Graphs, Linear Equations, and Functions

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Graphs, Linear Equations, and Functions

• 3-1 The Rectangular Coordinate System
• 3-2 The Slope of a Line
• 3-3 Linear Equations in two variables
• 3-5 Introduction to Functions

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3-1 The Rectangular Coordinate System

• Plotting ordered pairs.
• An ordered pair of numbers is a pair of
  numbers written within parenthesis in which the
  order of the numbers is important.
• Example 1 (3,1), (-5,6), (0,0) are ordered
  pairs.
• Note The parenthesis used to represent an
  ordered pair are also used to represent an open
  interval. The context of the problem tells
  whether the symbols are ordered pairs or an open
  interval.
• Graphing an ordered pair requires the use of
  graph paper and the use of two perpendicular
  number lines that intersect at their 0 points.
  The common 0 point is called the origin. The
  horizontal number line is referred to as the
  x-axis or abscissa and the vertical line is
  referred to as the y-axis or ordinate. In an
  ordered pair, the first number refers to the
  position of the point on the x-axis, and the
  second number refers to the position of the point
  on the y-axis.

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3-1 The Rectangular Coordinate System

• Plotting ordered pairs.
• The x-axis and the y-axis make up a rectangular
  or Cartesian coordinate system.
• Points are graphed by moving the appropriate
  number of units in the x direction, than moving
  the appropriate number of units in the y
  direction. (point A has coordinates (3,1), the
  point was found by moving 3 units in the positive
  x direction, then 1 in the positive y direction)
• The four regions of the graph are called
  quadrants. A point on the x-axis or y-axis does
  not belong to any quadrant (point E). The
  quadrants are numbered.

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3-1 The Rectangular Coordinate System

• Finding ordered pairs that satisfy a given
  equation.
• To find ordered pairs that satisfy an equation,
  select any number for one of the variables,
  substitute into the equation that value, and
  solve for the other variable.
• Example 2 For 3x 4y 12, complete the table
  shown
• Solution

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3-1 The Rectangular Coordinate System

• Graphing lines
• Since an equation in two variables is satisfied
  with an infinite number of ordered pairs, it is
  common practice to graph an equation to give a
  picture of the solution.
• Note An equation like 2x 3y 6 is called a
  first degree equation because it has no term
  with a variable to a power greater than 1.
• The graph of any first degree equation in two
  variable is a straight line.
• A linear equation in two variable can be written
  in the form Ax By C, where A,B and C are
  real numbers
• (A and B both not zero). This form is called the
  Standard Form.

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3-1 The Rectangular Coordinate System

• Graphing lines
• Example 3 Draw the graph of 2x 3y 6
• Step 1 Find a table of ordered pairs that
  satisfy the equation.
• Step 2 Plot the points on a rectangular
  coordinate system.
• Step 3 Draw the straight line that would pass
  through the points.
• Step 1 Step 2 Step 3

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3-1 The Rectangular Coordinate System

• Finding Intercepts
• In the equation of a line, let y 0 to find the
  x-intercept and let x 0 to find the
  y-intercept.
• Note A linear equation with both x and y
  variables will have both x- and y-intercepts.
• Example 4 Find the intercepts and draw the graph
  of 2x y 4
• x-intercept Let y 0 2x 0 4 2x 4
  x 2
• y-intercept Let x 0 2(0) y 4 -y 4
  y -4
• x-intercept is (2,0)
• y-intercept is (0,-4)

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3-1 The Rectangular Coordinate System

• Finding Intercepts
• In the equation of a line, let y 0 to find the
  x-intercept and let x 0 to find the
  y-intercept.
• Example 5 Find the intercepts and draw the graph
  of 4x y -3
• x-intercept Let y 0 4x 0 -3 4x -3
  x -3/4
• y-intercept Let x 0 4(0) y -3 -y
  -3 y 3

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3-1 The Rectangular Coordinate System

• Recognizing equations of vertical and
  horizontal lines
• An equation with only the variable x will always
  intersect the x-axis and thus will be vertical.
• An equation with only the variable y will always
  intersect the y-axis and thus will be horizontal.
• Example 6 A) Draw the graph of y 3
• B) Draw the graph of x 2 0 x -2
• A) B)

