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Fall 2023

 

           Precalc, Day-by-Day Course Content

Equation solving calculator
  https://www.symbolab.com/solver/equation-calculator/
Graphing Tool
  https://goodcalculators.com/graphing-calculator/
Applications Calculator
  https://goodcalculators.com/
Functions: pix, fx & ops, library, other lessons
  fx pix  fx & ops  fx.library  class
Functions, Vocabulary     Quadratic Functions Exponent & Log Laws
Orientation, ... Higher Order Polynomials Trig Functions, "Before Unit Circle"
Calculator Resources Rationals Trig Functions, "Unit Circle & On"
Lines Log & Exponential Functions     Trig Cheat Sheet

Download Geometer's Sketchpad for Free!

Day #, Chapter.Section    Text Page & Content/Topic
 
1       2.1, 2.2
2.1 The Rectangular Coordinate Systems and Graphs, pg. 74
      coord sys @ 74, plot @ 76, TI graph @ 78, intercepts @ 79, distance @ 80, midpoint @ 82, circle @ 83, exercises @ 84
  • plot, graph
    Cartesian -- definition and analytic history web page
    graphing "What's A Graph?"
  • compute
    distance.gsp   -- start w/a segment, coordinatize it, build triangle, use Pythagorean Theorem to find distance
    fx & ops - midpoint, slope, distance, ...
  • circles, distance formula, tangents, normals
    circle.tangent.normal.gsp
        1 drag center & radius 2 create equation
        2 enter h, k, r
        3 input coord, m=, line=, perp=
        4 slide, write tangent & normal
    just circleFormula   -- writes the formula for the depicted circle
    just circleTangent   -- writes the tangent to a given circle
  • trace   -- creates point defined by an x-intercept and a calculated value for y of point
    then traces the path of the point controlled by the x-intercept
2.2 Linear Equations in One Variable, pg. 87
      lines @ 88, solve linear equ. @ 88, solve rational (fraction equation @ 89, Solving Rational Equations with a Binomial in the Denominator @ 91,
      slope @ 83, write equ of lines @ 94, parallel, perpendicular @ 96, exercises @ 100
  • lines
    fx.library
    LINES -- Notes, Discussion, Activities
    Line Vocabulary
    line.gsp   -- slide points A and B and generate equation of the line in 4 forms.
    line.gsp   -- line segment by dragging & by coordinates, midpoint, distance formula, point-slope equation, perpendicular, parallel lines
    -- slide points A and B and generate equation of the line in 4 forms.
    -- computes slopes and writes lines by sliding and by parameters
        1 - drag segment endpoints to compute midpoint and distance
        2 - show how the distance formula computes
        3 - find a perpendicular w/input coordinates
        4 - find parallel w/input coordinates
    linear.xls for: graphs, slope
    SLOPE SONG .htm
    solve equ w/solver -- find x-intercepts
    solve graphically
    equations of lines .htm
    fx & ops .htm - midpoint, slope, distance, ...
  • circles, distance formula, tangents, normals
    circle.tangent.normal.gsp
        1 drag center & radius 2 create equation
        2 enter h, k, r
        3 input coord, m=, line=, perp=
        4 slide, write tangent & normal
Distance Formula, Midpoint Formula, Circles, Solving equations graphically
 
 
2       2.5, 2.6
2.5 Quadratic Equations, pg. 119
     
Graphing, Functions Vocabulary
domain range Symbols & words in gif format
Big Picture of thr Graphs of "all" functions Details/Summary of "all" functions in pdf format Library of Functions as a sequence of web pages
real zeros - say the same thing about roots in 4 ways ODD/EVEN functions by definition

odd/even/neither functions viewed w/a smile
odd by definition even by definition




Basic Functions
identity function opposite reciprocal function squaring function square root function absolute value function and absolute value
piecewise defined greatest integer function




