PC Chapter 1 notes 2009 by DWt7kRZs

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									                 Pre-Calculus

                        Chapter 1 notes

           Name:_____________
           Period:_____________
           Teacher:    Mr. Staley




A special thanks to Mrs. Sohalski and Mrs. Nesbitt for sharing their PC materials.
                                 Pre-Calculus/Trig – 1
                                      Mr. Staley
                                Assignment Schedule for
                               August 11 to September 22


    Date                                      Reading
                         Chapter                            Problems Assigned
 Blue   Gold                                 Assignment
                                                            p. 9-11: 2, 4,14, 15, 20,
                 1.1 Graphs of
                                         p. 2-8             21,22, 25, 50, 58, 61, 66, 70
 8/11     8/12   equations
                                         p. 12-20           p. 21-26: 10, 16, 19, 22, 24,
 Tues     Wed    1.2 Linear equations in
                                                            40, 50, 61, 62, 72, 87, 92,
                 Two Variables
                                                            116, 136
                                           p. 27-34         p. 35-40: 24, 27, 28, 32, 34,
                 1.3 Functions             p. 41-46         35, 47, 48, 52, 56, 64, 77,
8/13      8/14
                 1.4: Analyzing Graphs                      78, 80
Thurs      Fri
                 of Functions                               p. 47-50: 3-24(multiples of
                                                            3), 32,
                                                            p. 47-50: 34, 41, 44, 50, 53,
 8/17     8/18
                 Wrap up 1.1 to 1.4                         58, 62, 64, 66, 68, 69
 Mon      Tues
                                                            Worksheet 1.1 to 1.4
               Quiz 1.1 to 1.4             p. 52-55         p. 56-58: 2, 3, 13, 30, 32,
 8/19    8/20
               1.5: Library of                              44, 47, 64, 74, 76
 Wed     Thurs
               Functions
               1.6: Shifting,              p. 59-63         p. 64-68: 2, 6(a, c, d, e), 10,
 8/21    8/24
               Reflecting and                               20, 25, 28, 34. 40, 41, 44,
  Fri    Mon
               Stretching Graphs                            48, 80, 82
 8/25     8/26   Finish 1.6                                 Worksheet 1.1 to 1.6
 Tues     Wed
                                           p. 69-73         Read: 1.7: Combinations of
8/27      8/28                                              Functions
                 Test 1.1 to 1.6
Thurs      Fri                                              p. 74: 1, 2, 7, 35 - just these
                                                            four questions.

Labor Day is Monday, September 7th. This is a tentative schedule for the first grading
period. Subject to change, if needed.

                                           p. 69-73        p. 74-75: 35-38, 40, 41, 44,
 8/31     9/1    1.7 Composition of
                                           p. 77-82        45, 51, 54, 55, 67, 70
 Mon     Tues    functions
                                           p. 87-92        p. 83-86: 1-4, 5, 6, 23,
 9/2      9/3                                              26,30,32,54,63,81,84
               1.8 Inverse Functions
 Wed     Thurs
 9/4     9/8   1.9: Mathematical        p. 110-115     p. 93-98: 2, 3, 7, 13, 14, 22,
 Fri    Tues   Modeling                                23, 27, 30, 31
                                        p. 121- 129    p. 93-98: 8, 15, 32, 38, 42,
                                                       45, 48, 50, 54, 55, 57, 58, 60,
9/9     9/10 Finish 1.9
                                                       62
Wed     Thurs Quiz 1.7 and 1.8
                                                       We will do 65, 71, and 73 in
                                                       class.
                                        Review         p. 116-120: 1-8 matching,12,
               2.1: Quadratic
                                        packet for     14, 15, 17, 23, 24, 26, 33, 34,
               Functions
9/11    9/14                            Test 2 on      39, 40, 42, 46, 50, 55, 59, 67,
               2.2: Polynomial
 Fri    Mon                             Monday         68, 71,78, 84
               Functions of Higher
                                        Sept. 18       p. 130-133: 1-8, 28, 34, 40,
               Degree
                                                       41, 50, 53, 55, 59
                                                       Synthetic Division section: p.
9/15    9/16
               Test 1.7 to 2.2                         140-142: 5, 8, 13, 21, 25, 34,
Tues    Wed
                                                       37, 40, 45, 49, 50, 58, 59
                                        p. 134 – 139   (section 2.2 ) p. 130: 9, 12,
9/17    9/18
               2.3 Synthetic Division                  14, 17, 23, 24,
Thurs    Fri
                                                       ,64,68,70,77,87,90
               2.3 Synthetic            Read section   p. 140-142: 5, 8, 13, 21, 24,
9/21    9/22
               Division                 2.3            25, 34, 37, 40, 45, 49, 50, 58,
 Fri    Mon
                                                       59




