More on Graphing Parametrics

The textbook presents the following parametric equations. x = 2 + t
y = 3 – t² These equations are followed by a table that gives t values ranging from ^–3 to 3, in steps of 1.

t	^–3	^–2	^–1	0	1	2	3
x	^–1	0	1	2	3	4	5
y	^–6	^–1	2	3	2	^–1	^–6

This is followed by the statement "The graph is a parabola", along with a graph of a parabola. However, the table only has seven sets of values, and a graph of those seven ordered pairs would appear as

The seven points graphed above certainly lie on a parabola. We could generate more points if we had expanded the table to have t increase in steps of 1/4, thus generating 25 points. Such points are graphed as

This is starting to look like a parabola. We could increase the number of points by making t increase in steps of 0.05 to generate the graph

This graph has even more points, but it just covers the x values between ^–3 and 3. In order for us to have the calculator generate a parabola similar to the one shown in the text we will need to have the value of t start at about ^–4 and increase up to about 4. Then, with the step increase set for 0.05, the calculator would produce the graph

This is indeed similar to the graph given in the text.

The text goes on to show that for the two parametric equations given above, we can eliminate the parameter and produce the function

y = ^–x² + 4x – 1 However, the graph of that function should be given as

Note the difference between the last two graphs. The parametric equation graph seems thicker in some places, and jagged at others. The graph of the function is the usual smooth, well-defined parabola, although it appears here as a series of disconnected dots. Why is there a difference?

When graphing the function, the calculator uses the x-values associated with the middle of each pixel moving across the entire screen. As a consequence, each column of pixels has at most one pixel that is turned "black" because it is associated with the corresponding y-value. The calculator used here was set to "DrawDot". Therefore, the graph is made up of one dot per pixel-column for this function. On the other hand, the parametric graph lets t run from ^–4 to 4 in steps of 0.05. For each t-value the calculator produces an x-value and a y-value. Then, the calculator identifies the appropriate pixel that contains the ordered pair (x,y). In doing this it is entirely possible that two successive t-values would produce x values that are associated with the same column of pixels, but where the corresponding y-values are associated with different rows of pixels. Thus, in the parametric graph, within a column of pixels we may find two pixels turned "black". It is worth noting that this graph was also made with the "DrawDot" setting. The large number of t-values means that the graph looks like a connected series of dots.

For completeness, the following graph was made for the function

y = ^–x² + 4x – 1 but with the setting "DrawLine" turned on.