Graph r = 2 − 2 cos θ r=2-2\cos\theta r = 2 − 2 cos θ

In this problem we're asked to graph r equals two minus two cosine theta. Now I know that this is a graph of a cardioid because I'm dealing with subtraction and both my a and b values are equal to each other. Now we know the general shape of a cardioid, this sort of heart shape, but it remains to find the orientation and the exact positioning of it. So let's go ahead and dive into our steps here, first starting with our symmetry. Now because this is a cosine function inside of my equation, I know that my graph will be symmetric about the polar axis, which I can keep in mind mind as we proceed into step two and plot our points. Now we want to plot points specifically at our quadrantal angles, so I'm going to start with zero and plug that into my equation for theta. Two minus two times the cosine of zero. Now I know that the cosine of zero is just one, so this is two minus two, which is just zero. Now plotting this on my graph at zero zero, that's just my pole. So I can proceed to plugging in pi over two, my next quadrantal angle. Two minus two times the cosine of pi over two. The cosine of pi over two is zero, so this is just two. So I can go ahead and plot that at pi over two, two, which will be right here. Now remember that we have some symmetry going on. So because we're symmetric about this polar axis, I can go ahead and just reflect that point over onto one of my other quadrantal angles, and I know that there exists a point at three pi over two two having used that symmetry. Now one final point here plugging in pi, two minus two times the cosine of pi. The cosine of pi is negative one, so this ends up being two plus two or four. So I plot that point at pi at four pi. So going out here, I end up with my point all the way at the end of this axis, and I have all of those points that I need. Now I can connect this with a smooth and continuous curve. Now remember, for a cardioid, the furthest point from our pole will always be the sort of rounded bottom of that cardioid, which in this case is this point over here. Then I have that sort of dimple in the middle that kind of makes our heart shape right here at the pole. So I'm going to go ahead and connect everything using a sort of cardioid heart shape rounding around, and this is the basic shape of my cardioid. We have graphed r equals two minus two times the cosine of theta. Let me know if you have any questions here. Thanks so much for watching.

16. Parametric Equations & Polar Coordinates

Polar Coordinates

Calculus

16. Parametric Equations & Polar Coordinates

Polar Coordinates

Struggling with Calculus?

Join thousands of students who trust us to help them ace their exams!Watch the first video

Multiple Choice

Graph $r=2-2\cos\theta$

Graph of r=2-2cosθ showing a heart-shaped polar curve.

Graph of r=2-2cosθ depicting a heart-shaped polar curve.

Graph of r=2-2cosθ illustrating a heart-shaped polar curve.

Graph of r=2-2cosθ representing a heart-shaped polar curve.

Verified step by step guidance

Step 1: Recognize that the given polar equation is r = 2 - 2cosθ. This equation represents a polar curve, and the goal is to graph it accurately.

Step 2: Analyze the equation. The term -2cosθ indicates that the curve is influenced by the cosine function, which oscillates between -1 and 1. The subtraction and scaling affect the shape and size of the curve.

Step 3: Determine key points by substituting specific values of θ (e.g., 0, π/2, π, 3π/2, 2π) into the equation to calculate corresponding values of r. This helps identify the shape of the curve.

Step 4: Note that the curve is symmetric about the horizontal axis due to the cosine function. Plot the calculated points on a polar graph and connect them smoothly to form the curve.

Step 5: Observe that the resulting graph is a heart-shaped curve, known as a cardioid. Compare the graph to the provided images to confirm the correct representation.