                September 18 is the end of the First Grading Period
1.1    PC Notes

       What you should learn:
          How to sketch graphs of equations
          How to find x- and y- intercepts of graphs of equations
          How to use symmetry to sketch graphs of equations
          How to use graph of equations in solving real-life problems

Vocabulary
       Equation in Two Variables

       Solution (or solution point)

       Graph of an equation

       Intercepts

       Symmetry

Example 1 – Graphing with a table of values (Point Plotting Method)
      Graph y  2 x  5 using a table of values
                                                                         10
                                                                          8
                                                                          6
                                                                          4
                                                                          2

                                                      -10 -8 -6 -4 -2            2   4   6   8 10
                                                                          -2
                                                                          -4
                                                                          -6
                                                                          -8
                                                                         -10
       Graph y  x  3 using a table of values
                    2



                                                                               10
                                                                                8
                                                                                6
                                                                                4
                                                                                2

                                                          -10 -8 -6 -4 -2            2   4   6   8 10
                                                                                -2
What might be a problem with this method?
                                                                                -4
                                                                                -6
                                                                                -8
                                                                               -10
Finding Intercepts

       To find the x-intercept

       To find the y-intercept



Example 2 – Finding the x- and y-intercepts

       Find the x- and y-intercepts for y  3x  2
                                              3




       Find the x- and y-intercepts for y  x  3
                                        2




Symmetry with respect to the x-axis                  Symmetry with respect to the y-axis




Symmetry with respect to the origin




How could knowing the symmetry of a graph beforehand be helpful in sketching the graph?




Tests for Symmetry

       Symmetric with respect to the x-axis



       Symmetric with respect to the y-axis



       Symmetric with respect to the origin
Example 3 – Testing for Symmetry

       Describe the symmetry of the graph of y  x  3
                                                      2




       Describe the symmetry of the graph of y  4 x  12
                                              2




Example 4 – Using symmetry to graph

       Use symmetry to sketch the graph of y  x  5 x  3
                                                  4       2




                                                                            10
                                                                                8
                                                                                6
                                                                                4
                                                                                2

                                                              -10 -8 -6 -4 -2        2   4   6   8 10
                                                                                -2
                                                                                -4
                                                                                -6
                                                                                -8
                                                                            -10




Example 5 – Graphing an absolute value equation
       Sketch the graph of   y  x 3  2
                                                                            10
                                                                                8
                                                                                6
                                                                                4
                                                                                2

                                                              -10 -8 -6 -4 -2        2   4   6   8 10
                                                                                -2
                                                                                -4
                                                                                -6
                                                                                -8
                                                                            -10
Circle:
Equation (from the definition)



Example 6 – Finding the Equation of a Circle
        The point (2,6) lies on a circle whose center is at (-1,5). Write the standard form of
this circle.




