Counter Examples
Mathematics Dictionary
Dr. K. G. Shih
Content by Index ........ Contents by index
Contents by Sections .... Contents by chapters
Program WebABC .......... Application of Math Dictionary on Web

Read Subtract of Quadratic Functions

Contents
    Read Symbol defintion

  • Q01 | - Expressions of quadratic functions
  • Q02 | - Definition and formula of quadratic function
  • Q03 | - Properties of y = a*x^2 + b*x + c :
  • Q04 | - Completing the square of y = a*x^2 + b*x + c
  • Q05 | - Intersections of y = a*x^2 + b*x + c with its inverse
  • Q06 | - Parabola and quadratic functions
  • Q07 | - Example : Three point define a parabola
  • Q08 | - Example : Difference of quadratic function
  • Q09 | - Intersections of y = a*x^2 + b*x + c with y = 1/x
  • Q10 | - Summary of quadratic functions
Answers


Q01. Expressions
    a. Polynomial form .... y = a*x^2+b*x+c = F(x)
    b. Factorial form ..... y = (x-r)*(x-s)
    c. Vertex form ........ y-k = a*(x-h)^2
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Q02. Defintion and formula for y = a*x^2 + b*x + c
    a. Curve name
      A. Parabola
      B. Curve of quadratic functions

    b. Discriminant
      A. Expression : Di = b^2-4*a*c
      B. Di < 0 and a > 0

        (1) No real roots
        (2) Range : yv to +infinite and yv > 0
        (3) No real factors

      C. Di = 0 and a > 0
        (1) Two same real roots
        (2) Range : 0 to +infinite and yv = 0
        (3) It is perfect square y = (x-r)^2

      D. Di > 0 and a > 0
        (1) Two different real roots
        (2) Range : yv to +infinite and yv < 0
        (3) It has factor as y = (x-r)*(x-s)

    c. Vertex (h,k)
      A. xv = h = -b/(2*a)
      B. yv = k = F(xv)

    d. Focus
      A. xf = xv
      B. yf = yv + 1/(2*D) = yv + 1/(4*a))
      C. D = Focus to directrix = 1/(2*a) if y = a*x^2 + b*x + c

    e. Directrix
      A. Equation of directrix is y = k - 1/(4*a)
      B. Distance to Focus : D = 1/(2*a)

    f. Quadratic formula
      A. r = (-b+sqr(b^2-4*a*c))/(2*a)
      B. s = (-b-sqr(b^2-4*a*c))/(2*a)

    g. Principal axis of y - k = a*(x-h) is x = h.
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Q03. Properties of y = a*x^2 + b*x + c :
    a. y-ingtercept = c at x = 0
    b. Discriminant Di = b^2-4*a*c
    c. zeros of y
      A. No zero if discriminant < 0
      B. One zero if discriminant = 0
      C. two zeros if discriminant > 0

    d. Curve and Slope y' = 2*a*x+b = first derivative of y
      A. Slope > 0 the curve is increasing
      B. Slope < 0 the curve is decreasing
      C. Slope = 0 the curve has minimum or maximum
        (1) Minimum if slope = 0 and a > 0
        (2) Maximum if slope = 0 and a < 0
      D. Notes
        (1) y' = 2*a + b = Slope
        (2) y' = 0 it has extreme points or critical points

    e. Concavity and Second difference = 2*a
      A. If second difference > 0 the curve is concave upward
      B. if second difference < 0 the curve is concave downward
      C. If second difference = 0 the curve has infexion
      D. Second difference = 2nd derivative of y" = 2*a

      E. Inflextion if y"=0

    f. Range
      A. Range > 0 if a > 0 and discriminant < 0
      B. Range >= 0 if a > 0 and discriminant = 0
      C. Range for a > 0 and discriminant > 0
        (1) Range => 0 if x <= r
        (2) Range => 0 if x >= s
        (3) Range < 0 if r < x < s

      Where a*x^2 + b*x + c = (x-r)*(x-s) and s > r
    g. Vertex
      A. xv = -b/(2*a)
      B. yv = F(xv) where F(x)=a*x^2+b*x+c

    h. Quadratic formula
      x1 = (-b + sqr(b^2-4*a*c)/(2*a)
      x2 = (-b - sqr(b^2-4*a*c)/(2*a)

    g. The roots of F(x)
      A. If b^2-4*a*c < 0 x1 and x2 are compex root and conjgate
      B. If b^2-4*a*c = 0 then x1=x2 (two same real roots)
      C. If b^2-4*a*c > 0 then there two different real roots

    h. What is conjugate ?
      A. x1 = m + n*i
      B. x2 = m - n*i
      C. hence x1+x2 = real and x1*x2 = real where i=sqr(-1)

    i. Roots and coefficient a b c
      A. Sum of roots = -b/a
      B. Product of roots = c/a