Function Sketchpads
reciprocal gsp, 1/x, etc.   -- reciprocal functions
    * VERTICAL ASYMPTOTE MUST BE POSITIONED BY USER!
absolute value gsp  -- |x|, functions controlled by parameters and NOT controlled by parameters
2graphs.gsp  -- 1,2 graphed functions, piece-wise, composite ready to edit, link to "coordinate" and more history, table of f(x) gps files
graphs gsp  -- 1 or 2 graphed functions ready to edit
squaring gsp, parabola   -- Multi page: 2 parabolas: 1 in general form, 1 in standard forms, ...
square root gsp   -- 2 square root functions defined by easy-edit parameters
piece-wise gsp   -- graphs a piecewise defined function when left function and right function
absValu.gsp  -- 2 absolute value functions controlled by parameters
greatest integer function gsp, [[x]] -- also a[[bx+c]] + k
reciprocal functions   -- graph of parameter-defined reciprocal function,
    * with slide-able tangent line,
    * VERTICAL ASYMPTOTE MUST BE POSITIONED BY USER!
composite functions gsp  -- build 2 functions using other functions
inverse gsp -- take an inverse of an invertable fdunction, horizontal & vertical line tests
parentFX gsp   -- function "manipulatives"
parentFX2 gsp>   -- revised function "manipulatives"


2.6 Other Types of Equations -- Solving Equations Algebraically, pg. 131
     
 
 
3       2.7
2.7 Linear Inequalities and Absolute Value Inequalities -- Solving Quadratic and Rational inequalities, pg. 142
     
  quadratic.xls
  • Enter h, k, and a to generate general form and x-intercepts.
  • Multiply 2 binomials to find the product, vertex, discriminant.
  • Solve a quadratic equation by entering the required constants and coefficients.
  • Write general form in quadratic form.
composite, polynomial, rational functions
polynomial
linear
quadratic
quadratic formula
discriminant
 
 
4       3.1-3.3, 4.1
3.1 Functions and Function Notation, pg. 160
     
3.2 Domain and Range, pg. 180
     
3.3 Rates of Change and Behavior of Graphs, pg. 196
     
4.1 Review of slopes and functions, pg. 280
     
  • ssong.gsp -- slope from history, to sound, to words & numbers, through examples, to computation
    0 - toc, history
    1 - only 1 function
    2 - parabola w/tangent line
    3 - cubic w/tangent line
    4 - polynomial
    5 - rational
    6 - compute slope drag 2 points
    7 - dy/dx by definition
    8 - m= given A, B
    9 - 3 points define a parabola
    10 - notes on 3 points define a parabola
  • 2graphs.gsp  -- 1,2 graphed functions, piece-wise, composite ready to edit, link to "coordinate" and more history, table of f(x) gps files
  • Cartesian -- definition and analytic history web page
  • ssong.gsp -- slope from history, to sound, to words & numbers, through examples, to computation
    0 - toc, history
    1 - only 1 function
    2 - parabola w/tangent line
    3 - cubic w/tangent line
    4 - polynomial
    5 - rational
    6 - compute slope drag 2 points
    7 - dy/dx by definition
    8 - m= given A, B
    9 - 3 points define a parabola
    10 - notes on 3 points define a parabola
 
 
5       TEST#1 (10% of course grade and covers sections 2.1,2.2,2.5,2.6,2.7.3.1-3.3, 4.1)
5       3.5
3.5 Transformation of Functions - Graphing functions; Shifting, Reflecting, Stretching and Shrinking graphs, pg. 222
     
 
 
6       3.4, 3.7
3.4 Composition of Functions, pg. 209
     
  • inverse.gsp
    0 - vertical, horizontal line tests
    1 - square root fx
    2 - any function
    3 - sqrt fx by parameters
    4 - restricted domain on inverse
    5 - arcsine
    6 - arctangent
    7 - f and inverse
    8 - f, inverse, tangents --
  • compositeFX.gsp --
  • parabola.gsp --
  • compositeFx.poly.ratl.gsp
    0 - graph of 1 function
    1 - graphs of 2 functions
    2 - composite of 4 functions
    3 - polynomial. links to polynomial, dilations gif, quatratic.htm, equation, discriminant
    4 - rational, rational function features & how to find them
    5 - end behavior, battle of (Ax^m)/(Bx^n)
  • compositeFX.gsp  -- build 2 functions using other functions
  • inverse.gsp -- take an inverse of an invertable fdunction, horizontal & vertical line tests
3.7 Inverse Functions, pg. 254
     