Three Common Approaches to solving an application




Example 7 – Application
A rectangle of length x and width w has a perimeter of 12 meters.
        a. Draw a rectangle that gives a visual representation of the problem. Use the
specified variables to label the sides of the rectangle.




       b. Show that the width of the rectangle is   w  6  x and its area is A  x(6  x) .



       c. Use your graphing calculator to graph the area equation, sketch the graph below.


                                                                          10
                                                                           8
                                                                           6
                                                                           4
d. Use the graph to estimate the dimensions
of the rectangle that yield the maximum area                               2

                                                        -10 -8 -6 -4 -2         2   4    6     8 10
                                                                           -2
                                                                           -4
                                                                           -6
                                                                           -8
                                                                          -10
              1.2 PC Notes
              What you should learn:
                  How to use slope to graph linear equations in two variables
                  How to find slopes of lines
                  How to write linear equations in two variables
                  How to use slope to identify parallel and perpendicular lines
                  How to use linear equations in two variables to model and solve real-life problems
              Vocabulary
                     Linear Equation in Two Variables

                               Slope                                                           Slope-Intercept Form

                               Ratio                                                           Rate (Rate of Change)

                               Point-Slope Form                                                Two-Point Form

                               Linear Extrapolation

                               Linear Interpolation

                               General Form of the Equation

                               Parallel                                                        Perpendicular

                               Depreciation

                               Linear Depreciation

                               Book Value



              Example 1 – Sketching the Graph of a Linear Equation

              Sketch the graph of each of the equations below

                  y  3x  4                               x 8                                 y  2                            3x  2 y  6

              10                                            10                                        10                                        10
                  8                                             8                                         8                                         8
                  6                                             6                                         6                                         6
                  4
                                                                4                                         4                                         4
                  2
                                                                2                                         2                                         2
-10 -8 -6 -4 -2        2   4   6   8 10
                  -2                          -10 -8 -6 -4 -2        2   4   6   8 10   -10 -8 -6 -4 -2        2   4   6   8 10   -10 -8 -6 -4 -2        2   4   6   8 10
                                                                -2                                        -2                                        -2
                  -4
                                                                -4                                        -4                                        -4
                  -6
                                                                -6                                        -6                                        -6
                  -8
              -10                                               -8                                        -8                                        -8
                                                            -10                                       -10                                       -10
Example 2 – Application of Slope

Road Grade: From the top of a mountain road, a surveyor takes several horizontal
measurements x and several vertical measurements y, as shown in the table (x and y are
measured in feet).
   a. Sketch a scatter plot of the data.
                                                                      x         y
                                                                    300        -25
                                                                    600        -50
                                                                    900        -75
                                                                    1200      -100
                                                                    1500      -125
   b. Use a straightedge to sketch the                              1800      -150
      best-fitting line through the points.                         2100      -175

   c.   Find an equation for the line.




   d. Interpret the meaning of the slope of the line in part (c) in the context of the
      problem.

   e.   The surveyor needs to put up a road sign that indicates the steepness of the road.
        For instance, a surveyor would put up a sign that states “8% grade” on a road with a
        downhill grade that has a slope of -8/100. What should the sign state for the road
        in this problem?



Example 3 – Finding the slope of a line through two points
      Find the slope of the line passing through (4,5) and (2,6) .




Example 4 – Finding Equations for Lines
       Find the equation for each of the following lines in slope-intercept form, point-slope
form and two-point form.
   a. Passes through (2,6) with a slope of -3.




   b. Passes through (1,5) and (-2,7)




   c.   Has an x-intercept of -4 and a y-intercept of 3.
Equations of Lines
       General Form

       Vertical Line

       Horizontal Line

       Slope-intercept form

       Point-slope form

       Two-point form




Example 5 – Finding Parallel and Perpendicular Lines

        Find the slope-intercept form of the line that passes through (1,4) and is parallel to
3x  2 y  8 .




        Find the slope-intercept form of the equation of the line that passes through (-2,6)
and is perpendicular to 2 x  5 y  3 .




