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

4.3 Parametric functions and rates of change

Lesson
Lesson

Introduction

Learning objective

  • 4.3.A - Identify key characteristics of a parametric planar motion function that are related to direction and rate of change.

Direction and rate of change

The direction and rate of change in parametric planar motion functions, are mathematical models that describe the position of a particle in a plane as a function of a parameter, typically time (t). To analyze the direction of a particle's motion, we need to consider the x and y components of the motion independently.

As the parameter (t) increases, the direction of the motion can be deduced by examining the x(t) and y(t)functions. A table can be helpful for visualizing the behavior of x(t)and y(t) as t increases:

tx(t)x\text{-direction}y(t)y\text{-direction}
t_{1}x_{1} y_{1}
t_{2}x_{2} y_{2}
t_{3}x_{3} y_{3}
-2
-1
1
2
3
x
-4
-3
-2
-1
y

If x(t) is increasing as t increases, the motion is to the right, while if x(t) is decreasing, the motion is to the left. Similarly, if y(t) is increasing, the motion is upward, and if y(t) is decreasing, the motion is downward.

It's important to note that the direction of planar motion may vary at any given point in the plane for different values of the parameter t. As time progresses, the particle's motion can change direction, demonstrating that a particle's path isn't necessarily fixed or linear.

It is crucial to understand that at any given point in the plane, the direction of planar motion may be different for different values of the parameter (t). This means that the particle's motion can change direction as time progresses. Moreover, it's possible for the same curve in the plane to be parameterized in different ways, leading to different directions of motion for different parametric functions. This highlights the flexibility and versatility of parametric functions in modeling particle motion.

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.
Direction of motion

The orientation of a particle's movement in the plane, which can be analyzed independently for the x and y components.

Rate of change

A measure of how quickly a variable changes with respect to another variable.

Examples

Example 1

Consider the parametric equations representing a particle's motion in the plane: x(t) = t^{2} - 2t and y(t) = 2t + 1.

a

Determine the direction of motion of the particle in the interval [0,\, 3].

Worked Solution
Create a strategy

To determine the direction of motion of the particle, we can examine the values of t in the interval [0,\, 3] and observe how the particle moves along the x and y axes.

Apply the idea

  • Evaluate x(t) and y(t) at the endpoints of the interval:

\displaystyle x(t)\displaystyle =\displaystyle t^{2} - 2tWrite the original first parametric equation
\displaystyle x(0)\displaystyle =\displaystyle 0^{2} - 2(0)Substitute t = 0
\displaystyle x(0)\displaystyle =\displaystyle 0Evaluate
\displaystyle y(t)\displaystyle =\displaystyle 2t + 1Write the original second parametric equation
\displaystyle y(0)\displaystyle =\displaystyle 2(0) + 1Substitute t = 0
\displaystyle y(0)\displaystyle =\displaystyle 1Evaluate
\displaystyle x(t)\displaystyle =\displaystyle t^{2} - 2tWrite the original first parametric equation
\displaystyle x(3)\displaystyle =\displaystyle 3^{2} - 2(3)Substitute t = 3
\displaystyle x(3)\displaystyle =\displaystyle 3Evaluate
\displaystyle y(t)\displaystyle =\displaystyle 2t + 1Write the original second parametric equation
\displaystyle y(3)\displaystyle =\displaystyle 2(3) + 1Substitute t = 3
\displaystyle y(3)\displaystyle =\displaystyle 7Evaluate

  • Compare the values of x(t) and y(t) at t = 0 and t = 3:

The x-coordinate increases from 0 to 3, indicating rightward motion.

The y-coordinate increases from 1 to 7, indicating upward motion.

b

Find another pair of parametric equations that represent the same path but with the particle moving in the opposite direction.

Worked Solution
Create a strategy

To find a pair of parametric equations that represent the same path but with the particle moving in the opposite direction, we can simply negate the coefficients of x(t) and y(t).

Apply the idea

New parametric equations:

\displaystyle x(t)\displaystyle =\displaystyle t^{2} - 2tWrite the original first parametric equation
\displaystyle x(t)\displaystyle =\displaystyle -(t^{2} - 2 t)Negate the coefficients of x(t)
\displaystyle x(t)\displaystyle =\displaystyle -t^{2} + 2tEvaluate
\displaystyle y(t)\displaystyle =\displaystyle 2t + 1Write the original second parametric equation
\displaystyle y(t)\displaystyle =\displaystyle -(2t + 1)Negate the coefficients of x(t)
\displaystyle y(t)\displaystyle =\displaystyle -2t - 1Evaluate

These new equations will represent the same path but with the particle moving in the opposite direction.

c

Show how the two sets of parametric equations produce the same path using a graph.

Worked Solution
Create a strategy

We can graph the original parametric equations x(t) = t^{2} + 2t and y(t) = 2t + 1 and the new parametric equations x(t) = -t^{2} + 2t and y(t) = -2t - 1.

Apply the idea

  1. Original Parametric Equations:

    • Given the original parametric equations: x(t) = t^2 - 2t and y(t) = 2t + 1.

    Choose a Range for t:

    • Select a range for t that allows for a comprehensive view of the particle's motion, such as [-3,\, 3].

    • This range ensures that we cover a significant portion of the particle's path.

  2. Calculate the (x,\, y) coordinates:

    • For various values of t within the chosen range, substitute the values into the original parametric equations to calculate the corresponding (x,\, y) coordinates.

    • Compute the coordinates for multiple values of t to obtain a set of coordinate pairs.