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3-1 The Rectangular Coordinate System

• Graphing a line that passes through the origin
  
• Some lines have both the x- and y-intercepts at
  the origin.
• Note An equation of the form Ax By 0 will
  always pass through the origin. Find a multiple
  of the coefficients of x and y and use that value
  to find a second ordered pair that satisfies the
  equation.
• Example 7
• A) Graph x 2y 0

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3-2 The Slope of a Line

• Finding the slope of a line given two points on
  the line
• The slope of the line through two distinct
  points
• (x1, y1) and (x2, y2) is
• Note Be careful to subtract the y-values and
  the x-values in the same order.
• Correct Incorrect

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3-2 The Slope of a Line

• Finding the slope of a line given two points on
  the line
• Example 1) Find the slope of the line through the
  points
• (2,-1) and (-5,3)

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3-2 The Slope of a Line

• Finding the slope of a line given an equation
  of the line The slope can be found by solving
  the equation such that y is solved for on the
  left side of the equal sign. This is called the
  slope-intercept form of a line. The slope is the
  coefficient of x and the other term is the
  y-intercept. The slope-intercept form is
• y mx b
• Example 2) Find the slope of the line given 3x
  4y 12

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3-2 The Slope of a Line

• Finding the slope of a line given an equation
  of the line
• Example 3) Find the slope of the line given y 3
  0
• y 0x - 3 ?The slope is 0
• Example 4) Find the slope of the line given x 6
  0
• Since it is not possible to solve for y, the
  slope is Undefined
• Note Being undefined should not be described
  as no slope
• Example 5) Find the slope of the line given 3x
  4y 9

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3-2 The Slope of a Line

• Graph a line given its slope and a point on the
  line Locate the first point, then use the slope
  to find a second point.
• Note Graphing a line requires a minimum of two
  points. From the first point, move a positive or
  negative change in y as indicated by the value of
  the slope, then move a positive value of x.
• Example 6) Graph the line given
• slope passing through (-1,4)
• Note change in y is 2

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3-2 The Slope of a Line

• Graph a line given its slope and a point on the
  line Locate the first point, then use the slope
  to find a second point.
• Example 7) Graph the line given
• slope -4 passing through (3,1)
• Note
• A positive slope indicates the line
  moves up from L to R
• A negative slope indicates the line moves
  down from L to R

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3-2 The Slope of a Line

• Using slope to determine whether two lines are
  parallel, perpendicular, or neither
• Two non-vertical lines having the same slope are
  parallel.
• Two non-vertical lines whose slopes are negative
  reciprocals are perpendicular.
• Example 8) Is the line through (-1,2) and (3,5)
  parallel to the line through (4,7) and (8,10)?

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3-2 The Slope of a Line

• Using slope to determine whether two lines are
  parallel, perpendicular, or neither
• Two non-vertical lines having the same slope are
  parallel.
• Two non-vertical lines whose slopes are negative
  reciprocals are perpendicular.
• Example 9) Are the lines 3x 5y 6 and 5x - 3y
  2 parallel, perpendicular, or neither?

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3-2 The Slope of a Line

• Solving Problems involving average rate of
  change The slope gives the average rate of
  change in y per unit change in x, where the value
  of y depends on x.
• Example 10) The graph shown approximates the
  percent of US households owing multiple pcs in
  the years 1997-2001.
• Find the average rate of change between years
  2000 and 1997.

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3-2 The Slope of a Line

• Solving Problems involving average rate of
  change The slope gives the average rate of
  change in y per unit change in x, where the value
  of y depends on x.
• Example 11) In 1997, 36.4 of high school
  students smoked. In 2001, 28.5 smoked.
• Find the average rate of change in percent per
  year.

