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

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

Key Concepts

Parametric Equations

Parametric equations describe curves by expressing the coordinates as functions of a parameter, rather than as a direct relationship between x and y. This method enables the representation of more complex curves, including those that loop or cross themselves, by allowing the separate control of the x and y components of the curve as the parameter varies.

Trigonometric Functions in Parametric Curves

Trigonometric functions like sine and cosine are frequently used in parametric equations to generate periodic or circular patterns. Their inherent periodicity helps model curves with loops, oscillations, or symmetries, making them ideal for generating a wide variety of shapes such as circles, ellipses, and other harmonic curves.

Graph Transformations

Graph transformations involve operations that change the appearance of a curve, such as rotations, translations, reflections, or scaling. In the context of parametric equations, altering coefficients or signs of terms can lead to a reflected or rotated version of the original curve, which affects both the shape and the orientation of the graph.

Effect of Coefficient Sign Change

Changing the sign of a coefficient in a parametric equation, as done by replacing a positive constant with a negative one, constitutes a transformation that often results in a reflection or reversal of the curve. This type of change can alter the direction of traversal, modify the symmetry properties, and produce a graph that is essentially a mirror image of the original, highlighting the sensitivity of parametric representations to coefficient values.

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