  3. New Parametric Equations from part (b):

    • Calculate the (x,\, y) Coordinates for the New Equations

tx(t)=t^{2}-2ty(t) =2t + 1x(t) = -t^2 +2ty(t) = -2t - 1
-315-5-155
-28-3-83
-13-1-31
0010-1
1-131-3
2050-5
337-3-7
-15
-10
-5
5
10
15
x
-8
-6
-4
-2
2
4
6
8
y
Idea summary

The rate of change of the x(t) and y(t) components of a parametric function can be used to determine the direction of motion in the plane.

The same curve can be parametrized in different ways, with different parametric equations representing the same path but with different directions of motion. By using substitution and graphing, we can show that two sets of parametric equations represent the same path in the plane but with opposite directions of motion.

Average rate of change

The average rate of change of a function over an interval gives us an idea of how the function is changing over that specific interval. For parametric functions, the average rate of change provides insights into the particle's motion in the plane.

We compute the average rate of change using the following formula:

\text{Average rate of change} = \dfrac{(f(t_2) - f(t_1))}{(t_2 - t_1)}, where f(t) represents the function (x(t) or y(t)) and t_1 and t_2 are the endpoints of the interval.

To analyze the motion of the particle, we calculate the average rate of change for both x(t) and y(t) independently. This gives us information about the horizontal and vertical changes in the particle's position over time.

Now, to understand the overall direction and steepness of the curve representing the particle's motion, we take the ratio of the average rate of change of y(t) to the average rate of change of x(t), as long as the average rate of change of x(t) is not equal to 0. This ratio represents the slope of the graph between the points on the curve corresponding to t_1 and t_2.

The slope is an essential characteristic of a curve because it tells us how steep the curve is and the direction in which it is inclined. In the context of a particle's motion, it provides valuable information about the particle's behavior in the plane over the interval [t_1,\, t_2].

Average rate of change

The ratio of the change in a function's output to the change in its input over a given interval.

Examples

Example 2

Consider the parametric equations x(t) = 2t^{2} + t and y(t) = t^{3} - 4t. Compute the average rate of change of x(t) and y(t) over the interval [1,\, 3], and determine the slope of the graph between the points corresponding to t = 1 and t = 3.

Worked Solution
Create a strategy

First recall the formula for the average rate of change: \dfrac{(f(t_2) - f(t_1))}{(t_2 - t_1)}. Next, apply the formula to both x(t) and y(t) functions separately over the given interval and finally calculate the slope using the computed average rates of change.

Apply the idea

Step 1: Find the average rate of change for x(t) over the interval [1,\, 3].

\displaystyle x(t)\displaystyle =\displaystyle 2t^{2} + tWrite the first parametric equation
\displaystyle x(1)\displaystyle =\displaystyle 2(1)^{2} + 1Substitute t=1
\displaystyle =\displaystyle 3Evaluate
\displaystyle x(t)\displaystyle =\displaystyle 2t^{2} + tWrite the first parametric equation
\displaystyle x(3)\displaystyle =\displaystyle 2(3)^{2} + 3Substitute t=3
\displaystyle =\displaystyle 21Evaluate
\displaystyle \text{Average rate of change}\displaystyle =\displaystyle \dfrac{(x(3) - x(1))}{(3 - 1)}Write the formula
\displaystyle =\displaystyle \dfrac{21 - 3}{3 - 1}Substitute x(1)=3,\,x(3)=21
\displaystyle =\displaystyle 9Evaluate

Step 2: Find the average rate of change for y(t) over the interval [1,\, 3].

\displaystyle y(t)\displaystyle =\displaystyle t^{3}-4tWrite the second parametric equation
\displaystyle y(1)\displaystyle =\displaystyle 1^{3}-4(1)Substitute t=1
\displaystyle =\displaystyle -3Evaluate
\displaystyle y(t)\displaystyle =\displaystyle t^{3}-4tWrite the second parametric equation
\displaystyle y(3)\displaystyle =\displaystyle 3^{3}-4(3)Substitute t=3
\displaystyle =\displaystyle 9Evaluate
\displaystyle \text{Average rate of change}\displaystyle =\displaystyle \dfrac{(y(3) - y(1))}{(3 - 1)}Write the formula
\displaystyle =\displaystyle \dfrac{9 - (-3)}{3 - 1}Substitute y(1)=-3,\,y(3)=9
\displaystyle =\displaystyle 6Evaluate

Step 3: Calculate the slope using the average rate of change of x(t) and y(t).

\displaystyle \text{Slope}\displaystyle =\displaystyle \dfrac{\text{Average rate of change of }y(t)}{\text{Average rate of change of }x(t)}Write the formula
\displaystyle =\displaystyle \dfrac{6}{9}Substitute the values of Average rate of change
\displaystyle =\displaystyle \dfrac{2}{3}Simplify

So, the slope of the graph between the points corresponding to t = 1 and t = 3 is \dfrac{2}{3}.

Idea summary

The key formula for average rate of change is \dfrac{(f(t_2) - f(t_1))}{(t_2 - t_1)}. By applying this formula to both x(t) and y(t) functions independently over a given interval, we can compute the average rate of change for each function.

To determine the slope of the graph between the points corresponding to two values of t, we can calculate the ratio of the average rate of change of y(t) to the average rate of change of x(t), provided the average rate of change of x(t) is not equal to 0. This helps us analyze the direction and rate of change of a parametric planar motion function.

Outcomes

4.3.A

Identify key characteristics of a parametric planar motion function that are related to direction and rate of change.

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