Precalculus

1
Precalculus MAT 129

• Instructor Rachel Graham
• Location BETTS Rm. 107
• Time 8 1120 a.m. MWF

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Chapter One

• Functions and Their Graphs

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1.1 Lines in the Plane

• Slope
• Equation of a line
• Point-Slope form
• Slope-intercept form
• General form
• Parallel and Perpendicular Lines
• Applications

4
Slope

• Formula (y2-y1)/ (x2-x1)
• Denoted m
• In words Change in y over change in x

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Slope Generalizations

• Positive slope rises (left to right)
• Negative slope falls (left to right)
• Horizontal line slope 0
• Vertical line slope undefined

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Example 1.1.1

• Pg. 11 7 (Just find the slope)
• Find the slope of the line from the given
  points.
• Points (0,-10) and (-4,0)

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Solution - Ex. 1.1.1

• Use slope formula!!!
• (0 10) / (-4 0)
• 10/(-4)
• m -5/2

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Line Equations

• Point-Slope form
• Y-intercept form
• General form

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Point-Slope Form

• Formula (y-y1) m(x-x1)
• one point and the slope.
• OR
• two points.
• study tip on pg. 5

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Y-Intercept Form

• Most common
• Formula y mx b
• Can be found by solving for y in the point-slope
  form or the general form.
• Easy to sketch a line from this equation.

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General Form

• Formula Ax By C
• Can also be found from either of the other forms.

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Example 2.1.1

• Pg. 12 25
• Find the general form of the line from the given
  point and slope.
• Point (0,-2) Slope 3

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Solution - Ex. 2.1.1

• Use the point-slope formula!!!
• (y 2) 3(x 0)
• y 2 3x
• -3x y -2

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Example 3.1.1

• Pg. 12 29
• Find the general form of the line from the given
  point and slope.
• Point (6,-1) Slope undefined

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Solution - Ex. 3.1.1

• If the slope is undefined we know that it is a
  vertical line. A vertical line only crosses the
  x-axis. So all we need is the x-value from the
  point given.
• x 6
• This is also in general form.

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Parallel and Perpendicular Lines

• If two lines are parallel they have the same
  slope.
• If two lines are perpendicular their slopes are
  negative reciprocals of each other.
• Change the sign and flip it!

17
Example 4.1.1

• Pg. 12 57
• Write the slope-intercept form of the equation
  of the line through the give point
• a) parallel to the line given.
• b) perpendicular to the line given.
• Point (2,1) Line gt 4x -2y 3

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Solution - Ex. 4.1.1

• Solving the given equation for y we get
• y 2x 3/2
• same slope (use the point-slope form)
• (y 1) 2(x 2)
• y 1 2x 4
• y 2x 3

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Solution - Ex. 4.1.1

• b) negative reciprocal slope (use the
  point-slope form).
• (y 1) (-1/2)(x 2)
• y 1 (-1/2x) 1
• y (-1/2x) 2

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Example 5.1.1

• Pg. 12 53
• Determine whether the lines L1 and L2 passing
  through the pairs of points are parallel,
  perpendicular, or neither.
• L1 ( 0,-1) and (5,9)
• L2 (0, 3) and (4,1)

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Solution - Ex. 5.1.1

• L1 Slope (9 1)/(5 0) 2
• L2 Slope (1 3)/(4 0) -2/4 -1/2
• m1 m2 -1 so these are negative reciprocals
• L1 and L2 are perpendicular.

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Example 6.1.1

• Pg. 13 71
• Looking at the graph
• a) Use the slopes to determine the year(s) in
  which the earnings per share of stock showed the
  greatest increase and decrease.
• b) Find the equation of the line between the
  years 1992 and 2002.

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Example 6.1.1

• Pg. 13 71 (cont.)
• c) Interpret the meaning of the slope from part
  (b) in the context of the problem.
• d) Use the equation from part (b) to estimate
  the earnings per share of stock for the year
  2006. Do you think this is an accurate
  estimation? Explain.

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Solution - Ex. 6.1.1

• a) The greatest increase was between 1998 and
  1999.
• b) 2 points (2, 0.58) and (12, 0.08) gives the
  slope -0.05.
• Use point-slope (y 0.58)-0.05(x-2)
• y - 0.05x 0.68

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Solution - Ex. 6.1.1

• c) For every year increase there is a 0.05
  decrease in earning per share.
• d) Plug in the appropriate number
• y -0.05(16) 0.68
• -0.12
• This is not accurate because our line does not
  accurately represent the data.

