AP Precalculus Unit 4 Notes: Understanding Parametric, Projectile, and Implicit Conic Models

Parametric function

A way to describe a relationship between variables (often x and y) by expressing both as functions of a third variable (a parameter), such as x=f(t) and y=g(t).

Parameter

An input variable (commonly t) that generates points (x(t), y(t)) on a curve; often represents time or an angle.

Vertical line test

A test for whether y is a function of x: if any vertical line intersects a graph more than once, then the relation is not a function of x.

Parametric equations

A pair of equations x=f(t) and y=g(t) that together define a curve by giving coordinates in terms of a parameter.

Ordered pair function

The notation (x(t), y(t)), meaning the point in the plane corresponding to a particular parameter value t.

Position vector

Vector form of planar motion: r(t)=⟨x(t), y(t)⟩, a vector from the origin to the moving point.

Tracing a parametric curve

The process of generating a path in the plane as t changes, including information about direction, speed changes, and whether the curve repeats.

Start point (parametric graph)

The point found by plugging the starting parameter value (e.g., t=0) into (x(t), y(t)).

Direction of motion (parametrics)

How the curve is traversed as t increases, determined by evaluating points for increasing t-values.

Retracing / looping

When different parameter values produce the same point, causing the curve to repeat parts of its path.

Unit circle (parametric form)

The parametrization x=cos(t), y=sin(t), which satisfies x^2+y^2=1 and traces the circle as t varies.

Eliminating the parameter

Rewriting a parametric curve as a Cartesian equation in x and y only, typically by solving for t (or an expression in t) and substituting.

Trig identity cos^2(t)+sin^2(t)=1

An identity used to eliminate t in trig-based parametrizations, converting them to an implicit Cartesian equation.

Planar motion

A model of motion in the plane using parametric functions x(t), y(t), with t interpreted as time.

Position (in planar motion)

The location of the object at time t, given by (x(t), y(t)) (or r(t)=⟨x(t), y(t)⟩).

Displacement vector

Change in position from t=a to t=b: ⟨x(b)−x(a), y(b)−y(a)⟩.

Average velocity vector

Displacement divided by elapsed time on [a,b]: ⟨(x(b)−x(a))/(b−a), (y(b)−y(a))/(b−a)⟩.

Average speed

Total distance traveled divided by elapsed time; generally not equal to the magnitude of displacement when motion curves or reverses.

Projectile motion (parametric modeling)

Motion of a launched object modeled with x(t) and y(t), where horizontal and vertical motions occur simultaneously and are described by different functions.

Standard projectile model

x(t)=x0+vx t and y(t)=y0+vy t−(1/2)gt^2, with constant horizontal velocity and downward gravitational acceleration.

Independence of components

In the simplified projectile model, horizontal and vertical positions are computed separately (x from x(t), y from y(t)) and then paired.

Range (projectile motion)

The horizontal distance traveled before landing, found by evaluating x(t) at the landing time (when y(t) returns to ground level).

Vertex time (maximum height)

The time when a downward-opening quadratic y(t)=at^2+bt+c reaches its maximum: t=−b/(2a); in projectile form, t=v_y/g.

Implicitly defined relationship

An equation involving x and y together (e.g., F(x,y)=0) that may represent curves not expressible as a single y=f(x).

Completing the square

An algebra technique used to rewrite quadratic expressions into squared-binomial form, especially to convert implicit conic equations into standard form.