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Q04. Completing the square

    a. Method : steps
      A. y = a*x^2 + b*x + c
      B. y = a*(x^2 + b*x/a + c/a)
      C. y = a*(x^2 + b*x/a + (b/2*a)^2-(b/(2*a)^2 + c/a)
      D. y = a*(x + b/(2*a))^2 - a*((b/2*a)^2 - c/a)
      E. y = a*(x+h)^2 - (b^2-4*a*c)/(4*a)
      F. y - k = a*(x-h)
      G. where h = -b/(2*a) and k = -(b^2-4*a*c)/(4*a)

    b. Application
      A. Find vertex of y = a*x^2 + b*x + c
      • Change to vertex form by completing squrare y - k = a*(x - h)^2.
      • Hence h = -b/(2*a) and k = -(b^2-4*a*c)/(4*a).
      B. Find center & radius of x^2 + y^2 + d*x + e*y + f = 0

      • Example : Prove that x^2 + y^2 - 4*x + 4*y - 8 = 0 is a circle.
      • (x^2 - 4*x + 4 - 4) + (y^2 + 4*y + 4 - 4) - 8 = 0
      • (x-2)^2 + (y+2)^2 = 4^2.
      • This a circle with radius 4 and center at (2,-2).
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Q05. Inverse of y = a*x^2 + b*x + c
What is inverse of y = a*x^2 + b*x + c ?
How many intersections of y = a*x^2 + b*x + c with its inverse ?
  • There are 4 answers : None, one, two or four intersections.
  • Example Intersections of Quadratic function with its inverse
    • Start the program ABH.
    • Click the Inverse command four times will see four diagrams.
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Q06. Parabola
Q6. Parabola Defintion and formula
[Example] Using definition of parabola prove that a = 1/(2*D)

    a. PF = Sqr((x-xf)^2+(y-yf)^2)
    b. PQ = (y-yf+D) = distance from P to directrix
    c. PF = PQ
    d. PF^2 = (x-xf)^2 + (y-yf)^2 = (y-yf+D)^2
    or PF^2 = (y-yf)^2 + 2*D*(y-yf) + D^2
    e. (x-xf)^2 - D^2 = 2*D*(y-yf)
    f. y = (x^2 - (2*xf)*x + (xf^2 + 2*D*yf))/(2*D)
    or y = a*x^2 + b*x + c
    g. Hence a = 1/(2*D)

[Example] Properteis of parabola
  • Expression y = a*x^2 + b*x + c.
  • Locus of parabola : PF = PQ.
    • P is moving point.
    • F is the focus.
    • Q is a point on a fixed line and PQ perpendicular to the line.
    • Equation of the parabola : y - k = (x - h)^2/(2*D).
    • Principal axis is x = h.
    • Focus is at x = h and y = k + D/2.
    • Equation of directrix is y = k - D/2.
    • D is the distance from focus to dirrectrix = 1/(2*a).
    • Digram of parabola in rectangular coordinates.
    • Digram of parabola in polar coordinates
[Example] Parabola has vertex at (3,-1) and passes (0;8) find function
    Using vetex form we have (y+1) = a*(x-3)^2
    Substitute (0;8) into above function : 9 = a*9 and a=1
    Hence y + 1 = 1*(x-3)^2
    or y = x^2 - 6*x + 8

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Q07. Example : Three points define a formula

Three points define a parabola
    a. Submite 3 points in to y=a*x^2+b*x+c
    b. We have 3 linear equations
    c. Solve these 3 linear equations
    d. Reference :

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Q08. Example : Difference of quadratic function
[Example 1] Show that
    x = 1...2...3...4...5
    y = 1...4...9..16..25

    is a quadratic function and find the function

Solution
    [Method 1]
      a. By observation : relation between x and y is y=x^2.
      b. Hence it is a quadratic equation a=1 b=0 c=0

    [Method 2] Use 2nd difference = common difference
      ............ x = 1...2...3...4...5...6 ....
      ............ y = 1...4...9..15..25..36 ....
      1st difference = ....3...5...7...9..11 ....
      2nd difference = ........2...2...2...2 ....

      a. 2nd diff = common diff)
      b. Hence it is a quadratic equation
      c. Since second difference = 2*a hence a = 1
      d. Hence the function is y = x^2 + b*x + c
      e. We substitute two points into above equation
      f. We get 2 linear equations
        Eq 1 .........1 = 1 + b + c
        Eq 2 .........4 = 4 + 2*b + c
        Eq 2 - Eq ....1 : 3 = 3 + b and b = 0
      g. Hence c = 0 and we have y = x^2
      h. Is it easier than solving 3 linear equations ?

[Example 2] Sequence start from n=0 we have
    ....... 0...1...2...3...4...5...6 .....
    ....... 1...3...7..13..21..31..43 .....

