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Mathematics Dictionary
Dr. K. G. Shih
Graphic Solutions
Questions


  • AL 21 00 | - How to use sketch program ?
  • AL 21 01 | - Polynomial Functions
  • AL 21 02 | - Functions in factor forms
  • AL 21 03 | - Rational functions
  • AL 21 04 | - Exponent and logarithm
  • AL 21 05 | - Functions with absolute operation
  • AL 21 06 | - Functions and its inverse
  • AL 21 07 | - Intersections of quadratic functions with other functions
  • AL 21 08 | - Solve x^7 + 2*x^6 - 5*x^5 - 13*x^4 - 13*x^3 - 5*x^2 + 2*x + 1 = 0
  • AL 21 09 | - Diagram of y = (x-1)*(x-2)*(x-3)
  • AL 21 10 | - Diagram of y = x + 1/x
  • AL 21 11 | - Diagram of y = x^2 + 1/x
  • AL 21 12 | - Diagram of y = x^3 + 1/x
  • AL 21 13 | - Diagram of y = (x^3)/(x^2 - 1)
  • AL 21 14 | - Diagram of y = (x^4)/(x^2 - 1)
  • AL 21 15 | - Diagram of e^x + e^(2*x) + e^y + e^(2*y) = 12
  • AL 21 16 | - Diagram of y = x^2 - 6*x + 8 and its inverse
  • AL 21 17 | - Diagram of y = Abs(x^2 - 6*Abs(x) + 8)
  • AL 21 18 | - Examples
    • Answers


    AL 21 01. Polynomial Functions and equations

    Diagram program     Functions
    • 01 01 y = a*x + b
    • 01 02 y = a*x^2 + b*x + c
    • 01 03 y = a*x^3 + b*x^2 + c*x + d
    • 01 04 y = a*x^4 + b*x^3 + c*x^2 + d*x + e
    • 01 05 y = a*x^5 + b*x^4 + c*x^3 + d*x^2 + e*x + f
    Example : Sketch y = x^2 - 6*x + 8
    • Start sketch programs
    • Click Menu command
    • Click polynomial functions in upper box
    • Click y = a*x^2 + b*x + c in lower box
    • Give coeffcient a, b, c : 1, -6, 8
    Demo examples
    • 01 11 Function highest Power is 7
    • 01 13 Quartic equation
    • 01 14 Quartic equation
    • 01 45 Find F(u+v*i) of example 01 11

    Go to Begin

    Q02. Functions in factor forms

    Diagram Programs     Functions
    • 02 06 y = (x-a)*(x-b)*(x-c)*(x-d) - e
    • 02 38 y = (x-a)*(x-b)*(x-c)
    • 02 39 y = (x-a)*(x-b)*(x-c)*(x-d)
    • 02 40 y = (x-a)*(x-b)*(x-c)*(x-d)*(x-e)
    • 02 41 y = (x-a)*(x-b)*(x-c)*(x-d) and y = e
    • 02 40 y = (x-a)*(x-b)*(x-c)*(x-d)*(x-e) and y = f
    Example : Find the domain of y = (x-1)*(x-2)*(x-3) when y is postive
    • Start sketch programs
    • Click Menu command
    • Click Section 2 of functions in factor form in upper box
    • Click program 38 y = (x-a)*(x-b)*(x-c) in lower box
    • Give constants a, b, c : 1, 2, 3
    Demo examples
    • 02 09 Function highest Power is 7
    • 02 10 Quartic equation
    • 02 12 Quartic equation

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    Q03. Rational functions

    Diagram Programs     Functions
    • 03 07 y = 1/(a^x^2 + b*x + c)
    • 03 08 y = 1/(a^x^3 + b*x^2 + c*x + d)
    • 03 24 y = 1/x
    • 03 30 y = ((x-1)^M)/(2*x)
    • 03 31 y = ((x-1)^M)/(2*x)
    • 03 47 y = x + 4/(x^2)
    Example : Find the asymptotes y = ((x-1)^3)/(2*x)
    • Start sketch programs
    • Click Menu command
    • Click Section 3 of functions in rational function in upper box
    • Click program 30 y = ((x-1)^3)/(2*x) in lower box
    • Give power M as 3
    Demo examples : Find asymptotes
    • 03 24 y = 1/x
    • 03 47 y = x + 4/x^2
    Sketch y = (a*x^3 + b*x^2 + c*x + d)/(p*x^3 + q*x^2 + r*x + s)
    • See rational function in graphic calculator : GC 03 10
    • GC Graphic Calculator : Y = F(x)
    • Example : Sketch y = x + 1/x
      • Let y = (x^2 + 1)/x
      • Use GC 03 08
      • Give coeffcients : 0,1,0,1 0,0,1,0
      • From graph we see that asymptotes : x = 0 and y = x.

