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

Using the identity \(\displaystyle x=rcos\theta\), we see the graph of \(\displaystyle rcos\theta=-2\) will have the same shape as the graph \(\displaystyle x=-2\), or a vertical line at \(\displaystyle x=-2\).

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

Using the identity \(\displaystyle y=rsin\theta\), we see the graph of \(\displaystyle rsin\theta=1\) will have the same shape as the graph \(\displaystyle y=1\), or a horizontal line at \(\displaystyle y=1\).

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

Substituting values of \(\displaystyle \theta\) (in radians) between \(\displaystyle 0\) and \(\displaystyle 2\pi\) into our expression, we find values of r. We then plot each ordered pair, \(\displaystyle (r,\theta)\), using the \(\displaystyle r\) value as the radius and \(\displaystyle \theta\) as the angle.  We get the graph below, a circle centered around \(\displaystyle (2,0)\) with a radius of \(\displaystyle 2\).

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

Substituting values of \(\displaystyle \theta\) (in radians) between \(\displaystyle 0\) and \(\displaystyle 2\pi\) into our expression, we find values of r. We then plot each ordered pair, \(\displaystyle (r,\theta)\), using the \(\displaystyle r\) value as the radius and \(\displaystyle \theta\) as the angle. We get the graph below, a circle centered around \(\displaystyle (0,-3)\) with a radius of \(\displaystyle 3\).

Graphing polar equations is different that plotting cartesian equations.  Instead of plotting an \(\displaystyle (x,y)\) coordinate, polar graphs consist of an \(\displaystyle (r,\theta)\) coordinate where \(\displaystyle r\) is the radial distance of a point from the origin and \(\displaystyle \theta\) 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 \(\displaystyle r=a\pm bcos\theta\) where \(\displaystyle a=b\).  Our cardioid is symmetric about the x-axis because our equation includes the \(\displaystyle cosine\) function  The point of the cardioid is at the origin. The y-intercepts are at \(\displaystyle (0,a)\) and \(\displaystyle (0,-a)\). The x-intercept is at \(\displaystyle (a+b,0)\).

We could also substitute values of \(\displaystyle \theta\) (in radians) between \(\displaystyle 0\) and \(\displaystyle 2\pi\) into our expression, to find values of r. We then plot each ordered pair, \(\displaystyle (r,\theta)\), using the \(\displaystyle r\) value as the radius and \(\displaystyle \theta\) as the angle.  

We get the graph below, a cardioid (heart shape) rotated \(\displaystyle 90^{\circ}\) left.

Graphing polar equations is different that plotting cartesian equations.  Instead of plotting an \(\displaystyle (x,y)\) coordinate, polar graphs consist of an \(\displaystyle (r,\theta)\) coordinate where \(\displaystyle r\) is the radial distance of a point from the origin and \(\displaystyle \theta\) 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 \(\displaystyle r=a\pm bsin\theta\) where \(\displaystyle a=b\).  Our cardioid is symmetric about the y-axis because our equation includes the \(\displaystyle sine\) function. The point of the cardioid is at the origin.  The x-intercepts are at \(\displaystyle (a,0)\) and \(\displaystyle (-a,0)\).  The y-intercept is at \(\displaystyle (0,a+b)\).

We could also substitute values of \(\displaystyle \theta\) (in radians) between \(\displaystyle 0\) and \(\displaystyle 2\pi\) into our expression, to find values of r.  We then plot each ordered pair, \(\displaystyle (r,\theta)\), using the \(\displaystyle r\) value as the radius and \(\displaystyle \theta\) 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 \(\displaystyle (x,y)\) coordinate, polar graphs consist of an \(\displaystyle (r,\theta)\) coordinate where \(\displaystyle r\) is the radial distance of a point from the origin and \(\displaystyle \theta\) 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 \(\displaystyle r=a\pm bsin\theta\) where \(\displaystyle a>b\). This limacon will have no loop because \(\displaystyle a/b>1\). Our limacon is symmetric about the y-axis because our equation includes the \(\displaystyle sine\) function.  The x-intercepts are at \(\displaystyle (a,0)\) and \(\displaystyle (-a,0)\).  The y-intercept is at \(\displaystyle (0,a+b)\).

