Download Unit 2: Using Algebra and Graphs to Describe Relationships

Unit 2: Using Algebra and
Graphs to Describe
Relationships
Learning Goals 2.3 and 2.4
Michelle A. O’Malley
League 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
Function
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 (xcoordinate) 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