 
 
7       3.4, 3.5, 3.7
3.4 Composition of Functions, pg. 209
     
3.5 Transformation of Functions, pg. 222
     
3.7 Inverse Functions, pg. 254
     
 
 
8       Test #2 (10% of course grade and covers sections 3.4, 3.5, 3.7)
8       5.1, 5.2
5.1 Quadratic Functions, pg. 344
     
5.2 Power Functions and Polynomial Functions, pg. 360
     
quadratic formula
discriminant
 
 
9       5.3, 5.4, 5.5
5.3 Graphs of Polynomial Functions, pg. 375
     
5.4 Dividing Polynomials, pg. 393
     
5.5 Zeros of Polynomial Functions, pg. 402
     
zeros
Polynomial functions notes
poly.xls
  • Graph a polynomial defined by degrees and coefficients.
    y= anxn + an -1xn -1 + an - 2xn -2 + an - 3xn -3 + an - 4xn - 4 + an - 5xn - 5 + an-6xn - 6 + an -7xn -7
  • Graph a polynomial displayed in factored form.
    y = A(x - b)B(x - c)C(x - d)D
  • Synthetically divide and then view the quotient polynomial and remainder.
    Divide (anxn + an -1xn -1 + an - 2xn -2 + an - 3xn -3 + an - 4xn - 4 + an - 5xn - 5 + an-6xn - 6 + an -7xn -7) รท (x-c)
  • Solves linear or quadratic equations. Multiplies 2 binomials or 3 binomials.
  • parabola   -- Multi page: 2 parabolas: 1 in general form, 1 in standard forms,
        * both generated by equations defined by parameters.
        * WITH SHOW/HIDE ACTION BUTTON
        * graphs of 2 parabolas: 1 in general form, 1 in standard forms,
        * 1 parabola, set a, then generat parabola w/2 points.
        * 1 parabola controlled by 3 points, used for determinant analysis
        * 1 vertical axis of symmetry, 1 horizontal "axis of symmetry
  • TracePoly   -- drag point f(x) to state and show new ( x, f(x))
  • recipfx   -- graph of parameter-defined reciprocal function,
        * with slide-able tangent line,
        * VERTICAL ASYMPTOTE MUST BE POSITIONED BY USER!
 
 
10     5.6
5.6 Rational Functions: Asymptotes and Graphing, pg. 414
      ratl.xls
  • Graph and table of (Axm)/(Bxn)
    Watch the power of the top or bottom equal or dominate the other.
    Examine hoirzontal asymptotes or infinite increase or decrease.
    Think endbehavior and explore.
  • Graph and table of rational function, polynomial A(x) divided by B(x).
    Examine asymptotes.
    Examine the role of factors A(x) and 1/B(x).
  • Graph and table of rational functions written as A(x) / B(x) + C(x)
    Really play with C(x), the asymptote
  • Complete synthetic division and rewrite of quotient and remainder.
 
 
11 twice     6.1-6.4 (Two-Days)
6.1 Exponential Functions, pg. 464
     
6.2 Graphs of Exponential Functions, pg. 479
     
6.3 Logarithmic Functions, pg. 491
     
6.4 Graphs of Logarithmic Functions, pg. 499
     
  • logs   -- Multi page: 2 logs, log w/exponential
  • exp2.xls
  • Compute yearly, semiyearly, quarterly, monthly, daily, and every minute given initial principal, rate, and time.
  • Solve for P, P0, r, t, but not n, for instantaneous and non-instantaneous interest.
  • Graph two exponential curves w/user entry of parameters.
 