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3-3 Linear Equations in Two Variables

• Writing an equation of a line given its slope
  and y-intercept The slope can be found by
  solving the equation such that y is solved for on
  the left side of the equal sign. This is called
  the slope-intercept form of a line. The slope is
  the coefficient of x and the other term is the
  y-intercept.
• The slope-intercept form is y mx b, where m
  is the slope and b is the y-intercept.
• Example 1 Find an equation of the line with
  slope 2 and y-intercept (0,-3)
• Since m 2 and b -3, y 2x - 3

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3-3 Linear Equations in Two Variables

• Graphing a line using its slope and
  y-intercept
• Example 2 Graph the line using the slope and
  y-intercept y 3x - 6
• Since b -6, one point on the line is (0,-6).
• Locate the point and use the slope (m ) to
  locate a second point.
• (01,-63)
• (1,-3)

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3-3 Linear Equations in Two Variables

• Writing an equation of a line given its slope
  and a point on the line The Point-Slope form
  of the equation of a line with slope m and
  passing through the point (x1,y1) is
• y - y1 m(x - x1)
• where m is the given slope and x1 and y1 are the
  respective values of the given point.
• Example 3 Find an equation of a line with slope
  and a given point (3,-4)

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3-3 Linear Equations in Two Variables

• Writing an equation of a line given two points
  on the line The standard form for a line was
  defined as Ax By C.
• Example 4 Find an equation of a line with
  passing through the points (-2,6) and (1,4).
  Write the answer in standard form.
• Step 1 Find the slope
• Step 2 Use the point-slope
• method

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3-3 Linear Equations in Two Variables

• Finding equations of Parallel or Perpendicular
  lines
• If parallel lines are required, the slopes are
  identical.
• If perpendicular lines are required, use slopes
  that are negative reciprocals of each other.
• Example 5 Find an equation of a line passing
  through the point (-8,3) and parallel to 2x - 3y
  10.
• Step 1 Find the slope Step 2 Use the
  point-slope method
• of the given line

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3-3 Linear Equations in Two Variables

• Finding equations of Parallel or Perpendicular
  lines
• Example 6 Find an equation of a line passing
  through the point (-8,3) and perpendicular to 2x
  - 3y 10.
• Step 1 Find the slope Step 3 Use the
  point-slope method
• of the given line
• Step 2 Take the negative reciprocal
• of the slope found

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3-3 Linear Equations in Two Variables
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3-3 Linear Equations in Two Variables

• Writing an equation of a line that models real
  data If the data changes at a fairly constant
  rate, the rate of change is the slope. An initial
  condition would be the y-intercept.
• Example 7 Suppose there is a flat rate of .20
  plus a charge of .10/minute to make a phone
  call. Write an equation that gives the cost y for
  a call of x minutes.
• Note The initial condition is the flat rate of
  .20 and the rate of change is .10/minute.
• Solution y .10x .20

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3-3 Linear Equations in Two Variables

• Writing an equation of a line that models real
  data If the data changes at a fairly constant
  rate, the rate of change is the slope. An initial
  condition would be the y-intercept.
• Example 8 The percentage of mothers of children
  under 1 year old who participated in the US labor
  force is shown in the table. Find an equation
  that models the data.
• Using (1980,38) and (1998,50)

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3-5 Introduction to Functions

• Defining and Identifying Relations and Functions
  If the value of the variable y depends upon the
  value of the variable x, then y is the dependent
  variable and x is the independent variable.
• Example 1 The amount of a paycheck depends upon
  the number of hours worked. Then an ordered pair
  (5,40) would indicate that if you worked 5 hours,
  you would be paid 40. Then (x,y) would show x as
  the independent variable and y as the dependent
  variable.
• A relation is a set of ordered pairs.
• (5,40), (10,80), (20,160), (40,320) is a
  relation.

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3-5 Introduction to Functions

• Defining and Identifying Relations and Functions
  
• A function is a relation such that for each
  value of the independent variable, there is one
  and only one value of the dependent variable.
• Note In a function, no two ordered pairs can
  have the same 1st component and different 2nd
  components.
• Example 2 Determine whether the relation is a
  function (-4,1), (-2,1), (-2,0)
• Solution Not a function, since the independent
  variable has more than one dependent value.

