Unit 2: Using Algebra and Graphs to Describe Relationships Learning Goals 2'3 and 2'4

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Unit 2 Using Algebra and Graphs to Describe
Relationships Learning Goals 2.3 and 2.4

• Michelle A. OMalley
• League Academy of Communication Arts
• Greenville, South Carolina

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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.

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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.

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Unit 2 Learning Goal 2.3 Essential Question

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

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• 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.

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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.

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A Function
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Not a Function
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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.

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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 persons 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.

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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

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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

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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

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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.

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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.

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Unit 2 Learning Goal 2.4

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

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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.

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Unit 2 Learning Goal 2.4 Essential Question

• How are functions used in real-world situations?

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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.

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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)).

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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