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Mathematics Dictionary
Dr. K. G. Shih
Contents of Graphic Calulator
Subjects


  • GC 00 | - Outlines
  • GC 01 | - Linear Functions and Equations
  • GC 02 | - Quadratic Functions and Equations
  • GC 03 | - Rational Functions
  • GC 04 | - Trigonometric Functions
  • GC 05 | - Hyperbolic Functions
  • GC 06 | - Intersections of Functions and inverse
  • GC 07 | - Absolute operation
  • GC 08 | - Limit
  • GC 09 | - Polynomial Functions
  • GC 10 | - Graphic calculator for polar coordinates
  • GC 11 | - Graphic calculator for parametric equations
  • GC 12 | - Demo Examples
  • GC 13 | - Sketch Examples
    • Answers


    GC 01. Linear Functions and Equations

    Computer program     Programs
    • 01 y = a*x + b
    • 02 x = a*y + b
    • 03 y = a*x + b and inverse
    • 04 y = |a*x + b|
    • 05 Solve |a*x + b| = c
    Example : Solve Abs(2*x - 3) = 4
    • Use subject 1 and program 5.
    • Data : give 2,-3,4
    • Solution can be seen from intersection of y = Abs(2*x-1) and y = 4.

    Go to Begin

    GC 02. Quadratic Functions and Equations

    Computer programs     Programs
    • 01 y = a*x^2 + b*x + c
    • 02 x = a*y^2 + b*y + c
    • 03 y = a*x^2 + b*x + c and inverse
    • 04 y = abs(a*x^2 + b*x + c)
    • 05 y = abs(a*x^2 + b*abs(x) + c)
    • 06 y = a*x^2 + b*abs(x) + c
    • 07 RX of y = a*x^2 + b*x + c (RX = Reflection of x-axis)

    Example : Solve Abs(x^2 - 6*Abs(x) + 8) = 0 graphically. (Abs = Absolute).
    • Use subject 2 and program 5
    • Give data : 1,-6,8
    • From graph we can get 4 answers.

    Go to Begin

    GC 03. Rational Functions

    Computer program     Programs
    • 01 y = 1/(a*x + b)
    • 02 y = 1/(a*x^2 + b*x + c)
    • 03 y = 1/(a*x^3 + b*x^2 + c*x + d)
    • 04 y = ((x-1)^M)/(2*x)
    • 05 y = ((x+1)^M)/(2*x)
    • 06 y = ((x-1)^M)/(a*x^2+b*x+c)
    • 07 y = ((x+1)^M)/(a*x^2+b*x+c)
    • 08 y = (a*x^3+b*x^2+c*x+d)/(p*x^3+q*x^2+r*x+s)
    Example
    • 1. Find asymptote of y = ((x-1)^3)/(2*x)
    • 2. Sketch (x^2 + 1)/x

    Go to Begin

    GC 04. Trigonometric Functions

    Computer program GC     Programs
    • 01 y = sin(x)
    • 02 y = cos(x)
    • 03 y = tan(x)
    • 04 y = csc(x)
    • 05 y = sec(x)
    • 06 y = cot(x)
    Examples :
    • 1. Find the range of y = sin(x) if x = -2*pi to 2*pi.
    • 2. Find the asymptotes of y = tan(x) between x = -2*pi and 2*pi.

    Go to Begin

    GC 05. Hyperbolic Functions

    Computer program GC     Programs
    • 01 y = sinh(x)
    • 02 y = cosh(x)
    • 03 y = tanh(x)
    • 04 y = csch(x)
    • 05 y = sech(x)
    • 06 y = coth(x)
    • 07 y = exp(x)
    • 08 y = log(x)
    • 09 y = 1/x(x)

    Examples
    • What is formula of y = sinh(x) ?
      • The answer can be found in Hyperbolic Function in MD
    • Compare y = sinh(x) with y = sin(x)
      • The answer can be found in Hyperbolic Function in MD

