Problem 32 - Precalculus: Functions and Graphs

Question

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

Step-by-Step Solution

Answer

(a) Circle at \((-2, 3)\), radius 3; (b) Same circle, reversed traversal; (c) Same circle, vertically flipped path.

Step 1: Identify the Standard Parametric Equation

Recognize that the given parametric equations correspond to transformations of a circle. The standard parametric equation for a circle centered at the origin with radius 3 is \(x=3\sin t\), \(y=3\cos t\).

Step 2: Analyze Transformation of Part (a)

The transformations \(x=-2+3\sin t\) and \(y=3-3\cos t\) indicate that the circle is translated horizontally by \(-2\) and vertically by \(3\), resulting in a circle centered at \((-2, 3)\) with a radius of 3.

Step 3: Calculate Transformation Effect in Part (b)

For \(x=-2-3\sin t\) and \(y=3+3\cos t\), note that the signs of the trigonometric functions inside the transformation terms have changed. This inverts the direction of traversal: \(-3\sin t\) equates to reflecting across the \(x\)-axis, and \(+3\cos t\) leads to reflection across the \(y\)-axis compared to part (a). The circle's position remains unchanged at \((-2, 3)\).

Step 4: Examine Parametric Change in Part (c)

The parametrization \(x=-2+3\sin t\), \(y=3+3\cos t\) maintains the center at \((-2, 3)\) but changes the path, resulting in a reflection over the horizontal axis (\(y=0\)) compared to the original in part (a). The direction of tracing is reversed along the vertical axis.

Key Concepts

Circle Translation

When dealing with parametric equations that involve circles, understanding how translation affects the graph is key.
Given a standard parametric circle equation, if you translate it, meaning you move the entire circle, the translation can be seen directly from the equation.
In the exercise, the original equation \[ x = -2 + 3 \sin t, \quad y = 3 - 3 \cos t \]marks a circle's transformation.
Here, we translate the circle by

• -2 units in the horizontal direction (left)
• 3 units in the vertical direction (up)

This means the entire circle shifts and finds its new center at the point \((-2, 3)\).
It's important to note that the circle’s size, defined by its radius, remains unchanged. The circle still keeps its radius of 3, but its entire position on the grid changes due to the translation.

Graph Reflection

Graph reflection in parametric equations can alter the path along which the graph is traced out.
In part (b) of the problem, the equations are changed to \[ x = -2 - 3 \sin t, \quad y = 3 + 3 \cos t \]By changing the signs inside these functions, we cause a specific type of reflection.
This means:

• The \(-3 \sin t\) flips the graph across the \( x\)-axis, changing how the curve is traversed horizontally.
• The \(+3 \cos t\) flips the graph across the \( y\)-axis, altering the vertical tracing of the curve.

Even with these changes, the center of the circle remains at \((-2, 3)\).
Through this reflection, what we observe is more about how the graph is approached and traced rather than its location.

Parametric Curve Transformation

Transforming parametric curves can result in intriguing changes to how the shape behaves graphically.
In part (c), we have modified the equation to \[ x = -2 + 3 \sin t, \quad y = 3 + 3 \cos t \]This reflects only the \( y \)-coordinates of the points across the horizontal axis (around the line \( y = 0 \)).
This reflection alters how the circle is drawn by changing the tracing path.
The path taken by the parametric curve will now trace the circle differently, leading to a curve that is mirrored over the horizontal center.

• The location remains unchanged, centered at \((-2, 3)\)

The main takeaway is that the points are traversed in an opposite direction in the vertical aspect but the position stays the same.