Problem 72 - Algebra and Trigonometry

Question

How is point plotting used to graph a plane curve described by parametric equations? Give an example with your description.

Step-by-Step Solution

Answer

Point plotting for graphing a plane curve involved by parametric equations involves generating a range of values for a common parameter, substitifying them into the parametric equations to obtain corresponding x and y values, and then plotting and connecting these points on a plane to form the curve. For the example of \(x = t\), \(y = t^2\), the points generated by t values of -2, -1, 0, 1, 2 when plotted and connected form a parabolic curve.

Step 1: Understand Parametric Equations

Parametric equations are a set of equations in which the outputs are represented as functions of one or more independent variables, called parameters. For example, a pair of parametric equations to represent a plane curve might be \(x = t\) and \(y = t^2\), where t is a parameter.

Step 2: Generate Values for the Parameter

Choose some values for the parameter, t. Typically, it's a good idea to choose a range of values, including both positive and negative numbers. For instance, let's take t = -2, -1, 0, 1, 2.

Step 3: Substitute Parameter Values into Equations

Substitute the chosen values of t into both equations to get corresponding x and y values. For our example, when t = -2, x = -2 and y = 4. Repeat this process for all chosen t values.

Step 4: Plot the Points

Now plot the points (x, y) on a plane. For our example these points would be (-2,4), (-1,1), (0,0), (1,1), (2,4). When these points are plotted and connected, they form a curve.

Step 5: Connect the Points to Form the Curve

Connect the points in the order of the parameter values. This will give a curve on the plane. The direction in which the curve is traced as the parameter increases is indicated by an arrow.