Example 6 – Depreciation
       You purchase a car valued at $35,000. 3 years later you decide to sell the car and
when you look up the book value you discover that it is now worth $22,000. Write a linear
equation that describes the book value of the car in any given year.




Use the equation to determine the book value of the car after 1 year. Is this interpolation
or extrapolation?




Use the equation to determine the book value of the car after 5 years. Is this interpolation
or extrapolation?
1.3 PC Notes
What you should learn:
    How to determine whether relations between two variables are functions
    How to use function notation and evaluate functions
    How to find the domains of functions
    How to use functions to model and solve real-life problems
Vocabulary
       Relation

       Function:

       Domain:

       Range:

       Independent Variable:

       Dependent Variable:

       Function Notation:

       Piecewise-defined Function:

       Implied Domain:



Example 1 – Testing for Functions
      Determine if each of the following relations is a function.

       a. (3,5), (4,8), (1,-3), (7,-3)



       b. (2,6), (5,7), (2,9), (-3,13)




Example 2 – Testing for Functions Represented Algebraically
      Determine whether the equation represents y as a function of x.

       a. 2 x  3 y  4                  b. x  y  2               c. x  y  4
                                                2                       2   2
Function Notation:

Example 3 – Evaluating a Function
       If h( x)  3 x  2 x  5 find each of the following
                      2




       a. h(3)



       b. h(2)



       c. h( x  2)




Example 4 – Evaluating a Piecewise-defined Function
                                  x2 1       x2
       Given     f (x)                             evaluate the function when x  0,3,2,6
                                  3x  4      x2




Example 5 – Finding the Domain of a Function
      Find the domain of each of the functions below.

                                                                                  1
       a. f: {(3,5), (4,7), (8,0), (11,15)}                     b.   g ( x) 
                                                                                3x  2




       c.   f ( x)  3x 2  2 x  8                             d.   h( x)  x 2  9
Example 6 – Evaluating a Difference Quotient
      Difference Quotient




        Find the difference quotient for f ( x)  x  3 x  6
                                                   2




Example 7 – Application
Cost, Revenue, and Profit: A company produces a product for which the variable cost is
$12.30 per unit and the fixed costs are $98,000. The product sells for $17.98. Let x be the
number of units produced and sold.

   a.   The total cost for a business is the sum of the variable cost and the fixed costs.
        Write the total cost C as a function of the number of units produced.




   b. Write the revenue R as a function of the number of units sold.




   c.   Write the profit P as a function of the number of units sold. (Note: P = R – C.)
1.4 PC Notes
What you should learn:
    How to use the Vertical Line Test for functions
    How to find the zeros of functions
    How to determine intervals on which functions are increasing or decreasing
    How to identify even and odd functions

Vertical Line Test:

Zeros of a Function:

Increasing (function):                               Decreasing (function):



Constant (function):

Relative Minimum:                                    Relative Maximum:



Even (function):                                     Odd (function):



To find the domain of a function from its graph:

To find the range of a function from its graph:

Example 1 - Using the Vertical Line Test
      Determine if the graph represents y as a function of x




If the graph of a function of x has an x-intercept at (a,0), then a is a _____________ of
the function.



Example 2 - Finding the Zeros of a Function
                                                                                    3x  5
       Find the zeros of f ( x)  4 x  19 x  5
                                     2
                                                     Find the zeros of   f ( x) 
                                                                                     x6




       Find the zeros of   f ( x)  12  x 2
The point at which a function changes from increasing to decreasing is a relative
____________. The point at which a function changes from decreasing to increasing is a
relative ________________.



Example 3 - Determining Information from a Graph
      Use the graph below to determine each of the following

       Determine the interval(s) over which
                                                                      10
       the following function is increasing.
                                                                       8
                                                                       6
       Determine the interval(s) over which
                                                                       4
       the following function is decreasing.
                                                                       2

                                                    -10 -8 -6 -4 -2         2   4   6   8 10
       Determine the interval(s) over which                           -2
       the following function is constant.
                                                                      -4
                                                                      -6
       Determine any relative maximum(s).                             -8
                                                                     -10

       Determine any relative minimum(s).