 
12     6.5,6.6
6.5 Logarithmic Properties, pg. 516
     
6.6 Exponential and Logarithmic Equations, pg. 526
     
 
 
13     6.7
6.7 Growth and Decay Applications -- Exponential and Logarithmic Models, pg. 537
     
 
 
14     There is no day 14.
 
 
15     Test #3 (20% of course grade and covers sections 5.1-5.6, 6.1-6.7)
 
 
16     7.1
7.1 Angles and their measures, pg. 576
     
  • complsuppl   -- just complementary & supplementary angles
  • standardPosition.gsp
    1 standard position angle central angle, negative & positive angle measures
    2 angles coterminal?
  • radianSector.gsp
    1 intro angles degrees radians
    2 radian questions
    3 arc length angular speed sector area
    4 sector and area
  • veronica.gsp
    1 - 1 standard position angle
    2 - angles coterminal?
    3 - 1 veronica angles -- trig fx of angles is 4 quadrants
    4 - 2 veronica angles
    5 - extra 2 veronica angles
  • elevation.gsp - angle of elevation, depression, ladders w/h(x), y(x), and h(y), x(y), and h as variable, and h s a constant
  • complsuppl   -- just complementary & supplementary angles
  • cosplay   -- just right triangle cosine
  • sinplay   -- just right triangle sine
  • tanplay   -- just right triangle cosine
  • NewCentral   -- An angle in standard position.
  • standardPosition.gsp
    1 standard position angle central angle, negative & positive angle measures
    2 angles coterminal?
  • radianSector.gsp
    1 intro angles degrees radians
    2 radian questions
    3 arc length angular speed sector area
    4 sector and area
  • veronica.gsp
    1 - 1 standard position angle
    2 - angles coterminal?
    3 - 1 veronica angles -- trig fx of angles is 4 quadrants
    4 - 2 veronica angles
    5 - extra 2 veronica angles
  • elevation.gsp - angle of elevation, depression, ladders w/h(x), y(x), and h(y), x(y), and h as variable, and h s a constant
 
 
17     7.2
7.2 Right Triangle Trigonometry, pg. 593
     
    2TrigGraphs.gsp  -- 1 blank function, 1 parameter-driven function on a "trig plane"
  • sinplay   -- just right triangle sine
  • cosplay   -- just right triangle cosine
  • tanplay   -- just right triangle tangent
  • TraceSin   -- drag point f(x) to state and show new ( x, f(x))
  • TraceCos   -- drag point f(x) to state and show new ( x, f(x))
  • TraceTan   -- just graph with a slideable point to display coordinates
  • SinCosTan1stDay   -- Just sine, cosine, tangent of right triangle angles.
  • SinCosTan6fx   -- All 6 functions in right triangle only.
  • SinCosTanAllNumbers   -- All 6 functions in right triangle and forall numbers,
        * controlled by figure for right triangle,
        * parameter-defined angle measure in degrees for all measures, and
        * parameter-defined angle measure in decimal number times pi for radian measures
  • sinNcos.gsp   -- sine and cosine in right triangle only
  • 2 Sine Functions -- Asin(Bx-C) + D on 2 functions for comparison also h(x) = sine + cosine
 
 
18     7.3
7.3 Trigonometric Functions of any angle -- Unit Circle, pg. 604
     
 
 
19     8.1,8.2
8.1 Graphs of the Sine and Cosine Functions, pg. 642
     
8.2 Graphs of the Other Trigonometric Functions, pg. 659
     
  • jig1.xls -- UNIT CIRCLE JIG SAW PUZZLE
  • sine.xls
    • Graphs y=Asin(Bx-C) +D, where A, B, C, D are input values.
    • Scatter-Plots y=Atan(Bx-C)+D, where A, B, C, D are input values.
    • Graphs two sine functions where A, B, C, D are input values.
    • Graphs one sine function and one cosine function where where A, B, C, D are input values.
     
     
    20     8.3
    8.3 Inverse Trigonometric Functions & Their Applications, pg. 677
         
     
     
    21     9.1
    9.1 Using, Solving, and Verifying Trigonometric Identities, pg. 696
         
     
     
    22     9.5
    9.5 Solving Trigonometric Equations, pg. 739
          solvtrg.xls
    • Solve a right triangle.
    • Solve a 45-45-90 triangle.
      Input leg a. Seek a leg and hypotenuse.
      Input side c. Seek two legs.
    • Solve a 30-60-90 triangle.
      Input leg a. Seek a leg and hypotenuse.
      Input leg b. Seek a leg and hypotenuse.
      Input side c. Seek the hypotenuse.
    • Use Pythagorean Theorem and arithmetic and basic trig.
      Input leg a and leg b. Seek the hypotenuse and the angles.
      Input hypotenuse c and leg a. Seek a leg and the angles.
      Input angle A and side a. Seek a leg, side, and hypotenuse.
    • Solve any triangle.
      Use the Sine Law, if a side and the opposite angle are given.
      Input angles A , B, side a. Seek two sides and an angle.
      Input angle A , side a, side b. Seek two angles and a side.
    • Use the Law of Cosines.
      Input sides a, b, c. Seek each angle.
      Input angle A , sides b,c. Seek no solution, 1 solution, or 2 solutions.
     