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3-5 Introduction to Functions

• Defining and Identifying Relations and Functions
  
• Since relations and functions are sets of
  ordered pairs, they can be represented as tables
  or graphs. It is common to describe the relation
  or function using a rule that explains the
  relationship between the independent and
  dependent variable.
• Note The rule may be given in words or given as
  an equation.
• y 2x 4 where x is the independent
  variable and y is the dependent variable

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3-5 Introduction to Functions

• Domain and Range
• In a relation
• A) the set of all values of the independent
  variable (x) is the domain.
• B) the set of all values of the dependent
  variable (y) is the range.
• Example 3 Give the domain and range of each
  relation. Is the relation a function?
• (3,-1), (4,-2),(4,5), (6,8)
• Domain 3,4,6
• Range -1,-2,5,8
• Not a function
• Example 4 Give the domain and range of each
  relation. Is the relation a function?
• Domain 0,2,4,6
• Range 4,8,12,16

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3-5 Introduction to Functions

• Domain and Range
• In a relation
• A) the set of all values of the independent
  variable (x) is the domain.
• B) the set of all values of the dependent
  variable (y) is the range.
• Example 4 Give the domain and range of each
  relation. Is the relation a function?
• Domain
• 1994,1995,1996,1997,1998,1999
• Range
• 24134,33786,44043,55312,69209,86047
• This is a function

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3-5 Introduction to Functions

• Domain and Range
• In a relation
• A) the set of all values of the independent
  variable (x) is the domain.
• B) the set of all values of the dependent
  variable (y) is the range.
• Example 5 Give the domain and range of each
  relation.
• Domain (-?, ? ) Domain -4, 4
• Range (-?, 4 Range -6, 6

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3-5 Introduction to Functions

• Agreement on Domain
• Unless specified otherwise, the domain of a
  relation is assumed to be all real numbers that
  produce real numbers when substituted for the
  independent variable.
• The function
• has all real numbers except x 0
• Note In general, the domain of a function
  defined by an algebraic expression is all real
  numbers except those numbers that lead to
  division by zero or an even root of a negative
  number.
• The function is not defined for values lt
  

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3-5 Introduction to Functions

• Identifying functions defined by graphs and
  equations
• Vertical Line Test
• If every vertical line intersects the graph of a
  relation in no more than one point, the relation
  represents a function .

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3-5 Introduction to Functions

• Identifying functions defined by graphs and
  equations
• Vertical Line Test
• If every vertical line intersects the graph of a
  relation in no more than one point, the relation
  represents a function.
• Function Not a Function

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3-5 Introduction to Functions

• Identifying functions defined by graphs and
  equations
• Example 5 Decide whether the equations shown
  define a function and give the domain
• Function Function Not a
  Function
• Domain Domain Domain

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3-5 Introduction to Functions

• Using Function Notation
• When a rule or equation is defined such that y
  is dependent on x, the Function Notation y
  f(x) is used and is read as y f of x where
  the letter f stands for function.
• Note The symbol f(x) does not indicate that f
  is multiplied by x, but represents the y-value
  for the indicated x-value.
• If y 9x -5, then f(x) 9x -5 and f(2) 9(2)
  -5 13
• and f(0) 9(0) -5 -5
• Example 6

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3-5 Introduction to Functions

• Using Function Notation
• When a rule or equation is defined such that y
  is dependent on x, the Function Notation y
  f(x) is used and is read as y f of x where
  the letter f stands for function.
• Example 7 f(x) 5x -1 find f(m 2)
• f(m 2) 5(m 2) -1 5m 10 - 1
• f(m 2) 5m 9
• Example 8 Rewrite the equation given and find
  f(1) and f(a)

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3-5 Introduction to Functions

• Identifying Linear Functions
• A function that can be defined by f(x) mx b
  for real numbers m and b is a Linear Function
• Note The domain of a linear function is (-?,
  ?).
• The range is (-?, ?).
• Note Remember that m represents the slope of a
  line and (0,b) is the y-intercept.
• A function that can be defined by f(x) b is
  called a Constant Function which has a graph
  that is a horizontal line.
• Note The range of a constant function is b.