Topics
4A. Parameters, Vectors, and Matrices
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United States of America
AP Precalculus

4.4 Parametrically defined functions

Lesson
Lesson

Introduction

Learning objective

  • 4.4.A - Express motion around a circle or along a line segment parametrically.

Modeling motion around a circle

A unit circle is a circle with a radius of exactly 1 unit, centered at the origin point (0,\, 0) of a Cartesian coordinate plane. This circle has the special property that any point on the circle can be represented by the coordinates (\cos t,\, \sin t), where t is the angle formed by the positive x-axis and a line segment drawn from the origin to the point on the circle. The circle is traversed counterclockwise, which is the standard direction for positive rotation in mathematics.

Imagine we place a point at the location (1, 0) on this unit circle. If we let this point move in a counterclockwise direction around the circle, the path it takes can be described by the set of parametric equations (x(t),\, y(t)) = (\cos t,\, \sin t). Here, t is a parameter that represents the angle that the line segment from the origin to the point makes with the positive x-axis. As t varies from 0 to 2\pi, the point completes a full rotation around the circle, returning to its starting position.

The unit circle with the special angles labeled. Starting from the positive x axis then moving counterclockwise, the special angles are: 0, pi over 6, pi over 4, pi over 3, pi over 2, 2 pi over 3, 3 pi over 4, 5 pi over 6, pi, 7 pi over 6, 5 pi over 4, 4 pi over 3, 3 pi over 2, 5 pi over 3, 7 pi over 4, and 11 pi over 6. Speak to your teacher for more details.

Transformations allow us to adjust the basic parametric equations (x(t),\, y(t)) = (\cos t,\, \sin t) to represent any circular motion in the coordinate plane. Here's how:

  1. Scaling: By multiplying the \cos t and \sin t by a constant, we can change the radius of the circle.

  2. Translating: By adding a constant to \cos t and \sin t, we can shift the center of the circle anywhere in the plane.

  3. Reflecting: By changing the sign of \cos t or \sin t, we can reverse the direction of motion along the circle.

(\cos t,\, \sin t)

This is a parametric representation of a point on the unit circle, where t is the angle formed by the positive x-axis and a line segment from the origin to the point.

Transformations

These are operations that alter the size, position, or direction of a figure in a mathematical space. In the context of parametric equations, transformations can be used to modify the path represented by the equations.

Scaling

This is a type of transformation that changes the size of a figure without altering its shape. Scaling a circle changes its radius.

Translating

This is a type of transformation that moves every point of a figure by the same distance in a given direction. Translating a circle moves its center to a different location on the coordinate plane.

Reflecting

This is a type of transformation that flips a figure over a line (in 2D) or a plane (in 3D). Reflecting a circular path changes the direction of motion from counterclockwise to clockwise, or vice versa.

Examples

Example 1

Given a point P initially located at (1,\, 0) on a unit circle centered at the origin of a Cartesian coordinate plane. The point moves counterclockwise along the circumference of the circle.

Find the coordinates of the point P after it has moved an angle of \dfrac{\pi}{3} radians from its initial position, and then determine its coordinates after a full rotation.

Worked Solution
Create a strategy

We will use the parametric equations (x(t),\, y(t)) = (\cos t,\, \sin t) to find the coordinates of the point P at any given angle t. The angle is measured counterclockwise from the positive x-axis.

Apply the idea

  1. Find the coordinates of the point after moving an angle of \dfrac{\pi}{3} radians:

    \displaystyle x\left(\dfrac{\pi}{3}\right)\displaystyle =\displaystyle \cos\left(\dfrac{\pi}{3}\right)Substitute t=\dfrac{\pi}{3}
    \displaystyle =\displaystyle \dfrac{1}{2}Evaluate
    \displaystyle y\left(\dfrac{\pi}{3}\right)\displaystyle =\displaystyle \sin\left(\dfrac{\pi}{3}\right)Substitute t=\dfrac{\pi}{3}
    \displaystyle =\displaystyle \dfrac{\sqrt{3}}{2}Evaluate

    Hence, after an angle of \dfrac{\pi}{3} radians, the point P is at \left(\dfrac{1}{2},\, \dfrac{\sqrt{3}}{2}\right).

  2. Find the coordinates of the point after a full rotation:

    A full rotation around the circle corresponds to an angle of 2\pi radians.

