Glossary
Acceleration (Planar Motion)
The rate of change of an object's velocity with respect to time, describing how its speed or direction of motion is changing.
Example:
When a spacecraft fires its thrusters, it experiences Acceleration, changing its speed and direction.
Domain of t
The specific interval of time values for which the parametric functions are defined and model the motion of an object.
Example:
If a roller coaster ride lasts for 90 seconds, then the Domain of t for its parametric equations would be [0, 90].
Extrema
The maximum or minimum values that a function attains within a given interval, representing the highest/lowest or farthest left/right points.
Example:
Finding the Extrema of a projectile's height function tells you its maximum altitude.
Horizontal Extrema
The maximum and minimum values of the x(t) function, which represent the farthest left and right points an object reaches in its path.
Example:
For a pendulum swinging, its Horizontal Extrema would be the points where it momentarily stops before changing horizontal direction.
Intercepts
Points where a graph crosses either the x-axis (x-intercepts) or the y-axis (y-intercepts).
Example:
When a robot's path crosses the starting line, it's hitting an Intercept.
Parametric Functions
Functions that express both the x and y coordinates of a point in terms of a third independent variable, typically time (t), allowing for the modeling of motion in two dimensions.
Example:
To describe the path of a rocket, we might use Parametric Functions like x(t) = 10t and y(t) = 5t - 0.5gt^2.
Position (Planar Motion)
The specific location of an object in a two-dimensional plane at a given time, determined by its (x(t), y(t)) coordinates.
Example:
At t=5, if a drone's coordinates are (10, 15), then (10, 15) is its Position in the plane.
Real zeros of x(t)
The specific values of time (t) for which the horizontal position function x(t) equals zero, indicating when the particle is located on the y-axis.
Example:
If x(t) = t^2 - 16, the Real zeros of x(t) are t=4 and t=-4, meaning the particle is on the y-axis at these times.
Real zeros of y(t)
The specific values of time (t) for which the vertical position function y(t) equals zero, indicating when the particle is located on the x-axis.
Example:
If y(t) = 3t - 6, the Real zero of y(t) is t=2, indicating the particle is on the x-axis at that time.
Time (t)
The independent variable in parametric functions, typically representing time, which dictates the corresponding x and y coordinates of an object's position.
Example:
In a simulation of a planet orbiting a star, the variable t would represent the elapsed Time in years or days.
Velocity (Planar Motion)
The rate of change of an object's position with respect to time, indicating both its speed and direction of movement.
Example:
A car's Velocity might be described by how fast its x and y coordinates are changing over time.
Vertical Extrema
The maximum and minimum values of the y(t) function, which represent the highest and lowest points an object reaches in its path.
Example:
The highest point a basketball reaches after being shot is its Vertical Extrema.
x(t)
The component of a parametric function that represents the horizontal position of an object at a specific time t.
Example:
If a car's horizontal movement is given by x(t) = 3t + 5, at t=2 seconds, its horizontal position is 11 units.
x-intercepts (Parametric)
Points where the particle's path crosses the x-axis, which occur when the vertical position function y(t) equals zero.
Example:
For a projectile, its x-intercepts are the points where it hits the ground (y=0).
y(t)
The component of a parametric function that represents the vertical position of an object at a specific time t.
Example:
For a ball thrown upwards, its vertical height might be described by y(t) = 20t - 4.9t^2.
y-intercepts (Parametric)
Points where the particle's path crosses the y-axis, which occur when the horizontal position function x(t) equals zero.
Example:
If a particle's path is given by x(t) = t-5 and y(t) = t^2, its y-intercept occurs when t=5, at the point (0, 25).