    Go to Begin

    Q06. Intersections of Functions and inverse

    Computer program GC     Programs
    • 01 Intersections of y= x^2+ 2 with inverse
    • 02 Intersections of y= x^2+ 1/4 with inverse
    • 03 Intersections of y= x^2 with inverse
    • 04 Intersections of y= x^2- 2 with inverse
    • 05 Intersections of y= exp(x)with inverse
    • 06 Intersections of y= 2x+ 2 with inverse
    • 07 Intersections of y= 4*x^2- 3 with y=1/x
    • 08 Intersections of y= 4*x^2+ 2*x+ 3 with y=1/x
    • 09 Intersections of y= 4*x^2+ 2*x- 5 with y=1/x
    • 10 Intersections of y= a*x^2+ b*x+ c with inverse
    • 11 Intersections of y= a*x^2+ b*x+ c with y=1/x
    • 12 Intersections of y= a*x^2+ b*x+ c with y=m*x + n

    Examples
    • How many intersections of y = a*x^2 + b*x + c with its inverse ?
      • Answers can be found in program 1,2,3 and 4.
      • They are demo only
    • How many intersections of y = a*x^2 + b*x + c with y = 1/x ?
      • Answers can be found in program 7,8 and 9.
      • They are demo only
    • Can we sketch the graph on computer ?
      • Programs 10, 11 and 12 are given to sketch
      • Give the coefficients to sketch

    Go to Begin

    GC 07. Absolute operation

    Computer program GC     Programs
    • 01 y = abs(x-1) + abs(x+1)
    • 02 Solve abs(x-1) + abs(x+1) = c
    • 03 Solve abs(x+a) + abs(x+b) = c
    • 04 Solve abs(a*x^2+b*x+c)= d
    • 05 Solve abs(a*x^2+b*abs(x)+c) = d
    • 06 Solve a*x^2+b*abs(x)+c = d

    Example : How many real roots if Abs(x^2 - 6*Abs(x) + 8) = 0.5.
    • Use subject 7 and program 5
    • Give data : 1,-6,8,0.5
    • From graph we can get 8 real roots.

    Reference
    • See keywords Absolute in Webpage MD to find how to solve.

    Go to Begin

    GC 08. Limit

    Computer program GC     Programs
    • 01 Limit[sin(x)/x] = 1 when x ------>> 0
    • 02 Limit[sin(x)/(a*x)] = 1/a as x -->> 0
    • 03 Limit[sin(a*x)/x] = a as x ------>> 0
    • 04 limit[(1+x)^(1/x)] = e as x = 0
    • 05 limit[(1+a*x)^(b/c*x)] = exp(a*b/c) as x = 0
    • 06 limit[(1+1/x)^x] = e as x = infinite
    • 07 limit[(1+1/a*x)^(b*x) = exp(b/a)

    Examples
    • How to use Lim[sin(x)/x] = 1 to find derivative of sin(x) ?
      • Use the keyword Limit in MD to find the method.
    • How to use Lim[1+x)^(1/x)] = e to find derivative of ln(x) ?
      • Use the keyword Limit in MD to find the method.

    Go to Begin

    GC 09. Polynomial Functions

    Polynomial programs     Programs
    • 01 y = a*x + b
    • 02 y = a*x^2 + b*x + c
    • 03 y = a*x^3 + b*x^2 + c*x + d
    • 04 y = a*x^4 + b*x^3 + c*x^2 + d*x + e
    • 05 y = a*x^5 + b*x^4 + c*x^3 + d*x^2 + e*x + f

    Examples
    • 1. y = x^5 + x^4 - 7*x^3 - 7*x^2 + x + 1, find zeros of y
    • 2. y = x^5 + x^4 - 7*x^3 - 7*x^2 + x + 1 < 0, find domain
    • 3. y = x^5 + x^4 - 7*x^3 - 7*x^2 + x + 1 > 0, find domain
    • 4. Solve x^5 + x^4 - 7*x^3 - 7*x^2 + x + 1 = 0

