Problem 61 - Precalculus: Functions and Graphs

Question

Graph the given curves on the same coordinate plane, and describe the shape of the resulting figure. $$\begin{aligned} &C_{1}: x=2 \sin 3 t, \quad y=3 \cos 2 t, \quad-\pi / 2 \leq t \leq \pi / 2\\\ &C_{2}: x=\frac{1}{4} \cos t+\frac{3}{4}, \quad y=\frac{1}{4} \sin t+\frac{3}{2} ; \quad 0 \leq t \leq 2 \pi\\\ &C_{3}: x=\frac{1}{4} \cos t-\frac{3}{4}, \quad y=\frac{1}{4} \sin t+\frac{3}{2} ; \quad 0 \leq t \leq 2 \pi\\\ &C_{4}: x=\frac{3}{4} \cos t, \quad y=\frac{1}{4} \sin t, \quad 0 \leq t \leq 2 \pi\\\ &C_{5}: x=\frac{1}{4} \cos t, \quad y=\frac{1}{8} \sin t+\frac{3}{4} ; \quad \pi \leq t \leq 2 \pi \end{aligned}$$

Step-by-Step Solution

Answer

Graph forms an abstract figure with Lissajous-like curves and small circles, resembling interwoven loops.

Step 1: Analyze Curve C1

The parametric equations for curve \( C_1 \) are \( x = 2 \sin(3t) \) and \( y = 3 \cos(2t) \) for \( -\frac{\pi}{2} \leq t \leq \frac{\pi}{2} \). This curve represents a Lissajous figure, a complex oscillating path based on sine and cosine. The distinct multiplication factors in the arguments of the sine and cosine functions indicate that it would have an elliptic shape, but not aligned with the axes.

Step 2: Analyze Curve C2

The equations for curve \( C_2 \) are \( x = \frac{1}{4} \cos(t) + \frac{3}{4} \) and \( y = \frac{1}{4} \sin(t) + \frac{3}{2} \), defined for \( 0 \leq t \leq 2\pi \). These parametric equations describe a circle centered at \( (\frac{3}{4}, \frac{3}{2}) \) with radius \( \frac{1}{4} \). The circle is far right of the origin and quite high up.

Step 3: Analyze Curve C3

For curve \( C_3 \), the equations are \( x = \frac{1}{4} \cos(t) - \frac{3}{4} \) and \( y = \frac{1}{4} \sin(t) + \frac{3}{2} \), also for \( 0 \leq t \leq 2\pi \). This curve is a circle, like \( C_2 \), but centered at \( (-\frac{3}{4}, \frac{3}{2}) \). This circle mirrors \( C_2 \) across the y-axis.

Step 4: Analyze Curve C4

The parametric equations for curve \( C_4 \) are \( x = \frac{3}{4} \cos(t) \) and \( y = \frac{1}{4} \sin(t) \) for \( 0 \leq t \leq 2\pi \). These describe an ellipse centered at the origin. The semi-major axis is \( \frac{3}{4} \) aligned with the x-axis, and the semi-minor axis is \( \frac{1}{4} \) aligned with the y-axis, creating a horizontal ellipse.

Step 5: Analyze Curve C5

The equations for curve \( C_5 \) are \( x = \frac{1}{4} \cos(t) \) and \( y = \frac{1}{8} \sin(t) + \frac{3}{4} \) for \( \pi \leq t \leq 2\pi \). This curve is a part of an ellipse centered vertically shifted up to \( y = \frac{3}{4} \), with a semi-major axis along the x-axis of \( \frac{1}{4} \) and a semi-minor axis on the y-axis of \( \frac{1}{8} \). It's a small part at the lower vertical section.

Step 6: Combine and Graph the Curves

Plotting all curves on the same graph: \( C_1 \) adds complexity with its Lissajous-like shape centered around the origin. \( C_2 \) and \( C_3 \) contribute two small circles high above and to the right and left respectively. \( C_4 \) gives a horizontal ellipse centered at the origin. \( C_5 \) adds a small ellipse segment vertically above. Together, the figure consists of complex curves and small circles potentially creating a composite, abstract figure.

Step 7: Describe the Shape

The combined image on the graph from the plotted curves is an abstract shape formed from oscillatory and circular components. It resembles a complex arrangement of ellipses and circles, combining symmetry and varying centers to produce an artistic figure that may look like interwoven loops or petals.

Key Concepts

Lissajous Figures

Lissajous figures are fascinating curves that emerge when you plot two parametric equations together, often involving trigonometric functions such as sine and cosine with different frequencies. These curves are known for their intricate and oscillating patterns, which can appear as loops, figure-eights, or complex knot-like shapes.

The general parametric form for a Lissajous figure is:

• \( x(t) = A \sin(a t + \delta) \)
• \( y(t) = B \cos(b t) \)

where:

• \( A \) and \( B \) determine the amplitude or size of the figure in x and y directions, respectively.
• \( a \) and \( b \) are frequencies, indicating how many oscillations occur per unit of time for each axis.
• \( \delta \) is a phase shift that can rotate or change the alignment of the figure.

In the case of curve \( C_1 \) from the exercise, with \( x = 2 \sin(3t) \) and \( y = 3 \cos(2t) \), the different frequencies (3 and 2) result in the shape having an elliptic path. This difference in frequency is key to achieving the unique, non-repetitive pattern characteristic of Lissajous curves.

Parametric Curves

Parametric curves are a way to represent curves in the coordinate plane where both x and y coordinates are expressed as functions of a third variable, often \( t \), which is called the parameter. This contrasts with the usual Cartesian equations where x and y are directly related.

The beauty of parametric equations lies in their flexibility. They allow us to describe a wide variety of shapes that can be hard to represent using standard equation forms. For example, in this exercise, each of the curves \( C_2 \), \( C_3 \), \( C_4 \), and \( C_5 \) is parametrized over different intervals of \( t \), allowing for precise tracing of shapes like circles and ellipses. A parametric form can show paths that loop, oscillate, or even backtrack, something not typically achievable with standard y=f(x) forms.

Parametric equations can simplify computations and graphical interpretations of curves in motion or those defined by angles or periods, as often seen in mechanics and physics. They are a powerful tool for constructing complex shapes, especially in animations and computer graphics. By manipulating an additional parameter \( t \), we can align, rotate, and even distort curves in various inventive ways.

Ellipse and Circle Graphs

Ellipses and circles are simple yet prevalent shapes that commonly appear in geometry and various scientific applications. When described as parametric equations, they often reveal their smooth and uniform nature more clearly.

For a circle, the parametric equations can be expressed in a form such as:

• \( x(t) = r \cos(t) \)
• \( y(t) = r \sin(t) \)

where \( r \) is the radius, and \( t \) varies from 0 to \( 2\pi \). This example is clear in \( C_2 \) from the exercise, which illustrates a circle centered at a point away from the origin.

Ellipses use a similar form but allow different radii for the horizontal and vertical spreads:

• \( x(t) = a \cos(t) \)
• \( y(t) = b \sin(t) \)

Here \( a \) and \( b \) are semi-major and semi-minor axes, respectively. Unlike circles, ellipses are stretched either horizontally or vertically, providing a more elongated shape.

Ellipses and circles not only have aesthetic appeal but also feature in the natural world, from planetary orbits to atomic structures. Their smooth curves represent uniform motion or symmetry, which makes them fundamental in physics and engineering. Understanding parametric representation is crucial since you're often dealing with multivariable scenarios where circular and elliptical paths are natural outcomes of the system dynamics.