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    Q04. Exponent and logarithm

    Diagram Program     Functions
    • 04 15 y = sinh(x)
    • 04 16 y = cosh(x)
    • 04 17 y = tanh(x)
    • 04 18 y = csch(x)
    • 04 19 y = sech(x)
    • 04 20 y = coth(x)
    • 04 21 y = exp(x)
    • 04 22 y = exp(-x)
    • 04 23 y = ln(x) or log(x) base e
    • 04 33 y = exp(x) and y = ln(x)
    • 04 34 y = exp(x) and y = exp(-x)
    • 04 44 Equation : e^x + e^(2*x) + e^y + e^(2*y) = 12
    Example : Find the asymptotes of e^x + e^(2*x) + e^y + e^(2*y) = 12
    • Start sketch programs
    • Click Menu command
    • Click Section 4 of exponent and logarithm in upper box
    • Click program 44 in lower box
    • No data is required. Find asymptote from diagram.
    • Reference : See Exponent in Algebra
    Demo examples
    • All programs are demo
    Study example using 04 33
    • 1. Find intersections of y=exp(x) with its inverse
    • 2. Find intersections of y=ln(x) with its inverse
    Compare y = sinh(x) and y = sin(x)
    • See PM 14 00

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    Q05. Functions with absolute operation

    Diagram program     Functions
    • 05 25 y = Abs(a*x + b)
    • 05 26 y = a*Abs(x) + b
    • 05 27 y = a*x^2 + b*Abs(x) + c
    • 05 28 y = Abs(a*x^2 + b*x + c)
    • 05 29 y = Abs(a*x^2 + b*Abs(x) + c)
    • 05 46 Solve Abs(a*x^2 + b*Abs(x) + c) = d
    Example : How many real roots in Abs(x^2 - 6*Abs(x) + 8) = 0.5
    • Start sketch programs
    • Click Menu command
    • Click Section 5 in upper box
    • Click program 46 in lower box
    • Give data a, b, c, d : 1, -6, 8, 0.5
    • Reference : See Inverse in Algebra
    Demo examples
    • Click Menu and then click Demo
    • Select section 5
    • Select program

    Go to Begin

    Q06. Functions and its invsers

    Diagram program     Functions
    • 06 32 y = a*x + b
    • 06 33 y = a*x^2 + b*x + c
    • 06 35 y = exp(x)
    • 06 35 y = ln(x)
    Example : How many intersections of y = a*x^2 + b*x + c with its inverse
    • GC | Graphic Calculator : y = F(x).
    • Example 1 : None. See GC 06 01
    • Example 2 : One.. See GC 06 02
    • Example 3 : Two.. See GC 06 03
    • Example 2 : four. See GC 06 04

    Example : Find intersections of y = x^2 - 6*x + 8 with its inverse from graph
    • Sketch program in GC 06 10

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    Q07. Intersections of quadratic functions with other functions

    Diagram program     Functions
    • 07 36 with y = 1/x
    • 07 37 with y = p*x + q
    Example : How many intersections of y = a*x^2 + b*x + c with y = 1/x
    • GC | Graphic Calculator : y = F(x).
    • Example 1 : Two.... See GC 06 07
    • Example 2 : One.... See GC 06 08
    • Example 3 : Three.. See GC 06 09

    Example : Find intersections of y = x^2 - 6*x + 8 with y = 1/x from graph
    • Sketch program in GC 06 11
    Example : Find intersections of y = x^2 - 6*x + 8 with y = x - 2 from graph
    • Sketch program in GC 06 12

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    Q08. Solve x^7+ 2*x^6- 5*x^5- 13*x^4- 13*x^3- 5*x^2+ 2*x+ 1 = 0

    Method     Diagram
    Find solution using diagram
    • 1. Estimate the real roots
    • 2. Find the complex roots if we know real roots

    Go to Begin

    Q09. Diagram of y = (x-1)*(x-2)*(x-3)

    Questions : Using the diagrams
    • 1. Find domains if y is positive
    • 2. Find domains if the curve is increasing
    • 3. Find domains if the curve has maximum or minimum points
    Diagram

    Go to Begin

    Q10. Diagram of y = x + 1/x

    Questions : Using the diagrams
    • 1. Find domains if y is positive
    • 2. Find domains if the curve is increasing
    • 3. Find domains if the curve has maximum or minimum points
    • 4. Find equations of asymptotes
    Diagram

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    Q11. Diagram of y = x^2 + 1/x

