Question

Determine any differences between the curves of the parametric equations. Are the graphs the same? Are the orientations the same? Are the curves smooth? $$ \text { (a) } \begin{array}{l} x=\cos \theta \\ y=2 \sin ^{2} \theta \\ 0<\theta<\pi \end{array} $$ $$ \text { (b) } \begin{aligned} x &=\cos (-\theta) \\ y &=2 \sin ^{2}(-\theta) \\ 0 &<\theta<\pi \end{aligned} $$

Short answer

With respect to the given set of values for \(\theta\), the graphs of both sets of parametric equations are the same. They produce the curves of the same ellipse, are oriented from positive \(x\) to negative \(x\) (from right to left), and both are smooth.

Step-by-step solution

Graphing the First Set of Parametric Equations

To start off, recognize that \(x=\cos \theta\) and \(y=2 \sin ^{2} \theta\). The \(x\)-component is a standard cosine function, while the \(y\)-component is a scaled version of the square of sine. As \(\theta\) varies from 0 to \(\pi\), \(\cos \theta\) goes from 1 to -1. However, \(\sin^2(\theta)\) is always non-negative and its maximum value is 1. Hence, \(2sin^2(\theta)\) ranges from 0 to 2. Consequently, the curve described by these paramteric equations will be half an ellipse (open on the right and left), standing on the \(x\)-axis between -1 and 1.

Graphing the Second Set of Parametric Equations

The second set of equations are \(x=\cos (-\theta)\) and \(y=2 \sin^{2} (-\theta)\). Since cosine and sine squares are even functions, this means that \(x=\cos (-\theta)\) = \(\cos \theta\) and \(2 \sin ^{2}(-\theta)\) = \(2 \sin ^{2}\theta\). Hence, the \(x\) and \(y\) values will be the same as the previous set, and therefore the graph will not change.

Comparing the Graphs

Since both sets of parametric equations produce the same graph, then their graphs, orientations, and smoothness are all the same. The curve is smooth as there are no bumps, cusps, or sudden changes in direction. The curve goes from right to left, so its orientation can be described as being from positive \(x\) to negative \(x\).

Key concepts

Graphing Parametric Equations

Graphing parametric equations requires an understanding of how the individual parameters behave as they contribute to the creation of the curve. In a basic sense, a parametric equation expresses the coordinates of a point on a plane as functions of a variable, in many cases represented as \( t \) or \(\theta \). Unlike traditional Cartesian equations, parametric equations define both x and y independently in terms of another variable.

In the exercise, we work with \(\theta \) as our parameter and are given two equations: \(\text{a})\), where \(\begin{x} = \text{cos}\theta \
y = 2 \text{sin}^2\theta \
0<\theta<\text{π}\end{array}\), and \(\text{b})\), which essentially mirrors \(\text{a})\) due to the even properties of the trigonometric functions involved. Mapping these equations involves calculating the x and y coordinates at several values of \(\theta \) and plotting the corresponding points on the graph.

To enhance understanding, it's helpful to consider how the curve behaves as \(\theta \) varies. For example, as \(\theta \) increases within the given range, how do the values of cos and sin influence the resulting shape of the curve? Recognizing this relationship allows for a smoother graphing process and leads to a better grasp of the concepts.

Properties of Even Functions

The properties of even functions are essential in determining the symmetry of a graph and in simplifying complex equations. An even function is one that satisfies \(\text{f}(-x) = \text{f}(x)\) for every x in the function's domain. In our study material, sine squared \((\text{sin}^2)\) and cosine \((\text{cos})\) are highlighted as classic examples of even functions.

Understanding the implications of even functions on parametric equations allows us to simplify the graphing process. For graphing, these properties indicate that the curve will be symmetrical with respect to the y-axis. As seen in the solved exercise, \(x=\text{cos} (-\theta) = \text{cos}\theta\) and \(y = 2 \text{sin}^2(-\theta) = 2 \text{sin}^2\theta\), reinforcing that the graphs of the two given parametric equations will be identical due to the even properties of the sine squared and cosine functions. Recognizing even properties early in the graphing process can prevent redundant calculations and confirm the anticipated symmetry of the curve.

Comparing Parametric Curves

When comparing parametric curves, there are several key aspects to consider: the shape of the graphs, their orientation, and the smoothness of the curves. If the parametric equations are different but produce the identical set of points, the curves will overlay perfectly on one another.

In our exercise, after establishing that the trigonometric functions in both sets are even, we know that the resulting graphs will match point-for-point, confirming their sameness. However, orientation—a curve's direction as the parameter increases—is also crucial. In this case, as \(\theta\) increases, both curves are traversed from right to left, indicating that their orientations are also the same. Lastly, a smooth curve implies a continual, non-interrupted line without sharp turns or cusps. The solved problem affirms that both parametric curves are smooth as well as congruent. These comparisons are vital in many fields, such as physics and computer graphics, where understanding the trajectory and comparing paths is key to their practical application.