parametric equations formative (precalc. RV.MP.3.a)

Vocabulary

[b]Parametric equations[/b] are sets of equations that express a set of values as explicit functions of independent variables known as [b]parameters.[/b][br][br]The equation of a circle in Cartesian coordinates can be given by:[br]r[sup]2[/sup] = x[sup]2[/sup] + y[sup]2[br][/sup]where r is the radius.[br][br]A set of parametric equations for the same circle is:[br]x = r cos(t) [br]y = r sin(t)[br][br]A set of parametric equations with a [b]single parameter[/b] usually uses parameter [i]t[/i].[br][br]For parametric equations with [b]two parameters[/b], the symbols [i]u[/i] and [i]v[/i] are common.[br][br]Surfaces and curves graphed using parametric equations are known as [b]parametric surfaces[/b] and [b]parametric curves[/b].[br][br]Reference[br]Weisstein, Eric. W., et al. Wolfram MathWorld. (2022). https://mathworld.wolfram.com/ParametricEquations.html

#1) GRAPH THE CARTESIAN EQUATION: r^2 = x^2 + y^2 ... use the slider to experiment with the value of r.

#2) GRAPH THE SET OF PARAMETRIC EQUATIONS using Desmos:[br]x = r cos(t) & y = r sin(t).[br]Use the following input format: {x = cos(t), y = sin(t)}.[br]What happens when you change the value of r?

#3) Using the GeoGebra 2D "curve command" input format "Curve( <Expression>, <Expression>, <Parameter Variable>, <Start Value>, <End Value> )", graph the parametric set: Curve(2 cos(t), 2 sin(t), t, 0, 2π)

#4) Using the GeoGebra 3D "curve command" input format "Curve(<Expression>, <Expression>, <Expression>, <Parameter Variable>, <Start Value>, <End Value>)", graph the parametric set: Curve(cos(t), sin(t), t, t, 0, 10π).

#5) The [b]interval [/b]is defined by the last two entries in the formula (start value and end value). For the 3D curve graphed above in #4, what is the interval of the spiral?

#6) Using the GeoGebra 3D "curve command" input format "Curve(<Expression>, <Expression>, <Expression>, <Parameter Variable>, <Start Value>, <End Value>)", graph the parametric set: Curve(cos(t), sin(t), t, t, -10π, 10π).

#7) What is the interval of the 3D curve graphed above in #6?

#8) Using the same input formula as above, graph the parametric set: x(t) = (2 cos t + cos(2t)) and y(t) = (2 sin t - sin(2t)). (The shape is called a "tricuspoid.")