8.6 Parametric equations  (Page 2/6)

If $x\left(t\right)=t,$ then to find $y\left(t\right)$ we replace the variable $x$ with the expression given in $x\left(t\right).$ In other words, $y\left(t\right)=t^2-1.$ Make a table of values similar to [link] , and sketch the graph.

See the graphs in [link] . It may be helpful to use the TRACE feature of a graphing calculator to see how the points are generated as $t$ increases.

If $x(t)=t$ and we substitute $t$ for $x$ into the $y$ equation, then $y\left(t\right)=1-t^2.$ Our pair of parametric equations is

To graph the equations, first we construct a table of values like that in [link] . We can choose values around $t=0,$ from $t=-3$ to $t=3.$ The values in the $x(t)$ column will be the same as those in the $t$ column because $x(t)=t.$ Calculate values for the column $y(t).$

The graph of $y=1-t^2$ is a parabola facing downward, as shown in [link] . We have mapped the curve over the interval $[\mathrm{-3},3],$ shown as a solid line with arrows indicating the orientation of the curve according to $t.$ Orientation refers to the path traced along the curve in terms of increasing values of $t.$ As this parabola is symmetric with respect to the line $x=0,$ the values of $x$ are reflected across the y -axis.

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 $\mathrm{-5}$ meters and goes to 3 meters. This means the distance x has changed by 8 meters in 4 seconds, which is a rate of $\frac{\text{8 m}}{4\text{ s}},$ or $2\text{m}/\text{s}.$ We can write the x -coordinate as a linear function with respect to time as $x(t)=2t-5.$ In the linear function template $y=mx+b,2t=mx$ and $-5=b.$

Similarly, the y -value of the object starts at 3 and goes to $\mathrm{-1},$ which is a change in the distance y of −4 meters in 4 seconds, which is a rate of $\frac{-4\text{ m}}{4\text{ s}},$ or $-1\text{m}/\text{s}.$ We can also write the y -coordinate as the linear function $y(t)=-t+3.$ Together, these are the parametric equations for the position of the object, where $x$ and $y$ are expressed in meters and $t$ represents time:

Using these equations, we can build a table of values for $t,x,$ and $y$ (see [link] ). In this example, we limited values of $t$ to non-negative numbers. In general, any value of $t$ can be used.

From this table, we can create three graphs, as shown in [link] .