       Approximate any zeros




A function whose graph is symmetric with respect to the y-axis is a(n) _____________
function. A function whose graph is symmetric with respect to the origin is a(n)
______________ function.



Think! Can the graph of a nonzero function be symmetric with respect to the x-axis? Why
or why not??




Example 4 - Determining Even and Odd
       Determine if the function f ( x)  4 x  3x  1 is even, odd or neither. Explain your
                                               2


reasoning.
            1.5 PC Notes
            What you should learn:
                 How to identify and graph linear and squaring functions
                 How to identify and graph cubic, square root, and reciprocal functions
                 How to identify and graph step and other piecewise-defined functions
                 How to recognize graphs of common functions

            Linear Function:

            Constant Function:

            Squaring Function:

            Identity Function:

            Cubic Function:

            Square Root Function:

            Reciprocal Function:

            Step Function:

            Greatest Integer Function:

            Example 1 - Common Functions and Their Graphs
                  Identify each of the following common functions and then sketch their graphs.



             f ( x)  c                            f ( x)  x                            f ( x)  x




              10                                        10                                        10
                  8                                         8                                         8
                  6                                         6                                         6

                  4                                         4                                         4

                  2                                         2                                         2

-10 -8 -6 -4 -2        2   4   6   8 10   -10 -8 -6 -4 -2        2   4   6   8 10   -10 -8 -6 -4 -2        2   4   6   8 10
                  -2                                        -2                                        -2

                  -4                                        -4                                        -4

                  -6                                        -6                                        -6

                  -8                                        -8                                        -8

              -10                                       -10                                       -10
             f ( x)                x                          f ( x)  x 2                                         f ( x)  x 3



              10                                                     10                                                   10
                  8                                                      8                                                   8
                  6                                                      6                                                   6

                  4                                                      4                                                   4

                  2                                                      2                                                   2

-10 -8 -6 -4 -2             2   4       6   8 10       -10 -8 -6 -4 -2        2     4   6   8 10           -10 -8 -6 -4 -2        2   4   6   8 10
                  -2                                                     -2                                                  -2

                  -4                                                     -4                                                  -4

                  -6                                                     -6                                                  -6

                  -8                                                     -8                                                  -8

              -10                                                    -10                                                 -10




                                                                         f ( x)  x 
                                1
             f ( x) 
                                x


                       10                                                         10
                       8                                                           8
                       6                                                           6
                       4                                                           4
                       2                                                           2

  -10 -8 -6 -4 -2               2   4       6   8 10         -10 -8 -6 -4 -2            2   4   6   8 10
                       -2                                                          -2
                       -4                                                          -4
                       -6                                                          -6
                       -8                                                          -8
                   -10                                                            -10
The graph of a linear function f ( x)  ax  b is a line with slope ________ and y-intercept
_______.

Important features of the graph of a linear function f ( x)  ax  b




Important features of the graph of the constant function f ( x)  b




Important features of the graph of the identity function f ( x)  x




Important features of the graph of the squaring function f ( x)  x .
                                                                        2




Important features of the graph of the cubic function f ( x)  x .
                                                                 3




Important features of the graph of the square root function f ( x )           x.




                                                                         1
Important features of the graph of the reciprocal function   f ( x)       .
                                                                         x




Important features of the graph of the greatest integer function        f ( x)  x .
Example 2 - Writing a Linear Function
      Write a linear function for which f(2)=6 and f(-1)=3.