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Example 7.1.1

• Pg. 14 85
• A controller purchases a bulldozer for 36,500.
  The bulldozer requires an average expenditure of
  5.25/hr for fuel and maintenance, and the
  operator is paid 11.50/hr.
• a) Write a linear equation giving the total cost
  (C) of operating the bulldozer for t hours.
  (Include the purchase cost of the bulldozer)

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Example 7.1.1

• Pg. 14 85 (cont.)
• b) Assuming that customers are charged 27/hr
  of bulldozer use, write an equation for the
  revenue (R) derived from t hours of use.
• c) Use the profit formula (PR-C) to write an
  equation for the profit derived from t hours of
  use.

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Example 7.1.1

• Pg. 14 85 (cont.)
• d) Use the result of part (c) to find the
  break-even point. (The number of hours the
  bulldozer must be used to yield a profit of 0.

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Solution - Ex. 7.1.1

• a) C 16.75t 36,500
• b) R 27t
• c) P 10.25t 36,500
• d) t 3561 hours (graph and approx. where the
  lines cross)

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1.2 Functions

• Definitions
• Testing for functions
• Evaluating a function
• Domain of a function
• Applications
• Difference Quotients

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Definitions

• function a function is a relationship between
  two variables such that to each value of the
  independent variable there corresponds exactly
  one value of the dependent variable
• domain the domain of a function is the set of
  all values of the independent variable for which
  the function is defined.
• range the range of a function is the set of all
  values assumed by the dependent variable

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Definitions

• X independent variable
• Y dependent variable

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Testing for Functions

• If you are given the points check to see if
  there are any of the same x-values.
• If so, then it is not a function.
• Easiest way to test if a line is a function is to
  graph it and do the vertical line test.
• Solve for y
• Graph the line
• Do the vertical line test (if only touches once
  then it is a function)

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Evaluating a function

• This is where we are putting something in our
  bucket (the variable).
• At a given x-value what is the y-value?

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Example 1.1.2

• Pg. 19 Ex. 3 Evaluating a Function
• Let g(x) -x2 4x 1.
• Find (a) g(2)
• (b) g(t)
• (c) g(x2).

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Solution - Ex. 1.1.2

• a. Replace x with 2 in g(x)
• g(2) -(2)2 4(2) 1
• 5
• b. Replace x with t in g(x)
• g(t) -(t)2 4(t) 1
• -t2 4t 1

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Solution - Ex. 1.1.2

• c. Replace x with (x 2) in g(x)
• g(x 2) -(x 2)2 4(x 2) 1
• -(x2 4x 4) 4x 8 1
• -x2 5

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Piecewise-Defined Function

• A function that is defined by two or more
  equations over a specified domain.
• See example pg. 19 in beige box.

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Example 2.1.2

• On the board (19)

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Domain of a function

• Domain is the set of all real numbers for which
  the expression is defined.
• Its as easy as traveling along the x-axis on the
  road that is your function. You can also figure
  out where the function cannot be defined.

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Example 3.1.2

• Pg. 26 55
• Find the domain of the function.
• h(t) 4/t

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Solution - Ex. 3.1.2

• Pg. 26 55
• All real values of t except for t0.

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Applications

• Go over Example 8 on pg. 22

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Difference Quotients

• Basic definition in calculus
• ( f(x h) f(x)) / h, h ? 0

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Example 4.1.2

• Pg. 29 89
• Find the difference quotient and simplify your
  answer.
• f(x) x2 x 1, (f(2 h) f(2))/h, h?0.
• Work on the board!!

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Activities (23)

• 1. Evaluate f(x) 2 3x x2 for
• a. f(-3)
• b. f(x 1)
• c. f(x h) f(x)
• 2. Find the domain f(x) 3/(x1).

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1.3 Graphs of Functions

• The Graph of a Function
• Increasing and Decreasing Functions
• Relative Minimum and Maximum Values
• Graphing Step Functions and Piecewise-Defined
  Functions
• Even and Odd Functions

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The Graph of a Function

• x the directed distance from the y-axis.
• y the directed distance from the x-axis.
• Go over Example 2 pg. 31
• Note both Algebraic and Graphical solutions.

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Increasing and Decreasing Functions

• Increasing lt- the function is rising on the
  interval
• Decreasing lt- the function is falling on the
  interval
• Constant

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Example 1.1.3

• Pg. 39 21
• Determine the intervals over which the function
  is increasing, decreasing, or constant.
• f(x) x3 3x2 2
• Graph on calculator.
• Draw on the board.

