68 Parametric Equations: Graphs
Learning Objectives
In this section you will:
- Graph plane curves described by parametric equations by plotting points.
- Graph parametric equations.
It is the bottom of the ninth inning, with two outs and two men on base. The home team is losing by two runs. The batter swings and hits the baseball at 140 feet per second and at an angle of approximately[latex]\,45°\,[/latex]to the horizontal. How far will the ball travel? Will it clear the fence for a game-winning home run? The outcome may depend partly on other factors (for example, the wind), but mathematicians can model the path of a projectile and predict approximately how far it will travel using parametric equations. In this section, we’ll discuss parametric equations and some common applications, such as projectile motion problems.
Graphing Parametric Equations by Plotting Points
In lieu of a graphing calculator or a computer graphing program, plotting points to represent the graph of an equation is the standard method. As long as we are careful in calculating the values, point-plotting is highly dependable.
How To
Given a pair of parametric equations, sketch a graph by plotting points.
- Construct a table with three columns:[latex]\,t,x\left(t\right),\text{and}\,\,y\left(t\right).[/latex]
- Evaluate [latex]x[/latex] and [latex]y[/latex] for values of [latex]t[/latex] over the interval for which the functions are defined.
- Plot the resulting pairs[latex]\,\left(x,y\right).[/latex]
Sketching the Graph of a Pair of Parametric Equations by Plotting Points
Sketch the graph of the parametric equations [latex]x\left(t\right)={t}^{2}+1,\,\,y\left(t\right)=2+t.[/latex]
[hidden-answer a=”fs-id1165137938383″]
Construct a table of values for[latex]\,t,x\left(t\right),\,[/latex]and[latex]\,y\left(t\right),\,[/latex]as in (Figure), and plot the points in a plane.
| [latex]t[/latex] | [latex]x\left(t\right)={t}^{2}+1[/latex] | [latex]y\left(t\right)=2+t[/latex] |
|---|---|---|
| [latex]-5[/latex] | [latex]26[/latex] | [latex]-3[/latex] |
| [latex]-4[/latex] | [latex]17[/latex] | [latex]-2[/latex] |
| [latex]-3[/latex] | [latex]10[/latex] | [latex]-1[/latex] |
| [latex]-2[/latex] | [latex]5[/latex] | [latex]0[/latex] |
| [latex]-1[/latex] | [latex]2[/latex] | [latex]1[/latex] |
| [latex]0[/latex] | [latex]1[/latex] | [latex]2[/latex] |
| [latex]1[/latex] | [latex]2[/latex] | [latex]3[/latex] |
| [latex]2[/latex] | [latex]5[/latex] | [latex]4[/latex] |
| [latex]3[/latex] | [latex]10[/latex] | [latex]5[/latex] |
| [latex]4[/latex] | [latex]17[/latex] | [latex]6[/latex] |
| [latex]5[/latex] | [latex]26[/latex] | [latex]7[/latex] |
The graph is a parabola with vertex at the point[latex]\,\left(1,2\right),[/latex]opening to the right. See (Figure).[/hidden-answer]
Analysis
As values for[latex]\,t\,[/latex]progress in a positive direction from 0 to 5, the plotted points trace out the top half of the parabola. As values of[latex]\,t\,[/latex]become negative, they trace out the