    Questions : Using the diagrams
    • 1. Find domains if y is positive
    • 2. Find domains if the curve is increasing
    • 3. Find domains if the curve has maximum or minimum points
    • 4. Find equations of asymptotes
    Diagram

    Go to Begin

    Q12. Diagram of y = x^3 + 1/x

    Questions : Using the diagrams
    • 1. Find domains if y is positive
    • 2. Find domains if the curve is increasing
    • 3. Find domains if the curve has maximum or minimum points
    • 4. Find equations of asymptotes
    Diagram

    Go to Begin

    Q13. Diagram of y = (x^3)/(x^2 - 1)

    Questions : Using the diagrams
    • 1. Find domains if y is positive
    • 2. Find domains if the curve is increasing
    • 3. Find domains if the curve has maximum or minimum points
    • 4. Find equations of asymptotes
    Diagram
    Exercise : Sketch y = (x^3)/(x^2 - 1)
    Go to Begin

    Q14. Diagram of y = (x^4)/(x^2 - 1)

    Questions : Using the diagrams
    • 1. Find domains if y is positive
    • 2. Find domains if the curve is increasing
    • 3. Find domains if the curve has maximum or minimum points
    • 4. Find equations of asymptotes
    Diagram

    Go to Begin

    Q15. Diagram of e^x + e^(2*x) + e^y + e^(2*y) = 12

    Questions : Using the diagrams
    • 1. Find domains if y is positive
    • 2. Find domains if the curve is increasing
    • 3. Find domains if the curve has maximum or minimum points
    • 4. Find equations of asymptotes
    Diagram

    Go to Begin

    Q16. Diagram of y = x^2 - 6*x + 8 and its inverse

    Questions : Using the diagrams
    • 1. Find domains if y is positive
    • 2. Find domains if the curve is increasing
    • 3. Find domains if the curve has maximum or minimum points
    • 4. Find equations of asymptotes
    Diagram

    Go to Begin

    Q17. Diagram of y = Abs(x^2 - 6*Abs(x) + 8)

    Questions : Using the diagrams
    • 1. Solve Abs(x^2 - 6*Abs(x) + 8) = 0.5
    • 2. Solve Abs(x^2 - 6*Abs(x) + 8) = 1.0
    Diagram

    Go to Begin

    Q18. Examples

    Study notes of y = a*x^3 + b*x^2 + c*x + d
    • 1. Cubic functions : Highest power is 3.
    • 2. Slope of cubic function is y' = 3*a*x^2 + 2*b*x + c (a parabola).
      • Curve is increasing if y' is positive.
      • Curve is decreasing if y' is negative.
      • Curve has critical point if y' = 0
    • 3. Vertex
      • It has none vetex.
      • It has two vetices.
    • 4. Concavity
      • Concave upward upward if 2nd derivative is positive.
      • Concave upward downward if 2nd derivative is negative.
      • Point of inflexion if 2nd derivative is zero.
    • 5. Zero values of y
      • It has 3 zeros.
      • It has 2 zeros (One duplicate real roots)
      • It has 1 zero. (Two complex roots)
    • 6. Cubic formula : It is used in MD2002 Lesson 17.
    • 7. Example
      • What is the expression of the demo function ?
        • Polynomila form
        • Factor form
      • What is the y-intercept ?
      • what are the zero values of y ?
      • What are the coordinates of the critical points ?
      • What is the domian for y on graph when y is less than zero ?
    • 7. Exercises
      • 1. Sketch y = x^3
      • 2. Sketch y = x^3 -2*x^2 - x + 2
    Study notes of y = a*x^4 + b*x^3 + c*x^2 + d*x + e
    • 1. Slope of cubic function is y' = 4*a*x^3 + 3*b*x2 + c*x + d.
      • Curve is increasing if y' is positive.
      • Curve is decreasing if y' is negative.
      • Curve has critical point if y' = 0
    • 3. Vertex
      • It has one or three vetices.
    • 4. Concavity
      • Concave upward upward if 2nd derivative is positive.
      • Concave upward downward if 2nd derivative is negative.
      • Point of inflexion if 2nd derivative is zero.
    • 5. Zeros of y
      • It has no zero of y. (Four complex roots).
      • It has one zero of y. (One duplicate real and two complex).
      • It has three zeros of y. (One duplicate real and two different real)
      • It has four zeros of y. (Four different real roots)
    • 6. Quartic formula :
      • It used to find four complex roots
      • It is used in MD2002 Lesson 17.
  • Study the graph of demo question in Program 04
    • What is the expression of the demo function ?
      • Polynomila form
      • Factor form
    • What is the y-intercept ?
    • what are the zero values of y ?
    • What are the coordinates of the critical points ?
    • What is the domian for y on graph when y is less than zero ?
  • Exercises
    • 1. Sketch y = x^4 + 4*x^3 + 6*x^2 + 4*x + 1
    • 2. Sketch y = x^4 - 4*x^3 - x^2 + 16*x - 12
    Study notes of y = a*x^5 + b*x^4 + c*x^3 + d*x^2 + e*x + f
    • 1. Slope of quaint function is y' = 5*a*x^4 + 4*b*x3 + 3c*x^2 + 2*d*x + e.
      • Curve is increasing if y' is positive.
      • Curve is decreasing if y' is negative.
      • Curve has critical point if y' = 0
    • 3. Vertex
      • It has none, two or four vetices.
    • 4. Concavity
      • Concave upward upward if 2nd derivative is positive.
      • Concave upward downward if 2nd derivative is negative.
      • Point of inflexion if 2nd derivative is zero.
    • 5. Zeros of y
      • It has one zero of y. (One real root and Four complex roots).
      • It has two zeros of y. (One real, One duplicate real and two complex).
      • It has three zeros of y. (One duplicate real and one different real)
      • It has four zeros of y. (Three different real and one duplicate)
      • It has five zeros of y. (Five different real)
    • 6. Quaint formula : It may not be availble in mathematic field.
    • 7. Study the graph of demo question in Program 01 05
      • What is the y-intercept ?
      • what are the zero values of y ?
      • What are the coordinates of the critical points ?
      • What is the domian for y on graph when y is less than zero ?
    • 8. Exercises
      • 1. Change y = x^5 - 8*x^4 + 15*x^3 + 20*x^2 - 76*x + 48 to factor production
      • 2. Above function have roots p,q,r,s,t. Find p*q*r*s*t (Production)
      • 2. Above function have roots p,q,r,s,t. Find p+q+r+s+t (Sum)

