EMAT 6680

Assignment 10: Parametric Curves

Problem #9

Investigate each of the following pairs of parametric equations  for t from 0 to 2 pi.  Describe each when a=b, a<b, and a>b.

Part 1

Graphs of the equations

when a=b, a<b, and a>b.

Key:

a=1,b=1 Purple        a=2,b=2 Red        a=1,b=2 Blue

a=1,b=3 Green        a=2,b=1 Light Blue        a=3,b=1 Yellow

a=b

When a=b, the graph of the parametric equation is a circle. The circles are centered around the origin.  As the value of a and b increases from 1 to 2, the circle increases in size. In the first circle, when a and b are equal to 1, the circle extends to 1 and -1 on the x and y-axis.  In the second circle, the graph extends to 2 and -2 on the x and y-axis.

a<b

When a<b, the graph changes from a circle to an elliptical shape that is centered around the y-axis.  As the value of b increases, the ellipse increases in length but not in width.  On the first ellipse, the graph extends to 1 and -1 on the x-axis and 2 and -2 on the y-axis.  The x-coordinate is equal to a, and the y-coordinate is equal to b.  The same is true for the second ellipse where a=1 and b=3.  This ellipse extends to 1 and -1 on the x-axis and 2 and -2 on the y-axis.

a>b

When a>b, the graph has the same elliptical shape as the previous graphs, but the ellipse is centered around the x-axis instead of the y-axis.  As a increases, the ellipse again increases in length but not in width.  In the first graph, where a=2 and b=1, the ellipse extends to 2 and -2 on the x-axis and 1 and -1 on the y-axis.  When a=3 and b=1, the ellipse extends to 3 and -3 on the x-axis and 1 and -1 on the y-axis.  In both of these graphs, the x-coordinate is equal to a and the y-coordinate is equal to b.

Conclusion:

Therefore we have seen in the first set of parametric equations that when a=b, the graph is a circle.  When a<b, the graph is an ellipse that centers around the y-axis.  When a>b, the graph is an ellipse that centers around the x-axis.  In all of the graphs, as the value of a and b get larger, the graph increases in size.  In all of the graphs, the figure extends to the point on the x-axis that is equal to a and the point on the y-axis that is equal to b.  Therefore, we can conclude that this will be true for all values of a and b.

Part 2

Graph of the equations

when a=b, a<b, and a>b.

Key:

a=1,b=1 Purple        a=2,b=2 Red        a=1,b=2 Blue

a=1,b=3 Green        a=2,b=1 Light Blue        a=3,b=1 Yellow

a=b

When a=b, the graph of the equation is a segment that is linear.  Its path connects the x and y-coordinate on the x and y-axis that is equal to and b.  For example, when a=1 and b=1, the graph connects the points (0,1) and (1,0).  The same is true when a=2 and b=2.  The graph connects the points (0,2) and (2,0).  Therefore, we can conclude that when a and b are equal the graph will be a segment that connects two points on the x and y axis that are equal to a and b.

a<b

When a<b, the graph of the equation is also a linear segment.  Also, like the previous graphs, the graph connects two points on the x and y-axis.  In each of the graphs, a is equal to the x-intercept and b is equal to the y-intercept.  The graph of the segments are not straight across like they were when a=b, but are slanted to where the segment is higher on the y-axis.  This is because b is greater, and b gives us our y-coordinate.

a>b

When a>b, the graph is also a linear segment that is slanted just like the previous graphs where a<b.  This time the graph is further on the x-axis because a is greater than b.  Also, like in the previous graphs a gives us our x-intercept and b gives us our y-intercept.

Conclusion:

In each of the graphs of the parametric equations, the graph is a linear segment that connects two points on the x and y-axis.  In each set of graphs, a is equal to the x-intercept and b is equal to the y-intercept.  Therefore, we can conclude that this will be true for any values of a and b.

Part 3

Graphs of the equations

Key:

a=1,b=1 Purple        a=2,b=2 Red        a=1,b=2 Blue

a=1,b=3 Green        a=2,b=1 Light Blue        a=3,b=1 Yellow

a=b

When a=b, the graph is a inverted square shape where each side is curved inward.  In the first graph, where a and b are equal to 1, the graph extends to 1 and -1 on the x and y-axis.  When a and b are both equal to 2, the graph extends to 2 and -2 on the x and y-axis.

a<b

When a<b, the graph is a inverted rectangle where each side is curved inward.  The graph is similar to the graphs where a=b, except that the graph has been kind of stretched where one side is longer than the other.  Both of the graphs where a<b are centered around the x and y-axis and the part on the y-axis is longer than the part on the x-axis.  When a=1 and b=2, the graph extends to 1 and -1 on the x-axis and 2 and -2 on the y-axis.  When a=1 and b=3, the graph extends to 1 and -1 on the x-axis and 3 and -3 on the y-axis.

a>b

When a>b, the shape of the graph is the same as the graphs where a<b.  The difference is that the longer part here is on the y-axis, not the x-axis.  This is because a gives us our x-coordinates and is larger than b on each of the graphs.  When a=2 and b=1, the graph extends to 2 and -2 on the x-axis and 1 and -1 on the y-axis.  When a=3 and b=1, the graph extends to 3 and -3 on the x-axis and 1 and -1 on the y-axis.

Conclusion:

In each of the graphs, we have a kind of inverted box shape that is centered around the x and y-axis.  When a=b, the graph is a inverted square shape.  When a<b, we have an inverted rectangle shape that is longer on the y-axis because b is larger.  When a>b, we have an inverted rectangle shape that is longer on the x-axis because a is larger.  On all of the graphs of these parametric equations, a gives us the extension of the graph on the x-axis, and b gives us the extension of the graph on the y-axis.

On all of the graphs of these parametric equations, a gives us the extension on the x-axis of our graph, and b gives us the extension of the graph on the y-axis.  We can try the next pair of parametric equations to see if the same is true for its graph.

Graph of the equations

Key:

a=1,b=1 Purple        a=2,b=2 Red        a=1,b=2 Blue

a=1,b=3 Green        a=2,b=1 Light Blue        a=3,b=1 Yellow

In each of these graphs the same is true for the values of a and b as in the previous graphs.  Therefore we can conclude that this will be true for any graph of the parametric equations.

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