AP Precalculus Unit 4 Guide: Parametrics, Vectors, and Motion

Parametric Function

Defines a set of points (x, y) on a plane where both coordinates are functions of a parameter t.

Parameter

The independent variable, typically denoted as t, that drives parametric functions.

Plane curve

A curve traced out by a collection of points (x(t), y(t)) from parametric equations.

Eliminating the parameter

The process of converting parametric equations into a rectangular equation relating only x and y.

Rectangular equation

An equation relating y to x without involving the parameter t.

Domain Check

Ensuring that the parametric equations’ constraints align with the derived rectangular equations.

Pythagorean identity

Expresses the relationship sin²(t) + cos²(t) = 1, crucial for parametrizing circles.

Circle

A set of points in a plane at a fixed distance (radius r) from a center (h, k).

Ellipse

A shape described by a set of points where the sum of distances to two foci is constant.

Hyperbola

A set of points where the difference in distances to two foci is constant.

Projectile motion

The motion of an object thrown into the air with an initial velocity and influenced by gravity.

Initial velocity (v₀)

The speed at which an object is launched, affecting its trajectory.

Launch angle (θ)

The angle relative to the horizontal at which an object is launched.

Horizontal Position

Given by x(t) = (v₀ cos θ)t + x₀, it defines motion along the x-axis.

Vertical Position

Given by y(t) = -½gt² + (v₀ sin θ)t + y₀, it defines motion along the y-axis.

Gravity (g)

The acceleration acting on an object in projectile motion, approximately 9.8 m/s² or 32 ft/s².

Direction

The orientation of the motion of a parametric curve as the parameter t changes.

Calculator Mode Errors

Mistakes arising from using the wrong mode (Radians or Degrees) while calculating trigonometric functions.

Hidden Domain

The implicit restrictions on values of t that should be checked when eliminating the parameter.

Orientation of motion

The specific path taken by the object that must be verified by plotting values of t.

Conic Sections

Curves obtained by intersecting a plane with a double-napped cone, including circles, ellipses, and hyperbolas.

Standard Rectangular Form

The representation of conic sections in a Cartesian coordinate system.

Parametric Form

A specific representation of curves using one or more parameters.

Components of velocity

The breakdown of initial velocity into horizontal and vertical components.

Asymptotes

Lines that a hyperbola approaches as the distance increases.

Velocity magnitude

The speed of an object in a given direction; represented by the initial velocity v₀.

Trajectory

The path followed by an object in motion.