Sketch the curve by using the parametric equations to plot...

Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as $t$ increases.
$$
x=\cos ^{2} t, \quad y=1+\cos t, \quad 0 \leqslant t \leqslant \pi
$$

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Key Concepts

Parametric Equations

Parametric equations are used to define curves by expressing the coordinates as functions of an independent parameter. This approach allows for the representation of curves that might not be easily described by a single function in Cartesian coordinates, enabling the modeling of more complex shapes and behaviors.

Plotting Points on Parametric Curves

Plotting points involves evaluating the coordinate functions at selected values of the parameter to obtain specific (x, y) points on the curve. This method helps in constructing an accurate sketch of the curve by revealing its shape and key features as the parameter varies.

Direction of Tracing

The direction of tracing refers to the order in which points on the curve are traced as the parameter increases. Indicating the arrow of progression on a curve is important because it shows how the curve is generated and can reveal properties such as loops or oscillations that might not be evident from the static picture alone.

Trigonometric Parametrization

Trigonometric functions, such as sine and cosine, are frequently used in parametrization due to their periodic properties and smooth nature. They are particularly useful for describing curves with repetitive or circular characteristics, providing a convenient way to capture symmetry and periodicity in the curve's structure.

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