For Activity 1.1.1, we will examine the parametric equations given by \(x(t)=t-2\) and \(y(t)=4-t^2\) and the corresponding graph.
1.
In the table given on the left of Figure 1.1.1, you should enter at least seven values into the \(t\) column. How are the numbers you are entering relating to the points being graphed?
Figure1.1.1.A Desmos Example for use with Activity 1.1.1
Hint.
Use both positive and negative numbers for \(t\text{,}\) such as \(-3,-2,-1,0,1,2,3\text{.}\) You should also consider values that are not integers.
2.
Explain what shape you think the points given by these parametric equations will draw. Be sure to note any particular features of that graph that you can identify from examining your set of points in Figure 1.1.1.
3.
You should draw your seven points by hand on a set of axes and label each point with its corresponding \(t\) value. If you think of \(t\) as time, the value of \(t\) shows you how an object would move along this curve over time. This idea is often referred to as orientation.
Sometimes we can revert back to equations in terms of \(x\) and \(y\) by eliminating the parameter\(t\text{.}\) Notice that for the equation above we have
Substitute the result for \(t\) from the last part into the equation \(y=4-t^2\) to get an equation in terms of just \(x\) and \(y\text{.}\) Be sure to write down your equation and specify what shape the graph of this equation will have.
(c)
Graph the equation in \(x\) and \(y\) from the previous part using the input box that starts with \(y=\) (in Figure 1.1.1). Does your equation in \(x\) and \(y\) give you the same picture as the parametric equations? Be sure to explain any difference between your earlier prediction and what you see in the \(xy\)-graph.
