Parametric Equations and Cartesian Conversion: Study Notes for Calculus II

10.1 Homework

Parametric Equations and Cartesian Conversion

Introduction to Parametric Equations

Parametric equations are a way to represent curves by expressing the coordinates of the points on the curve as functions of a variable, typically t. This approach is especially useful for describing curves that cannot be represented as functions in the form y = f(x) or x = g(y).

• Parameter: A variable, often t, that both x and y depend on.

• Parametric Form: $x = f(t),\ y = g(t)$

• Cartesian Form: An equation relating x and y directly, eliminating the parameter.

Example: The parametric equations $x = t^2 - 4$, $y = \frac{1}{2}t$ for $t \in [-2,3]$ describe a curve in the plane as t varies.

Graphing Parametric Equations

Plotting Points and Intervals

To graph a parametric equation, calculate the x and y values for each integer value of t in the given interval. Plot the resulting points and connect them smoothly to visualize the curve.

• For $x = t^2 - 4$, $y = \frac{1}{2}t$, compute $(x, y)$ for $t = -2, -1, 0, 1, 2, 3$.

• Label each point with its corresponding t value.

Example Table:

t	x	y
-2	0	-1
-1	-3	-0.5
0	-4	0
1	-3	0.5
2	0	1
3	5	1.5

Converting Parametric Equations to Cartesian Form

Elimination of the Parameter

To convert a parametric equation to Cartesian form, solve one equation for t and substitute into the other.

• Step 1: Solve $y = \frac{1}{2}t$ for t: $t = 2y$

• Step 2: Substitute into $x = t^2 - 4$: $x = (2y)^2 - 4 = 4y^2 - 4$

• Result: The Cartesian equation is $x = 4y^2 - 4$

Example: For $x = e^{t} + 1$, $y = e^{2t}$, solve $x = e^{t} + 1$ for $e^{t}$: $e^{t} = x - 1$. Substitute into $y = e^{2t} = (e^{t})^2 = (x - 1)^2$. Thus, $y = (x - 1)^2$.

Parametric Equations for Common Curves

Circle

A circle with center $(h, k)$ and radius $r$ can be represented parametrically as:

• $x = h + r\cos(t)$

• $y = k + r\sin(t)$

• $t \in [0, 2\pi]$

Example: For center $(-3, 2)$ and radius $4$:

• $x = -3 + 4\cos(t)$

• $y = 2 + 4\sin(t)$

Ellipse

An ellipse with center $(h, k)$, semi-major axis $a$, and semi-minor axis $b$:

• $x = h + a\cos(t)$

• $y = k + b\sin(t)$

• $t \in [0, 2\pi]$

Example: For center $(-3, 2)$, vertices at $(-3, 7)$ and $(-3, -3)$, the major axis is vertical with length $10$($a = 5$), and minor axis can be determined from other points.

• Example: For an ellipse with center at (-3, 2), vertices at (-3, 7) and (-3, -3), and other points at (1, 2) and (-7, 2):

• The vertices (-3, 7) and (-3, -3) are vertically aligned, so the major axis is vertical.

• The distance from the center to a vertex is $|7 - 2| = 5$, so $a = 5$ (semi-major axis).

• The points (1, 2) and (-7, 2) are horizontally aligned with the center, so $b = |1 - (-3)| = 4$ (semi-minor axis).

Trigonometric Parametric Equations

Example: Sine and Cosecant

Given $x = \sin(t)$, $y = \csc(t)$, $t \in (0, \pi)$:

• $\csc(t) = \frac{1}{\sin(t)}$

• Express $y$ in terms of $x$: $y = \frac{1}{x}$, for $x \neq 0$

Summary Table: Parametric to Cartesian Conversion

Parametric Equations	Interval	Cartesian Equation
$x = t^2 - 4$, $y = \frac{1}{2}t$	$t \in [-2,3]$	$x = 4y^2 - 4$
$x = e^{t} + 1$, $y = e^{2t}$	All $t$	$y = (x - 1)^2$
$x = \sin(t)$, $y = \csc(t)$	$t \in (0, \pi)$	$y = \frac{1}{x}$

Applications and Graphing Tips

• Use graphing calculators or software (e.g., Desmos) to plot parametric curves.

• Label points with their parameter values for clarity.

• Parametric equations are essential for describing motion, curves, and shapes in physics and engineering.

Additional info: The problems also require students to graph circles and ellipses using parametric equations, and to convert parametric forms to Cartesian equations, which are key skills in Chapter 10 of Calculus (Parametric Equations and Polar Coordinates).