up a table of values and increment them by $π/2$:

$$\begin{array} {|c|c|}\hline \phantom{\_\_\_}\theta\phantom{\_\_\_} & \phantom{\_\_\_}r\phantom{\_\_\_} \\ \hline 0 & 3 \\ \hline \pi/2 & 1 \\ \hline \pi & -1 \\ \hline 3\pi/2 & 1 \\ \hline 2\pi & 3 \\ \hline 5\pi/2 & 1 \\ \hline 3\pi & -1 \\ \hline 7\pi/2 & 1 \\ \hline 4\pi & 3 \\ \hline \end{array}$$

When I plot these points I get:

When I go about graphing I think of $\theta$ as direction, and $r$ as distance. So I point in a direction, and go a certain distance. Then to the next one and I connect the dots. But there seems to be more going on than just connecting the dots when I look at the actual graph on Desmos. How am I supposed to know it's going to loop inwards, go through the $y$-axis, and go back to $1$ on the $x$-axis? Actual:

Thank you

Answer 1

One way to see that the polar curve loops inward is to graph $r$ versus $\theta$ on cartesian axes. You see that $r$ becomes negative when $\cos\theta < -\frac{1}{2}$; that is on the interval $\left(\frac{2\pi}{3},\frac{4\pi}{3}\right)$.

Answer 2

Firstly, to get a better picture of what the graph should look like, you should solve for ALL of the x intercepts (where theta equals 0 or pi) and y intercepts (where theta equals pi over two or three times pi over two). Also, solving for where the radius equals zero is always helpful.

Also calculus students are generally taught the general equation for this type of equations (r = k + Lcos(Θ)). When you see this form, you can use three general rules. 1) if k = L, then you will have a perfect cardioid 2) if k > L you will have something similar to a cardioid but closer to a circle 3) if k < L (like yours) then you will have a loop. (This also applies to when the function is a sine function instead of cosine)