Question

Graph the cardioids $r=1+\cos \theta ,r=2(1+\cos \theta ),andr=3(1+\cos \theta )$in the same polar coordinate system. What are the relationships among these three graphs? How does changing the constant $k$in the graph of the cardioid$r=k(1+\cos \theta )$ affect the graph? What happens when $k<0$?

Short answer

The graph is :

Step-by-step solution

Step 1. Given Information.

The given curves are :

$r=1+\cos \theta ,r=2(1+\cos \theta ),andr=3(1+\cos \theta )$.

Step 2. Graphically.

The graph of the curves will be :

Step 3. Theoretically.

The given equations are :

$r=k\left(1+\cos \theta \right)$

In equations, the sign$\left(+and-\right)$affects the direction of the cardioid. Changing the value of k changes the size of the cardioid but does not affect its shape. The larger the value of k, the larger the resulting cardioid. If the value of $k<0$ , then smaller will be the cardioid.