    A. Find T(n) and S(n)
    B. Find T(7) and S(7)
Solution : We find the 1st and 2nd difference

    ...... n =....0....1....2....3....4....5....6....7 ...
    Sequence =....1....3....7...13...21...31...43....? ...
    1st diff =.........2....4....6....8...10...12...14 ...
    2nd dife =..............2....2....2....2....2....2 ...

    Hence sequence fits y=a*x^2 + b*x + c and 2nd diff = 2*a
    Hence T(n) = n^2 + b*n + c
    Since T(0) = 1 and we have c = 1

    Now we have T(n) = n^2 + b*n + 1 where b is unknown
    Since T(1) = 3 and we have 3 = 1^2 + b*1 + 1 and b = 1
    The nth term : T(n) = n^2 + n + 1
    The sum of nterm is S(n) = Sum(n^2) + Sum(n) + Sum(1)
      S(n) = n*(n+1)*(2*n+1)/6 + n*(n+1)/2 + n
      S(n) = n*((n+1)*(2*n+1)/6 + (n+1)/2 + 1)
      S(n) = n*(2*n^2 + 3*n + 1 + 3*n + 3 + 6)/6
      S(n) = n*(n^2 + 3*n + 5)/3 and n is not equal 0

    If n start from zero S(n) = n*(n^2 + 3*n + 5)/3 + 1

*** Verify
    T(2) = 2^2 + 2 + 1 = 7. Correct.
    T(3) = 3^2 + 3 + 1 = 13. Correct.

    S(2) = 2*(2^2 + 3*2 + 5)/3 = 30/3 = 10. Not 11 why ?
    S(3) = 3*(3^2 + 3*3 + 5)/3 = 23. Not 24 why ?
    S(n) is for n > 0

[Example 3] Prove the 2nd difference of y=a*x^2+b*x+c is 2*a
    ..... x =....0......1..........2..........3...........4.....

    ..... y =....c..a+b+c..4*a+2*b+c..9*a+3*b+c..16*a+4*b+c

    1st dif =.........a+b......3*a+b......5*a+b.......7*a+b

    2nd dif =....................2*a........2*a.........2*a

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Q09. Intersection of y = a*x^2 + b*x + c with y = 1/x
[Example] if y = 1/x tangents to y = 4*x^2 + k
    1. find k
    2. find other intersection
    3. sketch the curve

[Solution]
    1. Finf k
      A. That is both curve has same slope at a point
      B. Slope of y = 1/x is y' = -1/x^2
      C. Slope of y = 4*x^2+k is y' = 8*x
      D. Hence -1/x^2 = 8*x or 8*x^3 + 1 = 0
      E. Hence x = -1/2 and y = -2 is the point
      F. Find k : -2 = 4*(-1/2)^2 + k
      G. Find k : k = -3

    2. Find other intersection
      A. y = 1/x and y= 4*x^2 - 3
      B. by observation : x = 1 and y = 1


      ** 2nd method :
        A. Since interception 1/x = 4*x^2 - 3
        B. Hence 4*x^3 - 3*x - 1 = 0
        C. Since y = 1/x touch y = 4*x^2-3 at (-1/2; -2)
        D. Equation B has roor -1/2 and -1/2
        E. Use synthetic divisition

          4....0....-3....-1....|...-1/2
          ....-2.....1.....1
          ---------------------
          4...-2....-2.....0....|...-1/2
          ....-2.....2
          ---------------------
          4...-4.....0

        Hence remainder is 4*x-4 = 0 and x = 1.

[Example] How many intersections of y = a*x^2 + b*x + c with y = 1/x
[Answer]
  • Examples on internet
  • After entering the program ABH.
  • Click Start command.
  • Click subject 6 in upper box.
  • Click program 07 it gives one intersection.
  • Click program 08 it gives two intersections.
  • Click program 09 it gives three intersections.
  • Click program 11 if we give data a,b,c for y = a*x^2 + b*x + c.
Go to Begin

Q10. Summary
    a. Keywords of quadratic functions
      Critical points : y' = 0
      Concavity and 2nd difference
      Derivative 1st and 2nd
      Difference 1st and 2nd
      Discriminat
      Distance between focus and directrix
      Equation of directrix
      Expression of discriminant
      Extreme points : y' = 0
      Focus
      Maximum : y' = 0 and y" < 0 (a < 0)
      Minimum : y' = 0 and y" > 0 (a > 0)
      Principal axis of paraboa : x = -b/(2*v)

      Quadratic formula
      Range and domain
      Roots and coefficients
      Slope and curve : y' < 0 y' = 0 y' > 0
      Vertex
      y-inercept
      zeros of y
 Go to Begin
Remarks
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    * For more information, Please contact Dr. K. G. Shih
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