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Polar equations are mathematical expressions that describe curves using the polar coordinate system. Unlike the Cartesian coordinate system, which is based on x and y coordinates, polar coordinates describe a point by its distance from a reference point (the pole or origin) and an angle from a reference direction (usually the positive x-axis).

The equation in the exercise, \(r=\dfrac{2}{2-\cos \theta}\), uses the variables \(r\) for the radius and \(\theta\) for the angle to define the curve. In polar equations for conic sections, the typical formula for a conic is \(r = \dfrac{ed}{1 \pm e\cos(\theta)}\) or \(r = \dfrac{ed}{1 \pm e\sin(\theta)}\). Here, \(e\) is the eccentricity, and \(d\) is the distance from the pole to the directrix. Understanding these variables and how they come together allows us to identify types of conics.

Eccentricity is a key parameter in understanding conic sections. It specifies the degree of deviation of a conic section from being circular. The eccentricity \(e\) is a non-negative number that can determine the specific type of conic:

• If \(e = 0\), the conic is a circle.
  
• If \(e\) is between 0 and 1, the conic is an ellipse.
  
• If \(e = 1\), the conic is a parabola.
  
• If \(e > 1\), the conic is a hyperbola.
  

In our exercise, the eccentricity \(e = 1\), which indicates that the conic section is a parabola. This is a crucial step in analyzing the type of graph that will be plotted from a polar equation.

A parabola is a U-shaped curve that has some unique properties. It has exactly one axis of symmetry and one vertex, which is the point at which the parabola changes direction. Parabolas can open upwards, downwards, left, or right depending on the orientation of the equation.

In the context of conic sections described using polar coordinates, a parabola occurs when the equation takes the form \(r = \dfrac{ed}{1 - e\cos(\theta)}\) or something similar, and the eccentricity \(e = 1\). The parabola's directrix (a guiding line outside of the parabola that helps in its construction) is crucial in its formation and depends on the other parameters of the equation. The vertex of the parabola is always equidistant from the directrix and the focus.

Graphing conics involves plotting equations of conic sections, such as ellipses, parabolas, and hyperbolas. Each conic section has unique graphing rules based on its equation and parameters like the eccentricity and directrix.

For our specific case, knowing that our polar equation represents a parabola, we need to focus on its orientation and location. The vertex of this parabola will be on the horizontal axis, with the curve opening to the right when graphed. This directional opening is due to the nature of its directrix being vertical to the left of the vertex.

It's important to start by identifying the general shape and orientation through the equation before graphing. By analyzing the polar form and computing key characteristics like the vertex and the axis, the conic can be accurately sketched.