If you have a parametric equation grapher, graph the...

If you have a parametric equation grapher, graph the equations over the given intervals.
$$x=3 \cos t+\cos 3 t, \quad y=3 \sin t-\sin 3 t ; \quad 0 \leq t \leq 2 \pi$$
What happens if you replace 3 with -3 in the equations for $x$ and
$y ?$ Graph the new equations and find out.

Key Concepts

Trigonometric Functions in Parametric Equations

Trigonometric functions like sine and cosine are commonly used in parametric equations because of their periodic and oscillatory nature. Adjusting the coefficients of these functions influences the amplitude, phase, and overall dynamics of the curve, allowing for a wide range of behaviors and symmetries to be modeled in graphical form.

Transformation of Parametric Equations

Transformations in parametric equations, such as changing the sign of coefficients, can result in reflections, rotations, or other modifications of the curve. Such modifications alter the graph's orientation and symmetry without necessarily changing the fundamental shape, demonstrating the powerful impact of algebraic changes on geometric representations.

Parametric Equations

Parametric equations describe curves by defining both x and y as functions of a third variable (or parameter), which allows the representation of curves that may not be easily expressed as a single function y = f(x). This method is useful for capturing more complex movements and shapes in the plane.

Graphing Parametric Curves

Graphing parametric curves involves plotting the set of points (x(t), y(t)) for a range of the parameter. This process not only reveals the shape of the curve but also provides insight into the direction and orientation of the curve, which can be essential for understanding its behavior.

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