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Parametric Equations

By: Amanda Sawyer

 

In this assignment, we are asked to investigate each of the following equations from 0 to 2 pi and describe each of the following when A = B, A < B, and A > B.

Description: equations1.tiff

 

Letís consider each of the three situations A) when A = B, B) when A < B and C) when A > B.

 

A)     When A = B

 

Letís consider the case when A = B and investigate what happens.Letís study the following equations and the graphs with the following colors.

 

Description: equations.tiff

To investigate this behavior, letís look at the graphs of the above equations.We will study the effects of the graph when the value of A=B is one, two, four and sixteen.†††††††††††††††††††††††

 

VALUE of A=B

GRAPH

A=B=1

Description: eq1.tiff

A=B=2

Description: eq2.tiff

A=B=4

Description: eq4.tiff

A=B=16

Description: eq16.tiff

 

We can see from the following pictures that the shape of the graph does not change when the value of A=B changes, rather the distance each shape is on the x and y axis changes to the selected value of A=B.Therefore, we have shown that the value of the distance on the x and y axis is dependent on the value of A and B.Also, the parametric equations with even exponents always are located in the first quadrant while the parametric equations with odd exponents complete the circuit and exist in all four quadrants.

 

B)     When A > B

 

Next letís consider what happens when A > B.For the following graphs, we will keep the same four equations and colors of the graphs as above.Letís investigate the behavior of each of the graphs when the values of A and B are A=1 and B = 1.5, A=1 and B =2, A=1 and B = 4, A=1 and B=16.

 

VALUE of A=B

GRAPH

A=1 and B =1.5

Description: b1.5.tiff

A=1 and B=2

Description: b2.tiff

A=1 and B= 4

Description: b4.tiff

A=1 and B=16

Description: b16.tiff

 

This time our shape did change, but it is clear that that the A value represents the solution on the x axis and the B value represents a solution on the y axis.For example when we set A=1 and B =4, this shows us that (1,0) is a solution for all four equations and (0,4) is a solution for all four equations.It also gives us the relative maximum values for all four equations.For example in the case where A= 1 and B = 16, the relative maximum value of each equation on the domain was at (1,0) and the relative maximum value on the range was at (0,16). Continually, the parametric equations with even exponents remain in the first quadrant while the parametric equations with odd exponents complete the circuit and exist in all four quadrants.

 

C)      When A > B

 

Our final case that we would like to study is when A > B.Again, let us keep the same four equations with the same colored graphs for each of our pictures.Letís consider the case when A=1.5 and B=1, A=2 and B=1, A= 4 and B= 1, and A=16 and B = 1.

 

VALUE of A=B

GRAPH

B=1 and A =1.5

Description: a1.5.tiff

B=1 and A=2

Description: a2.tiff

B=1 and A= 4

Description: a4.tiff

B=1 and A=16

Description: a16.tiff

 

 

As one might expect, similar things continue to happen. Like we say in the case where A < B, we can clearly see now that the A value does effect the x coordinates and the B value effects the y coordinates.For example when A=2 and B=1, we see a maximum x-value of each of the four equations is at (2,0) and the maximum y-values of each of the four equations is at (0,1).As we expect, the parametric equations with even exponents remain in the first quadrant while the parametric equations with odd exponents complete the circuit and exist in all four quadrants.

 

Through the use of Graphing Calculator, these properties are easy to observe, and it allows us to make obvious generalizations about the properties of these trigonometric functions.

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