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Solution - Ex. 1.1.3

• Pg. 39 21
• Increasing on (-8, 0),(2, 8)
• Decreasing on (0,2)

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Relative Minimum and Maximum Values

• A function value f(a) is called a relative
  minimum of f if there exists an interval
• (x1, x2) that contains a such that
• x1 lt x lt x2 implies f(a) f(x).
• Likewise if x1 lt x lt x2 implies f(a) f(x) then
  f(a) is called the relative maximum.
• See Figure 1.24 on pg. 33

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Example 2.1.3

• Pg. 39 31
• Use a graphing utility to approximate (to two
  decimal places) any relative minimum or maximum
  values of the function.
• f(x) x2 6x
• Graph on calculator.
• Use trace on calculator to approximate.

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Solution - Ex. 2.1.3

• Pg. 39 31
• Relative minimum (3, -9)

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Even and Odd Functions

• A function is even if, for each x in the domain
  of f, f(-x) f(x).
• These graphs are symmetric with respect to the
  y-axis.
• A function is odd if, for each x in the domain of
  f, f(-x) -f(x).
• These graphs are symmetric with respect to the
  origin.

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Example 3.1.3

• Pg. 37 Ex. 10
• Determine whether each function is even, odd, or
  neither.
• a. g(x) x3 x
• b. h(x) x2 1
• c. f(x) x3 1

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Solution - Ex. 3.1.3

• Pg. 37 Ex. 10
• See both algebraic and graphical solution on page
  37.

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Example 4.1.3

• Pg. 40 61
• Algebraically determine whether the function is
  even, odd, or neither.
• g(x) x3 5x

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Solution - Ex. 4.1.3

• Pg. 40 61
• f(-x) -f(x) so it is an odd function

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Example 5.1.3

• Pg. 40 75
• Use a graphing utility to graph the function and
  determine whether the function is even, odd, or
  neither.
• f(x) 3x - 2

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Solution - Ex. 5.1.3

• Pg. 40 75
• The graph is not symmetric to the y-axis or the
  origin so the function is neither.

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1.4 Shifting, Reflecting, and Stretching Graphs

• Summary of Graphs of Common Functions
• Vertical and Horizontal Shifts
• Reflecting Graphs
• Non-rigid Transformations

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Summary of Graphs of Common Functions

• Reading from pg. 42 at the top
• One of the goals of this text is to enable you
  to build your intuition for the basic shapes of
  the graphs of different types of functions.
• See the six graphs on pg. 42

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Vertical and Horizontal Shifts

• Let c be a positive real number.
• Vertical shift c units upward h(x) f(x)c
• Vertical shift c units downward h(x) f(x)-c
• Horizontal shift c units right h(x) f(x - c)
• Horizontal shift c units left h(x) f(x c)
• Do the exploration on pg. 43

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Example 1.1.4

• Pg. 48 3
• Sketch the graphs of the three functions by hand
  on the same rectangular coordinate system
• f(x) x2
• g(x) x2 2
• h(x) (x - 2)2

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Solution - Ex. 1.1.4

• Pg. 48 3
• See on the board and on the calculator.

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Reflecting Graphs

• Reflections in the coordinate axes of the graph
  of y f(x) are represented as follows.
• Reflection in the x-axis h(x) -f(x)
• Reflection in the y-axis h(x) f(-x)
• Do the exploration on pg. 45

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Example 2.1.4

• Pg. 49 15-25 odd
• We will do this together as a class.

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Non-rigid Transformations

• These are transformations that distort the graph
  by shrinking and stretching the graph.
• Given by y cf(x)
• The transformation is a vertical stretch if cgt1
• The transformation is a vertical shrink if 0ltclt1

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Example 3.1.4

• Pg. 49 37
• Compare the graph of the function with the graph
  of f(x) x3.
• p(x) (1/3x)3 2

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Solution - Ex. 3.1.4

• Pg. 49 37
• It will shrink the graph by 3 and vertical shift
  it up 2.

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Example 4.1.4

• Pg. 49 49 and 51
• G is related to one of the six parent graph
  functions on page 42.
• Identify the parent function
• Describe the transformation

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1.5 Combinations of Functions

• Arithmetic Combinations of Functions
• Compositions of Functions

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Arithmetic Combinations of Functions

• Let f and g be two functions with overlapping
  domains. Then, for all x common to both domains,
  the sum, difference, product, and quotient of f
  and g are defined as follows
• Sum (fg)(x) f(x) g(x)
• Difference (f-g)(x) f(x) - g(x)
• Product (fg)(x) f(x) g(x)
• Quotient (f/g)(x) f(x) / g(x), g(x) ? 0.

