Problem 62 - Precalculus: Functions and Graphs

Question

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

Step-by-Step Solution

Answer

The figure consists of two horizontal semi-ellipses connected by a vertical line, resembling an eye shape.

Step 1: Understanding the Parametric Equations for C1

The curve \(C_1\): \(x=\frac{3}{2} \cos t+1\), \(y=\sin t-1\) is a parametric equation where \(t\) varies from \(-\pi/2\) to \(\pi/2\). \(x=\frac{3}{2} \cos t+1\) represents a horizontal stretch and a translation to the right by 1 unit, while \(y=\sin t-1\) represents a vertical shift down by 1 unit. This describes a horizontally stretched and shifted semi-ellipse centered at (1, -1).

Step 2: Understanding the Parametric Equations for C2

The curve \(C_2\): \(x=\frac{3}{2} \cos t+1\), \(y=\sin t+1\) represents an ellipse where \(t\) varies from \(-\pi/2\) to \(\pi/2\). \(x=\frac{3}{2} \cos t+1\) is the same horizontal component as in \(C_1\), while \(y=\sin t+1\) indicates a vertical translation up by 1, making this a semi-ellipse centered at (1, 1).

Step 3: Understanding the Parametric Equations for C3

The curve \(C_3\): \(x=1\), \(y=2 \tan t\) has \(t\) ranging from \(-\pi/4\) to \(\pi/4\). As \(x=1\), this is a vertical line at \(x = 1\), and \(y=2 \tan t\) stretches the standard tangent function vertically by a factor of 2. This segment will be symmetric about the x-axis and extends vertically within \(y\) values of approximately \(-2\) to \(2\).

Step 4: Graphing the three curves

Plot the three curves on the same set of axes:
1. Semi-ellipse \(C_1\) stretching from \((1 - \frac{3}{2}, -2)\) to \((1 + \frac{3}{2}, 0)\).
2. Semi-ellipse \(C_2\) stretching from \((1 - \frac{3}{2}, 0)\) to \((1 + \frac{3}{2}, 2)\).
3. Vertical line \(C_3\) along \(x=1\) from \(y=-2\) to \(y=2\). These curves intersect at points \((1, -1)\) and \((1,1)\), forming the figure of two stacked, horizontally stretched semi-ellipses on a line.

Step 5: Describing the Shape of the Figure

The resulting figure consists of a vertical line segment connected between two horizontally stretched semi-ellipses. The top ellipse is centered at \((1, 1)\) and the bottom at \((1, -1)\), connected by the line \(x=1\) extending from \(y=-2\) to \(y=2\). This forms an ‘eye-like’ shape with the line as the ‘pupil’ and the ellipses as the ‘upper and lower eyelids’.

Key Concepts

Graphing Curves

Graphing curves involves translating abstract mathematical equations into visual representations on a coordinate plane. When dealing with parametric equations, each equation is defined by parameters that allow us to trace curves by varying a parameter, often denoted as "t." This allows us to graph complex shapes like ellipses and other intricate curves that are not easily represented by simple Cartesian equations.

• Parametric equations separate the variables, often representing 'x' and 'y' in terms of a third parameter.
• Visualizing curves helps in understanding the behavior and the relationships described by mathematical functions or equations.

By graphing each of the curves, we can see how they relate to one another, noting intersections and shared characteristics. This step provides a clear, visual tool that can make complex mathematical concepts more understandable.

Semi-Ellipses

A semi-ellipse is half of an ellipse, typically divided along its major or minor axis. In the context of the given parametric equations, the curves described are semi-ellipses lying along the x-axis.
The equations:

• For curve C1: \( x = \frac{3}{2} \cos t + 1 \), \( y = \sin t - 1 \)
• For curve C2: \( x = \frac{3}{2} \cos t + 1 \), \( y = \sin t + 1 \)

These equations portray two semi-ellipses, with the former centered at point (1, -1) and the latter shifted upwards to be centered at (1, 1). The horizontal stretching factor, \( \frac{3}{2} \), changes the width, making them appear more elongated.
Understanding these curves helps in visualizing how transformations can alter the shape and position of ellipses, aiding in developing a deeper grasp of geometric principles.

Vertical Line

A vertical line is a line parallel to the y-axis of a coordinate plane, where all points on the line have the same x-coordinate. In the provided exercise, the curve C3, given by:

• \( x = 1 \)
• \( y = 2 \tan t \)

represents a vertical line at \( x = 1 \). The function \( y = 2 \tan t \) dictates that this line stretches the tangent function vertically, extending from approximately \( y = -2 \) to \( y = 2 \).
Understanding vertical lines in parametric equations allows one to see how varying one parameter affects another, and how complex figures like this eye-like shape are formed when combined with other curves.

Coordinate Plane

The coordinate plane is a two-dimensional plane defined by a horizontal line, called the x-axis, and a vertical line called the y-axis. The intersection of these axes is called the origin, often denoted as (0,0). This plane allows for the plotting and studying of geometric figures and equations in a way that is visually accessible.

• On this plane, any point is given by an ordered pair (x, y) that shows its position relative to the origin.
• The plane is fundamental in graphing, allowing for the visual representation of solutions to equations and systems of equations.

In the graphing of the curves C1, C2, and C3, the coordinate plane serves as the canvas that holds these shapes, allowing us to clearly see how they intersect and interact with each other, forming an eye-like shape.