As we look at parametric equations and parametric curves, we explore the following equations and possible transformations using Graphing Calculator 3.5.
When graphing these equations in Graphing Calculator 3.5, we use the 2-vector function, so the equations entered in the program shows up as:
Graphing Calculator automatically assumes the range of t is from 0 to 1 and the following graph is shown:
We can conveniently change the range by changing the equation to:
Increasing the range results in the following graph:
By using the above equation and graph as a parent function, we can now look at various transformations using the 2-vector format to explore other graphs:
The above equations show an addition and subtraction outside of the trigonometric functions.† The addition of 1 to the x equation results in the circle shifting 1 unit in the positive direction along the x-axis, while the subtraction of 2 to the y equation results in the circle shifting 2 units in the negative direction along the y-axis.
What if we add a number inside the trigonometric function?
By adding 1 inside the cosine function, the circle is squeezed into the following shape:
Letís add 1 to the sine function:
By adding 1 inside the sine function, the circle is squeezed in the opposite direction:
By increasing the addend, the circle is continually squeezed and only returns to its normal circular shape when the addend is equal to pi or a multiple of pi.
What happens when we multiply one of the functions by a number?
By multiplying the x-equation by 2, the circle now takes on an elliptical shape with the distance from the center to each vertex on the x-axis being 2.
By multiplying the y-equation by 3, the circle takes on an elliptical shape with the distance from the center to each vertex on the y-axis being 3