Find a rectangular equation for a curve defined parametrically.
Find parametric equations for curves defined by rectangular equations.
Understanding Motion with Parametric Functions
An object travels at a steady rate along a straight path [latex]\left(-5,3\right)[/latex] to [latex]\left(3,-1\right)[/latex] in the same plane in four seconds. The coordinates are measured in meters. Find parametric equations for the position of the object.
The parametric equations are simple linear expressions, but we need to view this problem in a step-by-step fashion. The x-value of the object starts at [latex]-5[/latex] meters and goes to 3 meters. This means the distance x has changed by 8 meters in 4 seconds, which is a rate of [latex]\frac{\text{8 m}}{4\text{ s}}[/latex], or [latex]2\text{m}/\text{s}[/latex]. We can write the x-coordinate as a linear function with respect to time as [latex]x\left(t\right)=2t - 5[/latex]. In the linear function template [latex]y=mx+b,2t=mx[/latex] and [latex]-5=b[/latex].
Similarly, the y-value of the object starts at 3 and goes to [latex]-1[/latex], which is a change in the distance y of −4 meters in 4 seconds, which is a rate of [latex]\frac{-4\text{ m}}{4\text{ s}}[/latex], or [latex]-1\text{m}/\text{s}[/latex]. We can also write the y-coordinate as the linear function [latex]y\left(t\right)=-t+3[/latex]. Together, these are the parametric equations for the position of the object, where [latex]x[/latex] and [latex]y[/latex] are expressed in meters and [latex]t[/latex] represents time:
Using these equations, we can build a table of values for [latex]t,x[/latex], and [latex]y[/latex]. In this example, we limited values of [latex]t[/latex] to non-negative numbers. In general, any value of [latex]t[/latex] can be used.
[latex]t[/latex]
[latex]x\left(t\right)=2t - 5[/latex]
[latex]y\left(t\right)=-t+3[/latex]
[latex]0[/latex]
[latex]x=2\left(0\right)-5=-5[/latex]
[latex]y=-\left(0\right)+3=3[/latex]
[latex]1[/latex]
[latex]x=2\left(1\right)-5=-3[/latex]
[latex]y=-\left(1\right)+3=2[/latex]
[latex]2[/latex]
[latex]x=2\left(2\right)-5=-1[/latex]
[latex]y=-\left(2\right)+3=1[/latex]
[latex]3[/latex]
[latex]x=2\left(3\right)-5=1[/latex]
[latex]y=-\left(3\right)+3=0[/latex]
[latex]4[/latex]
[latex]x=2\left(4\right)-5=3[/latex]
[latex]y=-\left(4\right)+3=-1[/latex]
From this table, we can create three graphs.
(a) A graph of [latex]x[/latex] vs. [latex]t[/latex], representing the horizontal position over time. (b) A graph of [latex]y[/latex] vs. [latex]t[/latex], representing the vertical position over time. (c) A graph of [latex]y[/latex] vs. [latex]x[/latex], representing the position of the object in the plane at time [latex]t[/latex].
Analysis of the Solution
Again, we see that when the parameter represents time, we can indicate the movement of the object along the path with arrows.
A robot vacuum moves in a straight line from position [latex]\left(2, 1\right)[/latex] to position [latex]\left(8, 4\right)[/latex] in 3 seconds. Coordinates are measured in feet. Find parametric equations for the robot’s position.
