Unit 2: Using Algebra and Graphs to Describe

Relationships Learning Goals 2.3 and 2.4

Michelle A. O’MalleyLeague Academy of Communication Arts

Greenville, South Carolina

Unit 2: Learning Goal 2.3

• Determine if a relation is a function and identify the domain (independent variable) and range (dependent variable) of a function.

Unit 2: Learning Goal 2.3 Standards

• EA-3.1 Classify a relationship as being either a function or not a function when given data as a table, set of ordered pairs, or graph.

• EA-3.3 Carry out a procedure to evaluate a function for a given element in the domain.

• EA-3.4 Analyze the graph of a continuous function to determine the domain and range of the function.

• EA-5.10 Analyze given information to determine the domain and range of a linear function in a problem situation.

Unit 2: Learning Goal 2.3 Essential Question

• How can real-world situations be modeled using graphs and functions?

•A relation is a pairing between two sets of numbers to create a set of ordered pairs.•A function is a special type of relation.

It is a pairing between two sets of numbers in which each element of the first set is paired with exactly one element of the second set.

Learning goal 2.3

• The following does not represent a function because one input is paired with more than one output.

• In addition, the following does not represent a function because it does not pass the vertical line test.

A Function

Not a

FunctionFunction

Learning Goal 2.3

• In Functional Relationships one variable changes independently and the values of the other depend on those of the first.

• The Dependent Variable is the value of interest and is determined by the function rule acting upon the dependent variable.– For example, consider the number of

cricket chirps vs. the average temperature. The number of chirps is dependent upon how hot or cold it is and not the other way around.

– Therefore, the temperature is the independent variable (behavior is known) and the number of chirps (value of interest) is the dependent variable.

Learning Goal 2.3

• Occasionally, there are situations in which the independent and dependent variables could be interchanged.

• It is not always obvious which variable is dependent upon the other.– For example, although there is a

relationship between a person’s height and weight, changes in one do not cause changes in the other. In situations like this, the variable that is chosen as the independent variable is a matter of convenience.

Learning Goal 2.3

• The standard convention is to graph the independent variable on the horizontal axis (x-axis) and the dependent variable on the vertical axis (y-axis).

• Remember: DRY/MIX– Dependent-Responding Variable – Y-

axis– Manipulated-Independent Variable –

X-axis

Learning Goal 2.3

• The Domain of a function is:– The x-coordinates of a set of ordered

pairs (x, y)– A set of “input” values– A set of independent values

• Note: In Algebra 1 the Domain of a function is understood to be the set of real numbers.

• The Range of a function is:– The y-coordinates of a set of ordered

pairs (x, y)– The set of “output” values– A set of dependent values

Learning Goal 2.3

• There are many different ways to represent a function.

• Each representation is a way of communicating the same rule or relationship. – Examples:

• A mapping• A verbal description• A graph• A table• A set of ordered pairs• A “function machine”• An algebraic representation

Learning Goal 2.3• Vertical line test can be used to

determine whether the graph of a relation is a function.

• If a vertical line intersects two points on the graph, then the domain value (x-coordinate) has been assigned to two range values (y-coordinate). Therefore, the graph does not represent a function.

Learning Goal 2.3

• The domain and range of a function may be continuous or discrete depending on the situation.– If a variable can take all values within

some interval it is called a continuous variable.

• For example, distance traveled as a function of time is a continuous function since there would be no gaps in either the time or the distance variables.

• The total Cost of tickets as a function of the number of tickets purchased is an example of a discrete function since a fractional part of a ticket cannot be purchased.

• If the independent variable is discrete, the graph is a set of isolated points that are not connected with lines or curves. Depending on the constraints of the problem, the domain can either be finite or infinite.

Unit 2: Learning Goal 2.4

• Use Function notation to evaluate functions for a specific value of the domain.

Unit 2: Learning Goal 2.4 Standards

• EA-3.2 Use function notation to represent functional relationships.

• EA-3.3 Carry out a procedure to evaluate a function for a given element in the domain.

Unit 2: Learning Goal 2.4 Essential Question

• How are functions used in real-world situations?

Learning Goal 2.4

• Functional Notation is a method of writing a function in which the dependent variable is written in the form of f(x). – Note: f names the function and f(x)

represents the y-value of the function.

– The independent variable, x, is written inside the parentheses. • For example, instead of writing the

function as y = 2x + 4, it may be written as f(x) = 2x + 4.

– Note: Any variable may be used for function notation such as f(g), f(h) etc.

Learning Goal 2.4

• To determine functional values from an algebraic rule, substitute the given numbers for x into the function rule to determine the value for f(x).– For example, in the function f(x) = x + 3,

to find f(-2), substitute -2 into the expression to get a function value of 1. Therefore, f(-2) = 1.

• When finding function values, the result can be written as an ordered pair, (-2, 1).

• Functional values can be determined from a table, graph, or ordered pairs.

• Note: (x, y) is equivalent (the same) to (x, f(x)).

Work Cited

• Carter, John A., et. al. Glencoe Mathematics Algebra I. New York: Glencoe/McGraw-Hill, 2003.

• Greenville County Schools Math Curriculum Guide

• Gizmos http://www.explorelearning.com