We could also substitute values of \(\displaystyle \theta\) (in radians) between \(\displaystyle 0\) and \(\displaystyle 2\pi\) into our expression, to find values of r. We then plot each ordered pair, \(\displaystyle (r,\theta)\), using the \(\displaystyle r\) value as the radius and \(\displaystyle \theta\) 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 \(\displaystyle (x,y)\) coordinate, polar graphs consist of an \(\displaystyle (r,\theta)\) coordinate where \(\displaystyle r\) is the radial distance of a point from the origin and \(\displaystyle \theta\) 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 \(\displaystyle r=a\pm bcos\theta\) where \(\displaystyle a>b\).  This limacon will have no loop because \(\displaystyle a/b>1\). Our limacon is symmetric about the x-axis because our equation includes the \(\displaystyle cosine\) function. The y-intercepts are at \(\displaystyle (0,a)\) and \(\displaystyle (0,-a)\).  The x-intercept is at \(\displaystyle -(a+b,0)\).

We could also substitute values of \(\displaystyle \theta\) (in radians) between \(\displaystyle 0\) and \(\displaystyle 2\pi\) into our expression, to find values of r.  We then plot each ordered pair, \(\displaystyle (r,\theta)\), using the \(\displaystyle r\) value as the radius and \(\displaystyle \theta\) as the angle.  

We get the graph below, an limacon turned \(\displaystyle 90^{\circ}\) right.

Graphing polar equations is different that plotting cartesian equations.  Instead of plotting an \(\displaystyle (x,y)\) coordinate, polar graphs consist of an \(\displaystyle (r,\theta)\) coordinate where \(\displaystyle r\) is the radial distance of a point from the origin and \(\displaystyle \theta\) 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 \(\displaystyle r=a\pm bcos\theta\) where \(\displaystyle a< b\). This limacon will have a loop because \(\displaystyle a/b< 1\). The length of the loop is \(\displaystyle b-a\).  Our limacon is symmetric about the x-axis because our equation includes the \(\displaystyle cosine\) function. The y-intercepts are at \(\displaystyle (0,a)\) and \(\displaystyle (0,-a)\).  The x-intercept is at \(\displaystyle (a+b,0)\).

We could also substitute values of \(\displaystyle \theta\) (in radians) between \(\displaystyle 0\) and \(\displaystyle 2\pi\) into our expression, to find values of r. We then plot each ordered pair, \(\displaystyle (r,\theta)\), using the \(\displaystyle r\) value as the radius and \(\displaystyle \theta\) as the angle.  

We get the graph below, a limacon with a loop turned \(\displaystyle 90^{\circ}\) left.

Graphing polar equations is different that plotting cartesian equations. Instead of plotting an \(\displaystyle (x,y)\) coordinate, polar graphs consist of an \(\displaystyle (r,\theta)\) coordinate where \(\displaystyle r\) is the radial distance of a point from the origin and \(\displaystyle \theta\) 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 \(\displaystyle r=a\pm bsin\theta\) where \(\displaystyle a< b\).  This limacon will have a loop because \(\displaystyle a/b< 1\). The length of the loop is \(\displaystyle b-a\).  Our limacon is symmetric about the y-axis because our equation includes the \(\displaystyle sine\) function. The x-intercepts are at \(\displaystyle (a,0)\) and \(\displaystyle (-a,0)\). The y-intercept is at \(\displaystyle -(0,a+b)\).

We could also substitute values of \(\displaystyle \theta\) (in radians) between \(\displaystyle 0\) and \(\displaystyle 2\pi\) into our expression, to find values of r. We then plot each ordered pair, \(\displaystyle (r,\theta)\), using the \(\displaystyle r\) value as the radius and \(\displaystyle \theta\) as the angle.  

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