     
    23     9.2,9.3,9.4
    9.2 Sum and Difference Identities, pg. 706
         
    9.3 Double-Angle, Half-Angle, and Reduction Formulas, pg. 720
         
    9.4 Sum-to-Product and Product-to-Sum Formulas, pg. 732
         
     
     
    24 YELLOW -- NO SCHOOL!
    24 GREEN -- 7.1-7.3, 8.1-8.3, 9.1-9.5 Review & Catch Up
     
     
    25 Twice     10.1, 10.2
    10.1 Non-right Triangles: Law of Sines, pg. 762
         
    10.2 Non-right Triangles: Law of Cosines, pg. 776
         
     
     
    26     Test #4 (20% of course grade and covers sections 7.1-7.3, 8.1-8.3, 9.1-9.5, 10.1, 10.2)
     
     
    27     Review & Probable Course Grades
     
     
    28     Final Exam (20% of course grade and covers entire course)
     
     
    0       Calculator
    calculator
    window
    ZOOM
  • Calcu1: Change Mode, Store Number, Evaluate Expression .gif
  • Calcu2: Zooming Feature .htm
  • Calcu3: Calculator Solve (web page)
  • Calcu4: Solve an Equation w/Solver .gif (image)
  • Calcu5: Create a Program to Compute .gif
  • Calcu6: Create CHEAT Program .gif
  • Calcu7: Create SLOPE Program .gif
  • Calcu8: Character & PRGM Menu .gif
 
 

    To better see the big picture of the kinds of solutions to a quadratic equation, see "I Got Your Number".   To learn of the evolution of numbers, read "Imaginary Numbers Are Not Imaginary."

COMPLEX NUMBER   -- solves all ax² + bx + c = 0, but, there are different kinds of answers (easily identified by the discriminant). The kinds of answers are:
a + bi, where a is not 0, b is not 0, where i² = -1
a + bi, where a is 0, b is not 0, where i² = -1 -- These are pure IMAGINARY
a + bi, where a is not 0, b is 0, where i² = -1 -- These are pure REAL NUMBER -- solves x² - 4 = 0. There are two kinds of real numbers:
    IRRATIONAL -- solves x² = 3, NON-FRACTIONS, a non-repeating, non-terminating decimal number
    RATIONAL -- solves x² = 4, FRACTIONS, a repeating, terminating decimal number. There are two kinds of rational numbers:
                MIXED NUMBER - a rational number in which the denominator DOES NOT evenly divides the numerator
                INTEGER -- solves x + 4 = 3, a number in which the denominatore DOES evenly divide the numerator, a whole number or the opposite of a whole number.
                                WHOLE NUMBER -- solves x + 4 = 4. The NATURAL NUMBERS and also ZERO.
                                          NATURAL NUMBER -- the COUNTING NUMBERS, solves x + 4 = 6
More Extra Stuff
table   -- build and edit a table of data
pyth3.xls
  • Examine Pythagorean Triangles -- triangles with "nice" sides, generated because:
    If p > q, then p and q may be used to generate the right triangle with
          leg a = p2 - q2,
          leg b = 2pq, and
          hypotenuse c = p2 + q2.
    Includes Heronian Triangles
    Pythagorean Triples
Heronian Triangles -- 2 right triangles butted up together and sharing a common side.
for all x, generate triples
for all even x, generate triples
Heronian triangles are a pair of right triangles with a "shared side."
tiles.xls -- for the parent, teacher, professional educator contains ALL "manipulative graphics" and active hot cells.
© 2022, 2023, A. Azzolino