    Go to Begin

    Q00. How to use sketch program

    • Sketh Programs | Find graphic solutions.
    • 1. Start sketch program
      • Clcik sketch program
      • Select run at current location
      • Select yes to run
    • 2. Sketch y = x^2 - 6*x + 8 which is in 01 02
      • Click Menu command
      • Click polynomial function in section 1 in upper box
      • Click Program 02 in lowe box
      • Give coefficients a, b, c : That is 1, -6, 8
    • 3. How change scale and replot ?
      • After we get the graph, we select new xmax and ymax in left box
      • Click replot
    • 4. What is demo command ?
      • Click Menu then click Demo
      • Select a section number
      • Slect a program
      • It will give a plot using default values.
      • For example : What is the function of 01 04 ?
    • 5. What is the meaning of 01 04 ?
      • First number is the section number in upper box
      • Second number is the function number in lower box
    Factor form : y = (x-a)*(x-b)*(x-c) in Program 38
    • 1. Change to polynomial using program 38.
    • 2. Method 2 : Use multiplication of polynomials.
    • 3. Find y-intercept.
    • 4. Quick sketch : Use 3 zeros and y-intercept to plot the curve.
    Factor form : y = (x-a)*(x-b)*(x-c)*(x-d) in Program 39
    • 1. Change to polynomial using program 39.
    • 2. Method 2 : Use multiplication of polynomials.
    • 3. Find y-intercept.
    • 4. Quick sketch : Use 4 zeros and y-intercept to plot the curve.
    Factor form : y = (x-a)*(x-b)*(x-c)*(x-d)*(x-e) in Program 40
    • 1. Change to polynomial using program 40.
    • 2. Method 2 : Use multiplication of polynomials.
    • 3. Find y-intercept.
    • 4. Quick sketch : Use 5 zeros and y-intercept to plot the curve.
    Solve (x-a)*(x-b)*(x-c)*(x-d) = e
    • Study Program | Graphic Solutions of Polynomial Functions.
    • Graphic method 1 : Sketch y = (x-a)*(x-b)*(x-c)*(x-d) - e
      • Zeros of y are the solution (Program 06)
      • What is the question ? What are the solutions
    • Graphic method 2 : Sketch y = (x-a)*(x-b)*(x-c)*(x-d) and y = e
      • Intersections of line and curve the solution (Program ??)
      • What is the demo question ? What are the solutions ?
    • Examples
      • Program 09 : y = (x-5)*(x-7)*(x+4)*(x+6) - 504
      • Program 10 : y = (x+9)*(x-3)*(x-7)*(x+5) - 385
      • Program 12 : y = (28x-7)*(x-3)*(x+3)*(2*x+5) -91
      • Program 13 : y = 12*x^4 - 56*x^3 +59*x^2 - 56*x + 12
      • Program 14 : y = x^4 + x^3 - 4*x^2 + x + 1

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