Example 3 - Evaluating a Step Function
                                                                                10
                                           1
       Evaluate the function when x  1,3, ,                                     8
                                           2                                      6
then sketch the graph.
        f ( x)  x  2
                                                                                  4
                                                                                  2

                                                                -10 -8 -6 -4 -2        2   4   6    8 10
                                                                                  -2
                                                                                  -4
                                                                                  -6
                                                                                  -8
                                                                                -10

Example 4 - Graphing a Piecewise-defined Function
                       x2 1    x0
       Graph f(x)=
                       2x  1   x0                                      10
                                                                           8
                                                                           6
                                                                           4
                                                                           2

                                                         -10 -8 -6 -4 -2          2    4   6       8 10
                                                                           -2
                                                                           -4
                                                                           -6
                                                                           -8
                                                                        -10
1.6 PC Notes
What you should learn:
     How to use vertical and horizontal shifts to sketch graphs of functions
     How to use reflections to sketch graphs of functions
     How to use nonrigid transformations to sketch graphs of functions

Vertical Shift:

Horizontal Shift:

Reflection:

Rigid Transformation:

Nonrigid Transformation:

Vertical Stretch:

Vertical Shrink:

Horizontal Stretch:

Horizontal Shrink:



Shifting Graphs

       Graph the three graphs in the same viewing window (hint use different types of
graphs for each!)
       1.     f ( x)  x 2     g ( x)  ( x  3) 2            h( x)  ( x  3) 2  4
       2.     f ( x)  x 3     g ( x )  ( x  2) 3           h( x)  ( x  2) 3  3
       3.     f ( x)  x       g ( x)  x  3                 h( x)  x  3  5

       Given the graph of y  f (x) , write the equation that would give the following
transformations
       Vertical shift of c units upward:

       Vertical shift of c units downward:

       Horizontal shift of c units to the right:

       Horizontal shift of c units to the left:
Example 1 - Shifts of the Graph of a Function
       Let   f ( x)  x . Write the equation for the function resulting from a vertical shift
of 3 units downward and a horizontal shift of 2 units to the right.




       Use the graph of f ( x)  x to sketch the graph of h( x)  ( x  2)  3
                                   2                                           2



                                                                                   10
                                                                                    8
                                                                                    6
                                                                                    4
                                                                                    2

                                                               -10 -8 -6 -4 -2           2   4   6   8 10
                                                                                    -2
                                                                                    -4
                                                                                    -6
                                                                                    -8
                                                                                   -10




       A reflection in the x-axis is a type of transformation of the graph of y  f (x)
represented by h( x)  _____________ .
       A reflection in the y-axis is a type of transformation of the graph of y  f (x)
represented by h( x)  _______________ .

Example 2 - Reflections of Graphs
       Let   f ( x)  x . Describe the graph of g ( x)   x in terms of f .
Example 3 - Finding Equations from Graphs
      Given the following graph of y  f (x) , write the equation for y ' .


   y  f (x)                                                   y' 


                  10                                                          10
                   8                                                           8
                   6                                                           6
                   4                                                           4
                   2                                                           2

-10 -8 -6 -4 -2         2   4   6   8 10                   -10 -8 -6 -4 -2          2   4   6   8 10
                   -2                                                          -2
                   -4                                                          -4
                   -6                                                          -6
                   -8                                                          -8
                 -10                                                          -10




3 Types of Rigid Transformations
       1.

       2.

       3.



4 Types of Non-Rigid Transformations
       1.

       2.

       3.

       4.
Use your graphing calculator to graph each of the following and then fill in the blanks below.

       1.   f ( x)  x 2        g ( x)  3x 2
                                         1
       2.   f ( x)  x 3        g ( x)  x 3
                                         3
       3.   f ( x)  x          g ( x)  2x
                                                    2
                                         1 
       4.   f ( x)  x 2        g ( x)   x 
                                         4 

For y  f (x) and the real number c.
      A vertical stretch is represented by ________________ where
______________.

      A vertical shrink is represented by _________________ where
______________.

      A horizontal shrink is represented by _______________ where
______________.

