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Assignment 12:

Investigating Parametric Equations Using Spreadsheets

by

Ángel M. Carreras Jusino

Goal:

Given parametric equations use a spreadsheet to display their graphs.

Exploration 1.

First let explore parametric equations that we already know its graph.

Consider the parametric equations for the interval

Using a spreadsheet we use one column for the parameter t, one for x, and one for y.

For the parameter we decide to use 20 steps. To determine the step-size we need to know our initial t value, final t value, and the number n of steps that we want, then, we can use the formula .

Then the graph was constructed using a scatter plot.

As expected the graph of this parametric equations is a circle and the plotted points are equally spaced.

 

Exploration 2.

Now lets look to a more complex parametric equations

Consider the parametric equations . We are going to investigate these equations for -5 < t < 5 using 40 steps. Generating a table and a graph like in the previous example we get the following.

Apparently the graphical representation of this parametric equations is an unit circle, but as can be observed the points are not equally spaced, and the circle is not complete. Can be noticed from the table that as t increases the curve slowly approaches the point (-1, 0), and the same happen as t decrease. This can be explained by

Therefore, the complete circle can't be completed using a table based on the parametric equations provided. The equations can be manipulated, working with trigonometric functions, to make it easier to graph them using a spreadsheet.

Let t be tan(θ), then

Now the parametric equations are equivalent to therefore we obtain the parametric equations for a unit circle for 0 ≤ θ ≤ π.

Exploration 3.

Consider the equation . This equation is one difficult to graph with most graphing application, so we are going to parametrize it so it possible to graph it with a spreadsheet. We also need to determine an appropriate range of values for the parameter t.

Using Graphing Calculator we get the following graph for

For the parametrization let y = tx. Then substituting,

The parametric equations for this curve are .

Based on the graph, we can investigate what values of t produce coordinates in the first, second, and fourth quadrant. For this we are going to use a sign diagram.

Let begin with x(t),

t
3t	-	-	+
	-	+	+
x(t)	+	-	+

Now let check for y(t)

t
	+	+	+
	-	+	+
y(t)	-	+	+

From these two sign diagrams we can say that for t ∈ (-∞, -1) the graph lies in the fourth quadrant, for t ∈ (-1, 0) the graph lies in the second quadrant, and for t ∈ (0, ∞) the graph lies in the first quadrant.

t ∈ (-∞, -1)

t ∈ (-1, 0)

t ∈ (0, ∞)

Combining the three graphs we get the graph of the parametric equations.

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