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

4.1 Parametric functions

Lesson
Practice
Lesson

Introduction

Learning objective 4.1.A

  • Construct a graph or table of values for a parametric function represented analytically.

Understanding parametric functions

The coordinates (x_{i},\, y_{i}) at time t_{i} can be written as functions of t and can be expressed as the single parametric function f(t) = (x(t),\, y(t)), where in this case x and y are names of two functions. For example, consider the parametric function f(t) = (x(t),\, y(t)) where x(t) = t^2 and y(t) = 2t. In this parametric function, x and y are dependent on the parameter t. As t changes, the values of x and y change according to their respective functions: x(t) = t^{2} \\ y(t) = 2t

The parametric function: f(t)=(x(t),\, y(t)). Replace x(t) and y(t) in the expression f(t) = (x(t),\, y(t)), with the provided functions. The parametric function f(t) for the given x(t) and y(t) functions is f(t) = (t^{2},\, 2t).

Parametric function

A function that represents a relationship between two dependent variables, x and y, that are both dependent on a single independent variable, t, called the parameter. Parametric functions are typically represented as a set of two parametric equations

Parameter

The independent variable, t, in a parametric function, which influences the values of the dependent variables, x and y.

Parametric equations

A set of two equations that define a parametric function, where each equation expresses one of the dependent variables, x or y, in terms of the independent variable, t.

\Reals^{2}

The set of all ordered pairs of two real numbers, often used to represent the plane in which parametric functions are graphed.

Examples

Example 1

Given x(t) = 3t - 1 and y(t) = 2t + 1, express the parametric function f(t).

Worked Solution
Create a strategy

Replace x(t) and y(t) in the expression f(t) = (x(t),\, y(t)) with the provided functions.

Apply the idea
\displaystyle f(t)\displaystyle =\displaystyle (x(t),\, y(t))Write the parametric function
\displaystyle f(t)\displaystyle =\displaystyle (3t - 1,\, 2t + 1)Substitute the values of x(t) and y(t)
Idea summary

Parametric functions involve two dependent variables, x and y, dependent on a single independent variable, t.

Graphing and tabulating parametric functions

A graph of a parametric function can be created by plotting the points from the table of values and connecting them in order of increasing values of t. Visualizing parametric functions through graphs allows us to better understand the behavior and properties of these functions, making it easier to analyze and interpret the underlying relationships.

Table of values

A numerical table generated by evaluating the dependent variables, x_{i} and y_{i}, at several values of the independent variable, t_{i}, within the domain of a parametric function.

Graph of a parametric function

A visual representation of a parametric function, where the coordinates (x_{i},\, y_{i}) are plotted in the plane for various values of t_{i}, and the points are connected in order of increasing value of t.

Given the parametric function f(t) = (x(t),\, y(t)) where x(t) = t^{2} and y(t) = 2t, create a table of values and sketch the graph of the function for t in the interval [0,\, 3].

To create a table of values for a parametric function f(t) = (x(t),\, y(t)):

  1. Choose several values of t within the domain of the function.

  2. Calculate the corresponding x and y values using the given functions.

  3. Fill in the table with the calculated values.

The parametric function: f(t)=(x(t),\, y(t))

f(t) = (t^{2},\, 2t)

txy
000
112
244
396

Sketch the graph of a parametric function f(t)=(x(t),\, y(t)) in the interval [0,\, 3]:

  1. Plot the points from the table on a coordinate plane.

    1
    2
    3
    4
    5
    6
    7
    8
    9
    x
    1
    2
    3
    4
    5
    6
    y

  2. Connect the points in order of increasing value of t.

    1
    2
    3
    4
    5
    6
    7
    8
    9
    x
    1
    2
    3
    4
    5
    6
    y

Examples

Example 2

Given the parametric function f(t) = (3t - 1,\, 2t + 1) with t in the domain [0,\, 3].

a

Create a table of values.

txy
0
1
2
3
Worked Solution
Create a strategy

Evaluate x_{i} and y_{i} at several values of t_{i} within the domain, and create a table with the corresponding (x_{i},\, y_{i}) values.

Apply the idea
\displaystyle x(t)\displaystyle =\displaystyle 3t-1Write the first function
\displaystyle =\displaystyle 3(0)-1Substitute t=0
\displaystyle =\displaystyle -1Evaluate
\displaystyle x(t)\displaystyle =\displaystyle 3t-1Write the first function
\displaystyle =\displaystyle 3(1)-1Substitute t=1
\displaystyle =\displaystyle 2Evaluate
\displaystyle x(t)\displaystyle =\displaystyle 3t-1Write the first function
\displaystyle =\displaystyle 3(2)-1Substitute t=2
\displaystyle =\displaystyle 5Evaluate
\displaystyle x(t)\displaystyle =\displaystyle 3t-1Write the first function
\displaystyle =\displaystyle 3(3)-1Substitute t=3
\displaystyle =\displaystyle 8Evaluate
\displaystyle y(t)\displaystyle =\displaystyle 2t+1Write the second function
\displaystyle =\displaystyle 2(0)+1Substitute t=0
\displaystyle =\displaystyle 1Evaluate
\displaystyle y(t)\displaystyle =\displaystyle 2t+1Write the second function
\displaystyle =\displaystyle 2(1)+1Substitute t=1
\displaystyle =\displaystyle 3Evaluate
\displaystyle y(t)\displaystyle =\displaystyle 2t+1Write the second function
\displaystyle =\displaystyle 2(2)+1Substitute t=2
\displaystyle =\displaystyle 5Evaluate
\displaystyle y(t)\displaystyle =\displaystyle 2t+1Write the second function
\displaystyle =\displaystyle 2(3)+1Substitute t=3
\displaystyle =\displaystyle 7Evaluate

The complete table of values:

txy
0-11
123
255
387
b

Graph the parametric function.

Worked Solution
Create a strategy

Plot the points from the numerical table of values in order of increasing value of t and connect the points to sketch the graph.

Apply the idea
-2
-1
1
2
3
4
5
6
7
8
x
1
2
3
4
5
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7
y

Plot the points from the table on a coordinate plane.

-2
-1
1
2
3
4
5
6
7
8
x
1
2
3
4
5
6
7
y

Connect the points in order of increasing value of t.

Idea summary

A table of values can be created along with a sketch of the graph for parametric functions to be represented analytically. This will help them analyze and understand the behavior of the functions in different contexts.

Outcomes

4.1.A

Construct a graph or table of values for a parametric function represented analytically.

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