      A horizontal stretch is represented by _______________ where
______________.



Example 4 - Nonrigid Transformations
                                                                   1 3
       Compare the graph of y  x to the graph of
                                     3
                                                              y     x
                                                                   4




       Compare the graph of f ( x)  2  x          to f (2 x) .
                                                2
1.7 PC Notes
What you should learn:
    How to add, subtract, multiply, and divide functions
    How to find the composition of one function with another function
    How to use combinations of functions to model and solve real-life problems.

Arithmetic Combination of functions:

Composition of functions:



The domain of an arithmetic combination of functions f and g consists of….


Arithmetic Combination
       For two functions f and g with __________________________


        Sum:     ( f  g )(x) =

        Difference:      ( f  g )(x) 

        Product:         ( fg)(x) 

                         f
        Quotient:         (x) 
                         g
                          


Example 1 - Arithmetic Combinations of Functions
      Let f ( x)  7 x  5 and g ( x)  3  2 x , find ( f  g )(4) .



        Let f ( x)  3x  2 and g ( x)  x  3 x  2 , find ( f  g )(x) .
                                            2




        Let f ( x)  2 x  1 and g ( x)  3x  5 , find ( fg)(x) .




Example 2 - Finding the Domain of Arithmetic Combinations of Functions
                                                                             f
        Let f ( x )    x  2 and g ( x)  3  x 2 , determine the domain of  (x) and
                                                                             g
                                                                              
g
 (x) .
f
 
Example 3 - The Composition of Functions
         Let f ( x)  3x  4 and g ( x)  2 x  1 . Find:
                                                   2




   a.    ( f  g )(x)                          b. ( g  f )( x)                          c. ( f  g )(3)




Example 4 - Finding the Domain of Composite Functions
         Let f ( x)  x and
                        2
                                  g ( x)  4  x 2 , find the domain of ( f  g )(x) .



To "decompose" a composite function, look for an ___________ function and an
______________ function.

Example 5 - Finding Components of Composite Functions
         Express the function g ( x)  3( x  1)  2( x  1)  6 as a composition of two
                                                       2


functions.




Example 6 - Application
Health Care Costs: The table shows the total amount (in billions of dollars) spent on health
services and supplies in the U.S. (including Puerto Rico) for the years 1993 through 1999.
The variables y1 , y2 , and y3 represent out-of-pocket payments, insurance premiums, and
other types of payments, respectively. (Source: Centers of Medicare and Medicaid
Services)
  Year          y1           y2          y3
  1993        148.9         295.7       39.1
  1994        146.2         308.9       40.8
  1995        149.2         322.3       44.8
  1996         155          337.4       47.9
  1997        165.5         355.6        52
  1998        176.1         376.8       54.8
  1999        186.5         401.2       58.9

   a. Use the regression feature of a graphing utility to find a quadratic model for y1 and
      linear models for y2 and y3. Let t = 3 represent 1993.
   b. Find y1 + y2 + y3. What does the sum represent?
   c. Use a graphing utility to graph y1 , y2, y3 and y1 + y2 + y3 in the same viewing window.
   d. Use a model from (b) to estimate the total amount spent on health services and
      supplies in the years 2003 and 2005.
1.8 PC Notes
What you should learn:
     How to find inverse functions informally and verify that two functions are inverse
        functions of each other
     How to use graphs of functions to determine whether functions have inverse
        functions
     How to use Horizontal Line Test to determine if functions are one-to-one
     How to find inverse functions algebraically

Inverse Function:

Horizontal Line Test:

One-to-One Function:



Notation for Inverse Function

                                          1
For a function f and its inverse f             , the domain of f is equal to ________________ ,
and the range of f is equal to ________________.


To verify that two functions, f and g , are inverse functions of each other…..



Example 1 - Verifying that Functions are Inverses
                                                                        x3
       Verify that the functions f ( x)  2 x  3 and        g ( x)        are inverse functions of
                                                                         2
each other.




