Graphs, Equations and Inequalities

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CHAPTER 2

• Graphs, Equations and Inequalities

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Contents

• Graph
• Slops and equations of line
• Linear Models
• Linear inequalities

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2.1 Graph

• Cartesian coordinate system
• Origin
• x-axis, y-axis
• x- coordinate, y- coordinate
• Solution of an equation
• Graph of an equation
• x-intercept, y- intercept

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Cartesian coordinates system
y-axis
(-4,4)
(4,3)
Quadrant II
Quadrant I
(0,2)
(-5,0)
(3,0)
x-axis
origin
Quadrant IV
(-4,-3)
(5,-4)
Quadrant III
(0,-5)
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Solution of an Equation

• Example 1 Which of the followings is the
  solution of y -2x 5
• A) (1,3)
• B) (4,3)
• Example 2 Which of the followings is the
  solution of y x2 5x 6
• A) (1,1)
• B) (-6,0)

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Graph

• Example 3
• Graph y -2x 5
• Example 4
• Graph y x2 5x 6

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x-intercept and y-intercept

• x-intercept the x-coordinate of a point where
  the graph intercepts the x-axis
• y-intercept the y-coordinate of a point where
  the graph intercepts the y-axis
• Example 5 Find x and y intercept of graph y -2x
  5
• Example 6 Find x and y intercept of graph y x2
  5x 6

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2.2 Slope and Equations Of Lines

• Definition of slope
• Slope and intercepts
• Slope Intercept form
• Point Slope form
• Parallel and perpendicular lines
• Summary Table

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Definition of Slope

• ?x (change in x)
• ?y (change in y)
• Slope of a line
• Though the two points (x1, , y1) and (x2 , y2),
  where (x1?x2) is defined as the quotient of the
  change in y and the change in x, or

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Examples

• Example 1 Find the slope of the line through the
  point (4,6) and (1,2)
• Example 2 Find the slope of the line through the
  point (3,-5) and (2,-5)
• Example 3 Find the slope of the line through the
  point (3,5) and (3,-5)

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Slope and intercepts

• The slope of every horizontal line is 0.
• The slope of every vertical line is undefined.
• If k is a constant, then the graph of the
  equation y k is the horizontal line with
  y-intercep k.
• If k is a constant, then the graph of the
  equation x k is vertical line with x- intercept
  k.

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Slope Intercept form

• If a line has slope m and y-intercept b, then it
  is the graph of the equation
• y mx b.
• This equation is called the slope- intercept form
  of the equation of the line.

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Examples

• Example 1 Find an equation for the line with
  y-intercept 5/2 and slope-1/2
• Example 2 Find the equation of the horizontal
  line with y-intercept 5
• Example 3 Find the slope and y-intercept for the
  following line 12x-3y

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Point Slope form

• If a line has slope m and passes though the point
  (x1 , y1), then
• y y1 m(x x1 )
• is the point- slope form of equation of the line.

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Examples

• Example 1 Find the equation of the line
  satisfying the given condition
• Slope 3, the point (3,5)
• Example 2 Find an equation of the line through
  (-8,2) and (3,-6)

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Parallel and Perpendicular lines

• Two nonvertical line are parallel whenever they
  have same slope.
• Two nonvertical lines are perpendicular whenever
  the product of their slopes is 1.

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Examples

• Determine whether each of the following pairs are
  parallel, perpendicular, or neither
• A) x - 2y 6 and 2x y 5
• B) 3x 4y 8 and x 3y 2
• C) 2x y 7 and 2y 4x -5

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Summary
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2.3 Linear Models

• The profits of the General Electric Company in
  billions dollars, over a five-year period are
  shown in the following table

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Linear Models

• a) Let x0 correspond to 2000, and plot the
  points (x,y), where x is the year and y the
  profit
• The data points are (0,13), (1,14), (2,14),
  (3,15), and (4,17)
• b) Use data point (0,13) and (4,17) to find a
  line that models the data
• y x 13

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Linear Models

• c) Use data point (1,14) and (3,15) to find a
  line that models the data
• y .5x 13.5
• d) Use the model in part b) to estimate the
  profit in 2005
• y(5) 5 13 18 billion
• Data point (x,p) and point on the line (x,y),
  (p-y) is the error in the model

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Linear Models

• (p-y) residual
• Use the sum of the squares of the residual to
  measure how well a line fits the data points
• Perfect fit the error is zero
• Linear Regression For any set of data point,
  there is one and only one line for which the sum
  of square of the residual is as small as possible
• This line of best fit is the least squares
  regression line.
• The computational process for finding its
  equation is linear regression

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2.4 LINEAR INEQUALITIES

• Inequality statement that one expression is
  greater than (or less than) another.
• Example
• 4 3x 7 2x

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Properties of Inequalities

• For real numbers a, b, c
• If a lt b, then a c lt b c
• If a lt b, then
• If c gt 0, then ac lt bc
• If c lt 0, then ac gt bc
• Note the properties are valid not only for lt,
  but also for gt, , .

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Examples

• Solve 3x 5 gt 11
• Solve 4 3x 7 2x
• Solve -2 lt 5 3m lt 20
• The formula for converting from Celsius to
  Fahrenheit isF (9/5)C 32What Celsius range
  corresponds to the range from 32ºF to 77ºF?

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Inequalities with absolute values

• Assume a and b are real numbers with b positive.
• Solve a b by solving ab or a - b
• Solve a lt b by solving b lt a lt b
• Solve a gt b y solving a lt-b or a gt b

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Examples

• Solve x 2 lt 3
• Solve 2 7m - 1 gt 5
• Solve 2 5x gt -4
• Write statement using absolute value k is at
  least 4 units from 1
• Write statement using absolute valuep is within
  2 units of 5

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APPLICATIONS OF LINEAR EQUATIONS

• Example 1 Celsius and Fahrenheit temperature.
• 32?F corresponds to 0?C,
• 212?F corresponds to 100?C.
• The relationship between Celsius and Fahrenheit
  temperature is linear.
• Write the equation of this relationship.
• Find the Celsius temperature corresponding to
  68?F.

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APPLICATIONS OF LINEAR EQUATIONS

• Example 2 Break-even point.
• Profit Revenue Cost
• p r c
• Cost of producing x units c 20x 100
• The unit price is 24/unit r 24x
• Find the break-even point.
• Graph of cost and revenue equation.

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APPLICATIONS OF LINEAR EQUATIONS

• Example 3 Supply and demand.
• p price, q quantity
• Demand equation p 60 (3/4)q
• A) Find the demand at a price of 40 per unit
• B) Find the price if the demand is 32 units

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APPLICATIONS OF LINEAR EQUATIONS

• Example 4 Supply and demand.
• p price, q quantity
• Supply equation p .85q
• A) Find the supply if the price is 51 per unit
• B) Find the price per unit if the supply is 17
  per unit

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APPLICATIONS OF LINEAR EQUATIONS

• Example 5 Supply and demand.
• p price, q quantity
• Demand equation p 60 (3/4)q
• Supply equation p .85q
• Graph both equations
• Find the equilibrium price
• Find the equilibrium demand/supply