What is the primary difference between a Cartesian equation and a set of parametric equations?

A Cartesian equation defines a direct relationship between $x$ and $y$ (e.g., $y = x^2$), whereas parametric equations define both $x$ and $y$ in terms of a third independent variable called a parameter.

How does the information provided by parametric equations differ from that of a Cartesian graph?

Parametric equations provide the 'history' of the curve, showing the direction of motion and the specific 'time' (parameter value) at which each point is reached, which a static Cartesian equation lacks.

When comparing $x = \cos(t), y = \sin(t)$ and $x = \cos(2t), y = \sin(2t)$, how do the resulting graphs differ?

The Cartesian path (a circle) is identical for both, but the second set traces the circle twice as fast as the first set for the same range of $t$.

What is a common mistake when identifying the radius of a circle from $x = 5\cos( heta) + 2$ and $y = 5\sin( heta) - 3$?

A common error is confusing the radius with the center coordinates; the radius is the coefficient $5$, while the constants $2$ and $-3$ represent the horizontal and vertical shifts of the center.

What error occurs if a student plots parametric points but ignores the order of the parameter values?

The student loses the 'orientation' or direction of the curve, which is a critical feature of parametric representation showing how the path evolves.

Why is it incorrect to assume that $x = 2t$ and $y = 4t^2$ represents a full parabola if $t$ is restricted to $t > 0$?

The restriction on the parameter $t$ limits the domain and range of the resulting graph; in this case, only the right-hand branch of the parabola in the first quadrant would be traced.

Define the term 'parameter' in the context of coordinate geometry.

A parameter is an independent variable (often $t$ or $\theta$) upon which the values of other variables (like $x$ and $y$) depend, acting as an intermediary to define a curve.

What are the standard parametric equations for a circle with center $(a, b)$ and radius $r$?

The equations are $x = r \cos( heta) + a$ and $y = r \sin( heta) + b$, where $ heta$ is the parameter.

In the parametric form of a circle, what does the variable $\theta$ typically represent geometrically?

It represents the angle in radians (or degrees) measured from the positive $x$-axis to the line segment connecting the center of the circle to a point on its circumference.

How do you find the $y$-intercept of a curve defined parametrically?

Set the equation for $x$ equal to zero ($x = f(t) = 0$), solve for the parameter $t$, and then substitute those $t$-values into the equation for $y$.

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Basics of Parametric Equations

Summary

Parametric equations provide a way to define coordinates $x$ and $y$ independently through a third variable known as a parameter. This system is particularly useful for describing paths of motion where the position depends on an external factor like time or an angle, offering more flexibility than standard Cartesian equations.

1. Definition & Core Concepts

A graph showing a curve with points labeled at different values of the parameter t, illustrating how the curve is traced over time.

2. Underlying Principles

The fundamental principle is that $x$ and $y$ are dependent variables while the parameter $t$ is the independent variable.
For every value of the parameter within a specified domain, a unique pair of $(x, y)$ coordinates is generated, which corresponds to a single point on the Cartesian plane.
This system allows for the representation of curves that are not functions in the Cartesian sense (e.g., a circle or a figure-eight) because a single $x$ value can correspond to multiple $y$ values at different parameter values.

3. Methods: Plotting Parametric Curves

4. Parametric Equations of Circles

5. Key Distinctions

6. Exam Strategy & Tips