    Go to Begin

    GC 10. Graphic calculator in polar coordinates

    Polar coordinate     Programs
    • Chapter 1 : Introduction
    • Chapter 2 : Graph of R = sin(p*A/q)^M
    • Chapter 3 : Graph of R = tan(p*A/q)^M
    • Chapter 4 : Graph of R = sec(p*A/q)^M
    • Chapter 5 : Graph of R = 1 + 1*sin(p*A/q)^M
    • Chapter 6 : Graph of R = 2 + 4*sin(p*A/q)^M
    • Chapter 7 : Graph of R = 4 + 2*sin(p*A/q)^M
    • Chapter 8 : Graph of R = 1 + 1*sec(p*A/q)^M
    • Chapter 9 : Graph of R = 2 + 4*sec(p*A/q)^M
    • Chapter 10 : Graph of R = 4 + 2*sec(p*A/q)^M
    • Chapter 11 : Graph of R = 1 + 1*tan(p*A/q)^M
    • Chapter 12 : Graph of R = 2 + 4*tan(p*A/q)^M
    • Chapter 13 : Graph of R = 4 + 2*tan(p*A/q)^M
    • Chapter 14 : Hyperbolic function in polar form
    • Chapter 15 : Paramtric equations

    Examples :
    • 1. How many petals in R = sin(3*A/2) ?
      • Use chapter 2
      • Data for p, q, M : 3, 2, 1
    • 2. How to make a five points star ?
      • Use chapter 4
      • Find the values of p, q, M
    • 3. Compare y = sin(x) and y = sinh(x)
      • Use chapter 14
      • Describe the similarity and difference
    • 4. Compare graph of x=sec(t), y=tan(t) and graph of x=tan(t), y=sec(t)
      • Use chapter 15
      • Find the graphs of these two parametric equation
      • Reference : Keywords unit hyperbola in Webpage MD.

    Go to Begin

    GC 11. Graphic calculator in parametric equations

    Parametric equation programs Diagram Programs
  • Programs :
    • 36 diagrams of x = F(t) and y = G(t)
    • More examples in chapter 15 of PM
  • Examples
    • 1. Describe the graph of x = sec(t) and y = tan(t)
    • 2. Describe the graph of x = tan(t) and y = sec(t)
    • 3. Compare the similarity and difference of above two curves
    Parametric equations in AN 15 Parametric equations in CA 07 Parametric equations in TR 16
    Go to Begin

    Q12. Demo Examples

    Computer programs in GC
    Example 1 : Demo programs in Asymptote command
    • Asymptote : Find asymptotes of y = ((x-1)^M)/(2*x) for M=1,2,3,4,5
    • Click Command asymptote five times will give for M = 1, 2, 3, 4, 6
    Example 2 : Demo graphs in Inverse command
    • Demo graphs
      • 1. Intesection of y = x^2 + 2 and its inverse
      • 2. Intesection of y = x^2 + 1/4 and its inverse
      • 3. Intesection of y = x^2 and its inverse
      • 4. Intesection of y = x^2 - 2 and its inverse
      • 5. Intesection of y = e^x and y = ln(x)
      • 6. Intesection of y = 2*x + 2 and its inverse
      • 7. Intesection of y = 4*x^2 - 3 and y = 1/x
      • 8. Intesection of y = 4*x^2 + 2*x + 3 and y = 1/x
      • 9. Intesection of y = 4*x^2 + 2*x - 5 and y = 1/x
      • Graphs of y = e^x and y = ln(x)
      • Intesection of y=a*x^2 + b*x + c with y = 1/x
    • Mehtod : Click the inverse command 9 times
    • Purpose : Find the nomber of intersections

    Go to Begin

    GC 13. Sketch examples

    Example : Sketch y = (x^2 + 2)/x
    • Use GC 03 08
    • Click section 3 in upper box
    • Click program 8 in lower box
    • Give coefficients : 0, 1, 0, 2, 0, 0, 1, 0
    • Note : 0, 1, 0, 2 are coefficients of numerator
    • Note : 0, 0, 1, 0 are coefficients of denominator
    Sketch Examples : Q01 to Q10
    Go to Begin

    GC 00. Outlines

    Computer programs in GC
    • Subject | Open graphic calculator ABH
    • Select run at current location (No download)
    • Slect yes to run
    Start the menu
    • Click Start
    Back to menu
    • Click the Back command
    Select a program
    • Click section number in upper box
    • Click program number in lower box
    The demo command
    • Asymptote : Click it five times
    • Inverse : Click it nine times

    Go to Begin
    • Show Room of MD2002 Contact Dr. Shih Math Examples Room
    Copyright © Dr. K. G. Shih, Nova Scotia, Canada.
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