Graphing Parametric Equations

Parametric equations provide a different way of representing curves and functions in the Cartesian plane. Instead of expressing the relationship between x and y directly, parametric equations define x and y as separate functions of a third variable called the parameter. This allows for a more flexible and dynamic representation of curves and allows us to explore a wide range of shapes and patterns.

Plotting Points and Analyzing Symmetry

To graph parametric equations, we can plot points by substituting different values of the parameter into the equations. By choosing a range of values for the parameter, we can generate a set of corresponding points and connect them to form a curve. Let's consider an example:

Example 1: Plot the parametric equations x = 2t and y = t^2 - 1 for t ∈ [-2, 2].

To plot this curve, we can choose several values for t within the given range and substitute them into the equations:

For t = -2: x = 2(-2) = -4 and y = (-2)^2 - 1 = 3 For t = -1: x = 2(-1) = -2 and y = (-1)^2 - 1 = 0 For t = 0: x = 2(0) = 0 and y = (0)^2 - 1 = -1 For t = 1: x = 2(1) = 2 and y = (1)^2 - 1 = 0 For t = 2: x = 2(2) = 4 and y = (2)^2 - 1 = 3

Plotting these points on a graph, we can connect them to form a curve. In this case, we obtain a parabolic shape as shown below:

Analyzing the graph, we can observe that it is symmetric with respect to the y-axis. This symmetry indicates that for every point (x, y) on the curve, the point (-x, y) is also on the curve.

Domain and Range

The domain and range of a parametric curve are determined by the values of the parameter. In the example above, we defined our parameter t to be within the range [-2, 2]. Therefore, the domain of the curve is the set of x-values obtained from the parametric equations, which in this case is [-4, 4]. The range is the corresponding set of y-values, in this case, [-1, 3].

Special Cases of Parametric Equations

In addition to plotting general parametric equations, there are special cases that produce familiar geometric shapes. Some examples include:

1. Circle: For a circle of radius r centered at the origin, the parametric equations are: x = r * cos(t) y = r * sin(t)

   where t ranges from 0 to 2π. By varying the parameter t, we can plot points along the circumference of the circle.

2. Ellipse: To graph an ellipse centered at the origin with semi-major axis a and semi-minor axis b, we use the parametric equations: x = a * cos(t) y = b * sin(t)

   where t ranges from 0 to 2π. Again, by changing t, we can plot points on the ellipse.

These special cases provide a useful way to represent curves with parametric equations and allow us to explore various shapes and forms.