If the point (a,b) lies on the graph of f , then the point (_____, ______) must lie on the
              1
graph of f         and vice versa.
                     1
The graph of f            is a reflection of the graph of f in the line ___________.



Example 2 - Sketching the graph of the Inverse Function
      Using the graph of f ( x)  4 x  3 , sketch the graph of its inverse.                      10
                                                                                                      8
                                                                                                      6
                                                                                                      4
                                                                                                      2

                                                                                    -10 -8 -6 -4 -2        2   4   6   8 10
                                                                                                      -2
                                                                                                      -4
                                                                                                      -6
                                                                                                      -8
                                                                                                  -10
To tell whether a function has an inverse from its graph….




A function f has an inverse if and only if f is ……




Example 3 - Determining if a function has an inverse
      Does the graph of the function have an inverse? Explain.




To find the inverse of a function algebraically..
        1.

        2.

        3.

        4.

        5.




Example 4 - Finding the Inverse
      Find the inverse (if it exists) of f ( x)  4 x  5 .



                                                        4  2x
        Find the inverse (if it exists) of   f ( x) 
                                                           3


        Find the inverse (if it exists) of f ( x )         x4
                                                        3
1.9 PC Notes
What you should learn:
     How to use mathematical models to approximate sets of data points
     How to write mathematical models for direct variation
     How to write mathematical models for direction variation to the nth power
     How to write mathematical models for inverse variation
     How to write mathematical models for joint variation
     How to use the regression feature of a graphing utility to find the equation of the
        least squares regression line

Varies Directly (Directly Proportional to):

Constant of Variation (Constant of Proportionality):

Varies directly as the nth Power (Directly Proportional to the nth power):

Varies Inversely (Inversely Proportional to):

Varies Jointly (Jointly Proportional to):

Sum of Squares Differences:

Least Squares Regression Line:

Correlation Coefficient:



Example 1 - Direct Variation
      If y varies directly as x, and y is 6 when x is 4, find the value of y when x is 20.




Example 2 - Direct Variation, Application
       In Pennsylvania, the state income tax is directly proportional to gross income. You
are working in Pennsylvania and your state income tax deduction is $42 for a gross monthly
income of $1500. Find a mathematical model that gives the Pennsylvania state income tax in
terms of gross income.
Example 3 - Direct Variation as an nth Power
        If y is directly proportional to the third power of x, and y is 750 when x is 10, find
the value of y when x is 8.




Example 4 - Direct Variation as an nth Power, Application
        The distance a ball rolls down an inclined plane is directly proportional to the square
of the time it rolls. During the first second, the ball rolls 8 feet. (See Figure 1.87 on page
89)
    a. Write an equation relating the distance traveled to the time.

    b. How far will the ball roll during the first 3 seconds?



Example 5 - Inverse Variation
      If y varies inversely as x, and y is 4 when x is 16, find the value of y when x is 10.




Example 6 - Joint Variation
       If z varies jointly as x and y, and if z=10 when x=4 and y=15, find the value of z
when x=12 and y=7.




Example 7 - Putting it Together
       Suppose r varies directly as the square of m and inversely as s. If r=12 when m=6
and s=4, find r when m=4 and s=10.




        Let a vary directly as m and n and inversely as y . If a=9 when m=4, n=9 and y=3,
                                       2                    3


find a if m=6, n=2 and y=5.
Correlation Coefficient
       The closer   r is to ______, the better.



Example 8 - Finding a Least Squares Regression Line
        The numbers of U.S. Air Force personnel, p, on active duty for the years 1995
through 1999 are shown in the table. Use the regression capabilities of a graphing utility
to find a linear model for the data. Let t represent the year with t=5 corresponding to
1995. Find the correlation coefficient and explain what this tells us about our equation.

Year           1995            1996            1997            1998            1999
P              400             389             379             363             358

								
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