parametric equations formative (precalc. RV.MP.3.a)
Vocabulary
Parametric equations are sets of equations that express a set of values as explicit functions of independent variables known as parameters.
The equation of a circle in Cartesian coordinates can be given by:
r2 = x2 + y2
where r is the radius.
A set of parametric equations for the same circle is:
x = r cos(t)
y = r sin(t)
A set of parametric equations with a single parameter usually uses parameter t.
For parametric equations with two parameters, the symbols u and v are common.
Surfaces and curves graphed using parametric equations are known as parametric surfaces and parametric curves.
Reference
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: x = r cos(t) & y = r sin(t). Use the following input format: {x = cos(t), y = sin(t)}. 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 interval 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?