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Mathematics Dictionary
Dr. K. G. Shih
Rational Functions and Equations
Subjects

  • AL 16 00 | - Outlines
  • AL 16 01 | - y = 1/(a*x + b)
  • AL 16 02 | - y = 1/(a*x^2 + b*x + 1)
  • AL 16 03 | - y = 1/(a*x^3 + b*x^2 + c*x + d)
  • AL 16 04 | - y = (a*x^3 + b*x^2 + c*x + d)/(p*x^3 + q*x^2 + r*x + s)
  • AL 16 05 | - y = ((x - 1)^M)/(2*x)
  • AL 16 06 | - Example : y = x + 4/(x^2)
  • AL 16 07 | - Rationalize fraction ir-rational function
  • AL 16 08 | -
  • AL 16 09 | -
  • AL 16 10 | -
    • Answers


    AL 16 01. y = 1/(a*x + b) Sketch program     Example : Sketch y = 1/(x + 1)
    • Find asymptote
      • If x = -1, then y = infinite.
      • Hence x = -1 is a vertical asymptote.
      • If x goes to infinte, y goes to zero.
      • Hence y = 0 is a horizontal asysmptote.
    • Quick sketch
      • Draw the asymptote x = -1 and y = 0.
      • Find y-intercept. That is x = 0 and y = 1.
      • If x LT -1, then y is negative and goes to -infinite as x goes to -1.
      • If x goes to infinite, y goes to zero.
      • If x GT -1, then y is positve and from +infinite to 0.
      • Know we have some idea how to draw the curve.
      • To make the curve accuate, we can add two or more points
        • x = +0 and y = +1
        • x = +1 and y = +0.5
        • x = -2 and y = -1
        • x = -3 and y = -0.5
    • Find the sketch on computer
      • Open the Rational Function program
      • Click start in the program to get the menu
      • Click Subject 03 in lower box to open the program menu
      • Click program 01 to sketch y = 1/(a*x + b)
      • Give coefficients a and b as 1, 1 (i.e a = 1 and b = 1)

    Go to Begin

    AL 16 02. y = 1/(a*x^2 + b*x^2 + c*x + d)

    Sketch program     Example : Sketch y = 1/(x^2 - 1)
    • Find asymptote
      • Since x^2 - 1 = (x - 1)*(x + 1).
      • If x = -1, then y = infinite.
      • Hence x = -1 is a vertical asymptote.
      • If x = +1, then y = infinite.
      • Hence x = +1 is a vertical asymptote.
      • If x goes to infinte, y goes to zero.
      • Hence y = 0 is a horizontal asysmptote.
    • Quick sketch
      • Draw the asymptotes x = -1, x = +1 and y = 0 .
      • Find y-intercept. That is x = 0 and y = -1.
      • If x LT -1, then y is positive and from 0 increase to infinite as x increase.
      • If x goes to infinite, y goes to zero.
      • If x GT +1, then y is positive and from infinite to 0 as x increase.
      • If x is beteen -1 and +1, y form -infinite to -1 and from -1 to -infinite
      • Know we have some idea how to draw the curve.
      • To make the curve accuate, we can add two or more points
        • x = +0 and y = -1
        • x = +5 and y = +0.25
        • x = -5 and y = +0.25
    • Find the sketch on computer
      • Open the Rational Function program
      • Click start in the program to get the menu
      • Click Subject 03 in lower box to open the program menu
      • Click program 02 to sketch y = 1/(a*x^2 + b*x + c)
      • Give coefficients a, b, c as 1, 0, 1 (i.e a = 1, b = 0 and c = 0)

    Go to Begin

    AL 16 03. y = 1/(a*x^2 + b*x^2 + c*x + d) Sketch program     Example : Sketch y = 1/(x^3 -2*x^2 - x + 2)
    • Find asymptotes
      • Since x^3 - 2*x^2 - x + 2 = (x + 1)*(x - 1)(x - 2).
      • Hence vertical asymptotes are x = -1, x = 1 ans x = 2
      • If x goes to infinte, y goes to zero.
      • Hence y = 0 is a horizontal asysmptote.
    • Quick sketch
      • Draw the asymptotes x = -1, x = +1, x = 2 and y = 0 .
      • Find y-intercept. That is x = 0 and y = 1/2.
      • If x LT -1, then y =1/((-)*(-)*(-) = (-) and y is from 0 to -infinite
      • If x GT -1 and LT +1
        • y = 1/(+)*(-)*(-) = (+).
        • y is from infinite to finite and to infinite
      • If x GT +1 and LT +2
        • y = 1/(+)*(+)*(-) = (-).
        • y is from -infinite to -finite and to -infinite
      • If x GT +2 then y is positive and from infinite to 0 as x increase.
      • Know we have some idea how to draw the curve.
      • To make the curve accuate, we can add two or more points
        • x = -2 and y = 1/((-2+1)*(-2-1)*(-2-2)) = 1/(-12)
        • x = +0 and y = 1/2
        • x =0.5 and y = 1/((1.5)*(-0.5)*(-1.5) = 1/(-2.25)
        • x = +3 and y = 1/(4)*(2)*(1)) = 1/8
    • Find the sketch on computer
      • Open the Rational Function program
      • Click start in the program to get the menu
      • Click Subject 03 in lower box to open the program menu
      • Click program 03 to sketch y = 1/(a*x^3 + b*x^2 + c*x + d)
      • Give coefficients a,b,c,d as 1,-2, -1, 2