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Compositions of Functions

• The composition of the funciton f with the
  function g is
• (f o g)(x) f(g(x)).

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Example 1.1.5

• Pg. 55 Ex. 7
• Note both the Algebraic and Graphical Solutions.

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Example 2.1.5

• Pg. 59 39
• Find (f o g), (g o f), and the domain of (f o g).
• f(x) sqrt(x 4), g(x) x2

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Solution - Ex. 2.1.5

• Pg. 59 39
• (f o g)(x) sqrt(x2 4)
• (g o f)(x) x 4, x -4
• Domain all real numbers

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Example 3.1.5

• Pg. 57 Ex. 11
• Is the N(T(t)) composition
• Here we just substitute 2 into our composite
  function in part a
• Solving for t

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1.6 Inverse Functions

• Inverse Functions
• The Graph of an Inverse Function
• The Existence of an Inverse Function
• Finding Inverse Functions Algebraically

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Inverse Functions

• When a function f is composed with f-1 (called
  f-inverse) and vice versa they are equal to x.

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Example 1.1.6

• Pg. 62 Ex. 1
• Pg. 64 Ex. 4

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The Graph of an Inverse Function

• Reading from pg. 65 (top of the page)
• The graphs of a function f and its inverse
  function are related to each other in the
  following way. If the point (a,b) lies on the
  graph of f, then the point (b,a) must lie on the
  graph of f-inverse.
• The two will be reflections of each other across
  yx.
• See Figure 1.68 pg. 65

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The Existence of an Inverse Function

• A function f is one-to-one if, for a and b in its
  domain, f(a) f(b) implies that a b.
• A function f has an inverse function if and only
  if f is one-to-one.

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An easy test for one-to-one

• A function is one-to-one if it passes the
  horizontal line test.
• Two types of functions pass this test
• If f is increasing on its entire domain, then f
  is one-to-one.
• If f is decreasing on its entire domain, then f
  is one-to-one.

86
Example 2.1.6

• Pg. 70 39
• Use a graphing utility to graph the function and
  use the horizontal line test to determine whether
  the function is one-to-one and so has an inverse.
• h(x) sqrt(16 x2)

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Solution - Ex. 2.1.6

• Pg. 70 39
• Graph looks like a rainbow and does not pass the
  horizontal line test, therefore it is not
  one-to-one.

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Finding Inverse Functions Algebraically

• Use horizontal line test to decide whether f has
  an inverse function.
• Interchange x and y, and solve for y.
• Verify that the domain of f is equal to the range
  of f-inverse and f(f-1(x)) x.

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Example 3.1.6

• Pg. 70 59
• Find the inverse function of f. Use a graphing
  utility to graph both f and f-inverse in the same
  viewing window.
• f(x) 2x - 3

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Solution - Ex. 3.1.6

• Pg. 70 59
• f-1(x) (x 3)/2

91
Example 4.1.6

• Pg. 70 65
• Find the inverse function of f. Use a graphing
  utility to graph both f and f-inverse in the same
  viewing window.
• f(x) sqrt(4 x2), 0 x 2

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Solution - Ex. 4.1.6

• Pg. 70 65
• f-1(x) sqrt(4 x2)
• The graphs are the same.

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1.7 Linear Models and Scatter Plots

• Scatter Plots and Correlation
• Fitting a Line to Data

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Scatter Plots and Correlation

• When we graph a set of ordered pairs from a data
  set we call the collection of points a scatter
  plot.
• We use these to detect relationships (linear,
  quadratic, etc.)
• Correlation is a way to describe a positive or
  negative relationship between the variables.
• See Figure 1.77 on pg. 74

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Example 1.1.7

• Pg. 77 1
• We will do this on the calculators.

96
Example 2.1.7

• Pg. 78 3-6
• Determine whether there is positive correlation,
  negative correlation, or no discernable
  correlation between the variables.

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Solution - Ex. 2.1.7

• Pg. 78 3 - 6
• 3. Negative correlation
• 4. No correlation
• 5. No correlation
• 6. Positive correlation

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Fitting a Line to Data

• Those of you who took or will take a statistics
  class will cover this in detail. We will do
  example 5 on page 77.
• I want you to know how to use the regression
  feature of the calculator.