Graphing polar equations is different that plotting cartesian equations.  Instead of plotting an  coordinate, polar graphs consist of an  coordinate where  is the radial distance of a point from the origin and  is the angle above the x-axis.

From our equation, we know the shape of our graph will be a cardioid because our equation is in the form  where .  Our cardioid is symmetric about the x-axis because our equation includes the  function  The point of the cardioid is at the origin. The y-intercepts are at  and . The x-intercept is at .

We could also substitute values of  (in radians) between  and  into our expression, to find values of r. We then plot each ordered pair, , using the  value as the radius and  as the angle.  

We get the graph below, a cardioid (heart shape) rotated  left.

Graphing polar equations is different that plotting cartesian equations.  Instead of plotting an  coordinate, polar graphs consist of an  coordinate where  is the radial distance of a point from the origin and  is the angle above the x-axis.

 From our equation, we know the shape of our graph will be a cardioid because our equation is in the form  where .  Our cardioid is symmetric about the y-axis because our equation includes the  function. The point of the cardioid is at the origin.  The x-intercepts are at  and .  The y-intercept is at .

We could also substitute values of  (in radians) between  and  into our expression, to find values of r.  We then plot each ordered pair, , using the  value as the radius and  as the angle.  

We get the graph below, an upright cardioid (heart shape).

Graphing polar equations is different that plotting cartesian equations.  Instead of plotting an  coordinate, polar graphs consist of an  coordinate where  is the radial distance of a point from the origin and  is the angle above the x-axis.

From our equation, we know the shape of our graph will be a limacon  because our equation is in the form  where . This limacon will have no loop because . Our limacon is symmetric about the y-axis because our equation includes the  function.  The x-intercepts are at  and .  The y-intercept is at .

We could also substitute values of  (in radians) between  and  into our expression, to find values of r. We then plot each ordered pair, , using the  value as the radius and  as the angle.  

We get the graph below, an upside-down limacon.

Graphing polar equations is different that plotting cartesian equations.  Instead of plotting an  coordinate, polar graphs consist of an  coordinate where  is the radial distance of a point from the origin and  is the angle above the x-axis.

From our equation, we know the shape of our graph will be a limacon  because our equation is in the form  where .  This limacon will have no loop because . Our limacon is symmetric about the x-axis because our equation includes the  function. The y-intercepts are at  and .  The x-intercept is at .

We could also substitute values of  (in radians) between  and  into our expression, to find values of r.  We then plot each ordered pair, , using the  value as the radius and  as the angle.  

We get the graph below, an limacon turned  right.

Graphing polar equations is different that plotting cartesian equations.  Instead of plotting an  coordinate, polar graphs consist of an  coordinate where  is the radial distance of a point from the origin and  is the angle above the x-axis.

From our equation, we know the shape of our graph will be a limacon because our equation is in the form  where . This limacon will have a loop because . The length of the loop is .  Our limacon is symmetric about the x-axis because our equation includes the  function. The y-intercepts are at  and .  The x-intercept is at .

We could also substitute values of  (in radians) between  and  into our expression, to find values of r. We then plot each ordered pair, , using the  value as the radius and  as the angle.  

We get the graph below, a limacon with a loop turned  left.

Graphing polar equations is different that plotting cartesian equations. Instead of plotting an  coordinate, polar graphs consist of an  coordinate where  is the radial distance of a point from the origin and  is the angle above the x-axis.

From our equation, we know the shape of our graph will be a limacon  because our equation is in the form  where .  This limacon will have a loop because . The length of the loop is .  Our limacon is symmetric about the y-axis because our equation includes the  function. The x-intercepts are at  and . The y-intercept is at .

We could also substitute values of  (in radians) between  and  into our expression, to find values of r. We then plot each ordered pair, , using the  value as the radius and  as the angle.  

We get the graph below, an upright limacon with a loop.

The coordinate  goes to the right  units from the origin and is rotated  counter-clockwise, terminating in 

The coordinate  goes to the right  units from the origin and is rotated  counter-clockwise, terminating in 

The coordinate  goes to the left  units from the origin and is rotated  counter-clockwise, terminating in 

The polar coordinate

is graphed by moving  units to the left of the origin and rotating  counter-clockwise, resulting in