    Go to Begin

    AL 16 04. y = (a*x^3 + b*x^2 + c*x + d)/(p*x^3 + q*x^2 + r*x + s)

    Sketch program     Example : Sketch y = (x^2 - 2*x + 1)/x
    • Rational Functions
      GC 03 08
    • Find asymptotes
      • Since y = (x^2 - 2*x + 1)/x = x - 2 +1/x.
      • Hence asymptote is y = x - 2 as x goes to infinte
      • If x goes to 0, y goes to infinte.
      • Hence x = 0 is a vertical asysmptote.
    • Find the sketch on computer
      • Open the Rational Function program
      • Click start in the program to get the menu
      • Click Subject 03 in lower box to open the program menu
      • Click program 08 y = (a*x^3 + b*x^2 + c*x + d)/(p*x^3 + q*x^2 + r*x + s)
      • Give coefficients a,b,c,d and p,q,r,s as 0,1,-2,1, 0,0,1,0
    • Quick sketch
      • Draw asymptotes y = x - 2 and x = 0.
      • y = (x-1)^2/x
      • If x LT 0, then y = (-)
        • The curve is between x = 0 and y = x -2
        • From -infinite to -finite and from -finite to -infinte (at x=0)
      • if x GT 0 and LT 1, then y = (+)
        • The curve is between x = 0 and y = x -2
        • From +infinite to 0 (at x = 1)
      • If x GT 1, then y = (+)
        • The curve is between x = 0 and y = x -2
        • From y = 0 (at x = 1) to +infinite

    Go to Begin

    AL 16 05. y = ((x - 1)^M)/(2*x) Sketch program     Example : Sketch y = ((x - 1)^3)/(2*x) Solve x + 4/(x^2) > 3
    • Both sides times x^2
    • x^3 + 4 > 3*x^2
    • Both sides minus 3*x^2
    • x^3 - 3*x^2 + 4 > 0
    • Hence x = -1 or x > 0
    • Hence x is between -1 and zero and x GT 0.
    Use a + b + c > 3*((a*b*c)^(1/3))
    • let a = x/2, b = x/2, c = 4/(x^2)
    • Hence a + b + c = x + 4/(x^2)
    • Hence a*b*c = (x/2)*(x/2)*(4/x^2) = 1
    • Hence x + 4/(x^2) > 3
    Use y' and y"
    • y' = 1 - 8/(x^3) = 0 if there is critical point
    • Hence x = 2.
    • y" = + 24/(x^4) and y" is positive, Hence x = 2 it is minimum
    • Minimum = 2 + 4/(2^2) = 3

    Go to Begin

    AL 16 07. Rationalize fraction ir-rational function

    Method
    • Change ir-rational denominator to rational
    • Example : y = a/(b + Sqr(c))
      • Use (b - Sqr(c)) to multiply numberator and denominator
      • y = a*(b - Sqr(c))/((b + Sqr(c))*(b - Sqr(c)))
      • y = a*(b - Sqr(c))/(b^2 - c^2)

    Go to Begin

    AL 16 08.
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    AL 16 09. Answer

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    AL 16 10. Answer

    Go to Begin

    AL 16 00. Outlines
    Deifintion
    • Rational function : y = F(x)/G(x) where F(x) and G(x) are rational functions
    • Ir-rational function : y = F(x)/G(x) where F(x) or G(x) is ir-rational functions
    Topics
    • Rationalize ir-rational function
    • Horizontal asymptote : y = c if x goes to infinite
    • Vertical asymptote : x = c if y goes to infinite
    • Slant asymptote : y = a*x + b if x goes to infinite
    • Sketch program in GC 03
      • Sketch y = 1/(ax + b) in GC 03 01
      • Sketch y = 1/(ax^2 + b*x + c) in GC 03 02
      • Sketch y = F(x)/G(x) in GC 03 06
    • Sketch program in AL 21 03
      • Sketch y = 1/(ax + b) in
      • Sketch y = 1/(ax^2 + b*x + c) in

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