First, notice that since cos θ is always between -1 and +1, the quantity r = 1 + cos θ is always between 0 and 2. In particular, *r* will never be negative. We should find points in every quadrant. Find the value of *r* at some nice angles. %%%%%%DATA ENTRY This data is in a table. {c|c} θ&$r = 1 + cosθ$ \hline 0&2 $\frac{π}{2}$&1 π&0 $\frac{3π}{2}$&1 This gives us some points to start from. PICTURE polar funcs 24 Now we need to figure out what's going on in between these points. Remember that *r* is never negative. From θ = 0 to θ = \frac{π}{2}, the value of *r* will move from 2 to 1. PICTURE polar funcs 25 From $θ = \frac{π}{2}$ to $θ = π$, the value of *r* will move from 1 to 0. PICTURE polar funcs 26 From $θ = π$ to $θ = \frac{3π}{2}$, the value of *r* will move from 0 to 1. PICTURE polar funcs 27 From $θ = \frac{3π}{2}$ to $θ = 2π$, the value of *r* will move from 1 to 2. PICTURE polar funcs 28 We end up with a heart shape that looks nothing like the graph r = cosθ. The moral of the story is that we typically use calculators when making anything but the simplest of graphs in polar coordinates.. |