The following are a view of parametric equations given the range of t from 0 to 2 pi. We will look at the pair of functions noted below as the power is squared, cubed, and raised higher. We will also pay particular attention to a and b, when a is equal to b, when a is greater than b, and when a is less than b for each power indicated.
Notice in the figures given below, the values of a and b are the same and the power of the function is 1 (odd power). The figure is a circle and the values of a and b correspond with the length of the radii.
The following figures are for the pair of equations given below. Here the values of a are either greater than b or less than b, but not equal to b. Notice the figures resulting are that of an ellipse. The value given for a corresponds as an x value and the value for b corresponds as a y value.
The pair of equations below are squared (even power). Now the result becomes a straight line where once again the a value corresponds with x and the b value corresponds with y.
Now look at the pairs of equations below. Here is the cubed case (odd power). We get a closed figure with more of a concave curve to the shape. I think by now we can settle on the idea that the a values match with x and the b values match with y.
Here is the case with the 4th power (even power). Notice that we get a curved line.
The following are equations for you to try. What kind of shape do you think will result for each and where will each value correspond to the x and y axis?
Scroll below to see if your guesses are correct.
1.
2.
3.
CONCLUSION
After exploring graphs of parametric equations of the form there are several conclusions which can be stated:
POWERS
ODD POWER: the resulting graph will be a closed figure.
Power of 1: If a is equal to b, then the figure is a circle. If a is greater than or less than b then the figure is an ellipse.
EVEN POWER: the resulting graph will be a segment or curve connecting from the x and y axis for the values which
correspond with a (x-intercept) and b (y-intercept).
Power of 2: The graph will be a sement with the the x and y intercepts matching with the numerical value of a and b.
INCREASE OF POWER: As the powers increase above 2, the figure(s) take on a curve. The graphs become
increasingly more curved as the value of the power continues to increase
a and b
The values of a correspond with the x-intercepts for each graph.
The values for b correspond with the y-intercepts for each.
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