Problem 31 - Precalculus: Functions and Graphs

Question

(a) Describe the graph of a curve \(C\) that has the parametrization $$x=3+2 \sin t, \quad y=-2+2 \cos t ; \quad 0 \leq t \leq 2 \pi$$ (b) Change the parametrization to $$x=3-2 \sin t, \quad y=-2+2 \cos t, \quad 0 \leq t \leq 2 \pi$$ and describe how this changes the graph from part (a). (c) Change the parametrization to $$x=3-2 \sin t, \quad y=-2-2 \cos t, \quad 0 \leq t \leq 2 \pi$$ and describe how this changes the graph from part (a).

Step-by-Step Solution

Answer

(a) Circle centered at (3, -2) with radius 2. (b) Same circle, traced counter-clockwise. (c) Same circle, traced clockwise.

Step 1: Understanding the Parametrization for Part (a)

The given parametric equations are \( x = 3 + 2 \sin t \) and \( y = -2 + 2 \cos t \). Recognizing the form \( x = h + a\sin t \) and \( y = k + b\cos t \), we identify that this describes an ellipse centered at \((h, k) = (3, -2)\) with semi-major axis length of 2 in the \(x\)-direction and semi-minor axis length of 2 in the \(y\)-direction. This means the ellipse is actually a circle as both axes are of equal length.

Step 2: Identifying Shape for Part (a)

Since the parametric equations lead to \( (x - 3)^2 + (y + 2)^2 = 4 \), which is the equation of a circle with center \((3, -2)\) and radius 2, we deduce that the curve \(C\) in part (a) is a circle.

Step 3: Changing the Parametrization for Part (b)

The parametrization changes to \( x = 3 - 2 \sin t \) and \( y = -2 + 2 \cos t \). This change affects the \(x\)-coordinate equation, which now becomes \( x = 3 - 2 \sin t \), but the \(y\)-coordinate remains unchanged. Geometrically, reversing the sign of \(\sin t\) inverts the ellipse across the vertical line \(x = 3\). However, because the ellipse is a circle, the appearance remains the same—it is effectively the same circle, just traced in a different manner.

Step 4: Describing Effect of Change in Part (b)

The circle as described by \( x = 3 - 2 \sin t \) is identical to the circle in part (a) but traced in the opposite direction, counter-clockwise instead of clockwise.

Step 5: Changing the Parametrization for Part (c)

For this parametrization, \( x = 3 - 2 \sin t \) and \( y = -2 - 2 \cos t \). Here, both \(\sin t\) and \(\cos t\) have had their signs reversed. This will invert the circle both horizontally (across the line \(x = 3\)) and vertically (across the line \(y = -2\)). However, due to symmetry, this results in the same circle, but again traced in a different direction.

Step 6: Describing Effect of Change in Part (c)

The circle in part (c) is the same as in parts (a) and (b) but traced in a clockwise direction. It is effectively the same circle as in parts (a) and (b), because the curve's shape and center do not change with sign reversal in both sine and cosine.

Key Concepts

Circle

A circle is a perfectly round shape, where all points are equidistant from a central point. This distance is known as the radius. The equation of a circle in a standard coordinate system can be expressed as \( (x - h)^2 + (y - k)^2 = r^2 \), where

• \( (h, k) \) is the center of the circle, and
• \( r \) is the radius.

In the context of parametric equations, circles can be expressed with trigonometric functions like sine and cosine. This allows the circle to be drawn by varying a parameter, often denoted as \( t \), over a specific interval, typically from 0 to \( 2\pi \). For example, a circle centered at \( (3, -2) \) with a radius of 2 would be represented in terms of parametric equations as \( x = 3 + 2\cos t \) and \( y = -2 + 2\sin t \). Changing these parameters can allow the tracing of the circle to change direction, but the shape itself remains invariant.

Parametric Equations

Parametric equations are a pair or set of equations where both the x and y coordinates (or in higher dimensions, more coordinates) are expressed in terms of an independent parameter, generally \( t \). These equations are particularly useful for describing curves that cannot be easily represented by one function of x or y.
The general idea is to express both x and y as separate functions of \( t \), to describe paths or geometric curves effectively.

• For circles, the parametric equations might look like \( x = h + r\cos t \) and \( y = k + r\sin t \), where \( h \) and \( k \) define the circle's center, and \( r \) defines its radius.
• By varying \( t \), usually over the interval from 0 to \( 2\pi \), the full circle is traced.

This approach allows dynamic exploration of different configurations of the curve by simply adjusting the equations.

Ellipse

An ellipse is a more generalized form of a circle, essentially an elongated or squashed circle. It is defined by two axes, the major axis (longest diameter) and the minor axis (shortest diameter). Its equation is generally given by \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \), where

• \( (h, k) \) is the center,
• \( a \) is the semi-major axis,
• \( b \) is the semi-minor axis.

In parametric form, ellipses involve equations of the type \( x = h + a\cos t \) and \( y = k + b\sin t \). In the exercise described, while initial appearances suggest an ellipse, because the semi-major and semi-minor axes are equal (\( a = b = 2 \), it resolves to a circle centered at (3, -2).

Graphing Parametric Equations

Graphing parametric equations involves plotting points calculated by evaluating the parametric equations for a range of \( t \) values. As each \( t \) corresponds to a point on the graph, varying \( t \) traces out the curve or shape, like a circle or ellipse.
This process can effectively represent motion or paths that would be cumbersome or impossible to capture with standard y=f(x) plots.
To graph parametric equations, interpret the equations for particular \( t \) values:

• Calculate \( x \) and \( y \) separately for each chosen \( t \).
• Plot the resultant (x, y) coordinates on a graph.
• Continue to plot points until \( t \) has covered its full range (often from 0 to \( 2\pi \) for complete cycles).

Changes in the sign or coefficient in the parametric equations can adjust the path direction or orientation, often described as clockwise or counter-clockwise when graphing circular paths. This is an excellent way to visualize complex relationships and movements in mathematically pristine forms.