    \displaystyle x(2\pi)\displaystyle =\displaystyle \cos (2 \pi)Substitute t=2\pi
    \displaystyle =\displaystyle 1Evaluate
    \displaystyle y(2\pi)\displaystyle =\displaystyle \sin (2 \pi)Substitute t=2\pi
    \displaystyle =\displaystyle 0Evaluate

    Hence, after a full rotation, the point P returns to its starting position at (1,\, 0).

Reflect and check

Another way to solve this problem could be graphically. We could draw the unit circle, mark the initial position of the point, and then measure an angle of \dfrac{\pi}{3} radians counterclockwise from the positive x-axis to find the new position of the point. Similarly, we could trace a full rotation of the point to confirm that it returns to its starting position.

Idea summary

Parametric equations for a point on the unit circle: (x(t), y(t)) = (\cos t,\, \sin t)

Angle for a full rotation in radians: t varies from 0 to 2\pi.

Parametrizing a linear path

Parametric equations are incredibly versatile and can model more than just circles. In particular, they are excellent for describing linear motion, or motion along a straight path.

Let's consider a practical scenario. Imagine you want to model the movement of a car traveling in a straight line from point A\,(x_1,\, y_1) to point B\, (x_2,\, y_2) over time.

The first step is to establish the initial position of the car, which in this case, would be point A\,(x_1,\, y_1).

Next, we must consider how the car's position changes over time. If the car moves at a constant speed, then its position changes at a steady rate. This rate of change is often denoted by t, which can represent time or some other parameter.

We can represent the x-coordinate and y-coordinate of the car's position as functions of t. In general, these functions will take the form x(t) = x_1 + (x_2 - x_1)t and y(t) = y_1 + (y_2 - y_1)t. Here, (x_2 - x_1) and (y_2 - y_1) represent the changes in the x and y coordinates, respectively, from the initial position to the final position.

When t is 0, the car is at the initial position (x_1,\, y_1). As t increases, the car moves closer to the final position (x_2,\, y_2). When t is 1, the car has reached its final position.

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Examples

Example 2

Imagine we have a car starting at point A\, (-3,\, 2) on a Cartesian coordinate plane. The car is moving at a constant speed towards point B (5,\, 7). We want to model this movement using parametric equations. Find these parametric equations and determine the position of the car at time t = 0.5.

Worked Solution
Apply the idea

  1. The starting point A is (-3,\, 2), so x_1 = -3 and y_1 = 2.

  2. The ending point B is (5,\, 7), so x_2 = 5 and y_2 = 7.

  3. Substituting these values into the general form of the parametric equations gives us the specific equations for this situation:

    \displaystyle x(t)\displaystyle =\displaystyle -3 + (5 - (-3))t Substitute x_1 = -3 and x_2=5
    \displaystyle =\displaystyle -3 + 8tEvaluate
    \displaystyle y(t)\displaystyle =\displaystyle 2 + (7 - 2)t Substitute y_1 = 2 and y_2 = 7
    \displaystyle =\displaystyle 2 + 5tEvaluate

  4. To find the position of the car at t = 0.5, substitute 0.5 for t in both equations:

    \displaystyle x(t)\displaystyle =\displaystyle -3 + 8tWrite first parametric equation from step 3
    \displaystyle x(0.5)\displaystyle =\displaystyle -3 + 8(0.5)Substitute t=0.5
    \displaystyle =\displaystyle 1Evaluate
    \displaystyle y(t)\displaystyle =\displaystyle 2 + 5tWrite second parametric equation from step 3
    \displaystyle y(0.5)\displaystyle =\displaystyle 2 + 5(0.5)Substitute t=0.5
    \displaystyle =\displaystyle 4.5Evaluate
Reflect and check

The parametric equations for this linear motion are x(t) = -3 + 8t and y(t) = 2 + 5t. When t = 0, these equations give us the starting point of the car, A(-3,\, 2). When t = 1, they give us the ending point of the car, B(5,\, 7). Thus, our equations are consistent with the given conditions. The position of the car halfway through its journey, at t = 0.5, is (1,\, 4.5), which lies on the line segment connecting A and B. This suggests that our solutions are likely correct.

Idea summary

Parametric Equations: These are equations that express the coordinates of the points of a geometrical object as functions of a variable, called a parameter.

Linear Motion: This is movement in a straight line from one point to another.

Rate of Change: This represents how a quantity is changing over time.

The general form of parametric equations for linear motion is given by:

x(t) = x_1 + (x_2 - x_1)t \\ y(t) = y_1 + (y_2 - y_1)t

Outcomes

4.4.A

Express motion around a circle or along a line segment parametrically.

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