Contents

  • Fraction Rules
  • Exponent Rules
  • Factoring Rules
  • Inequality and Absolute Value Rules
  • Formulas
  • Logarithm Rules
  • Function Rules
  • Trigonometry Rules
  • Fraction Rules:


    1.
    ac
    bc
    = a
    b
    2.
    a
    b
    + c
    d
    = ad+bc
    bd
    3.
    a
    b
    c
    d
    = ac
    bd
    4.
    a
    b

    c
    d
    = ad
    bc
    5.
    a
    b
    = 0 if, and only if, a = 0 while b 0
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    Exponent Rules:


    1. x0 = 1, if x is not equal to 0.
    2. xpxq = xp+q
    3. (xy)q = xqyq
    4.
    xp
    xq
    = xp-q
    5.


    x
    y


    m

     
    = xm
    ym

    6.
    x(-y) = 1
    xy

    7. xyz = (xy)z

    8.

    9.


    10. xy = xz if, and only if, y = z
    11.
    If a > 1 and x < y then ax < ay
    12.
    If 0 < a < 1 and x < y then ax > ay
    13. elnx = x and lnex =x
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    Factoring Rules:


    1.
    xy + xz = x(y+z)
    2.
    x2 - y2 = (x + y)(x -y)
    3.
    x2 + 2xy + y2 = (x + y)2
    4.
    x2 - 2xy + y2 = (x - y)2
    5.
    x3 + y3 = (x + y)(x2 - xy + y2)
    6.
    x3 - y3 = (x - y)(x2 + xy + y2)
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    Inequality and Absolute Value Rules:


    1.

    x < y if, and only if, x+z < y+z and x-z < y-z.

    2.
    If x < y and if z > 0 then xz < yz
    If x < y and if z < 0 then xz > yz

    3.
    If     x < y     and     z > 0     then     x
    z
    < y
    z




    If     x < y     and     z < 0     then     x
    z
    > y
    z




    4.
    |a| = a if a is non-negative and |a| = -a if a is negative.

    5.
    |a - b|is the distance from point a to point b on the real line.


    6.




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    Formulas:


    1. The distance D between 2 points (a,b) and (c,d) in the plane is:

     

    2. The circle with center at (h,k) and radius r is given by:
    (x - h)2 + (y - k)2 = r2
    3. The Midpoint M of the line segment from (a,b) to (c,d) is:
    M =

    a + c
    2
    , b + d
    2


    4. The slope m of the line between the points (a,b) and (c,d) is:
    m = b - d
    a - c
    5. The point-slope formula for the straight line through the point (a,b) with slope m is
    y - b = m(x - a)
    6. A line perpendicular to the line y=mx + b will have slope
    -1
    m
    7. Quadratic Formula:
     


    8. If

    f(x) = a(x-h)2 + k
    then the graph is a parabola with vertex at (h,k) and axis of symmetry the line x = h.
    If
    a > 0
    then the parabola opens up and if
    a < 0
    then the parabola opens down.
    9. The equation of a circle with center at (h,k) and radius r is:
    (x - h)2 + (y - k)2 = r2
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    Logarithm & Exponential Rules:


    1.
    ln(e) = 1
    2.
    ln(xy) = ln(x) + ln(y)
    3.
    ln( x
    y
    ) = ln(x) - ln(y)
    4.
    ln(xr) = r·ln(x)
    5.
    ln(x) = y if, and only if, x = ey
    and:
    log(x) = y if, and only if, 10y = x

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    Function Rules:


    1.
    • A function f is Odd if f(-x) = -f(x) for every x in its domain.
    • A function f is Even if f(-x) = f(x) for every x in its domain.

      2.
    • The graph of f(x-a) for a positive is the graph of f(x) translated to the right by a units. For example (x-2)2 is the graph of x2 translated to the right 2 units.


    • The graph of f(x+a) for a positive is the graph of f(x) shifted to the left by a units. For example, (x+2)2 is the graph of x2 shifted to the left 2 units.


    • The graph of -f(x) is the graph of f(x) reflected about the x-axis.For example, -x2 is the graph of x2 reflected about the x-axis.


    • The graph of 2f(x) is the graph of f(x) stretched 2 units up at each point. For example, 2x2 is the graph of x2 stretched 2 units upward.


    • The graph of Af(x) is the graph of f(x) stretched up A units up at each point for positive A. (See, for example, the previous rule.)

    • The graph of f(x) + b is the graph of f(x) translated up b units if b is positive and down |b| units if b is negative. For example, x2+1 is the graph of x2 translated up by 1 unit.



      3.


      4. The definitions of the Trig Functions:
      Sin(x) = The 2d-coordinate of the point on the unit circle obtained by measuring counterclockwise around the circle beginning at the point (1,0) if x is non-negative, and measuring clockwise if x is negative. Cos(x) is the 1st-coordinate of the point so obtained.




      Tan(x) = Sin(x)
      Cos(x)
          Domain:     x p
      2
      ,    3p
      2
      ,    . . .
      Cot(x) = Cos(x)
      Sin(x)
          Domain:     x 0,   p,    2p,    . . .
      Sec(x) = 1
      Cos(x)
          Domain:     x p
      2
      ,    3p
      2
      ,    . . .
      Csc(x) = 1
      Sin(x)
          Domain:     x 0,   p,    2p,    . . .

      5. A physical quantity varying with time has exponential growth if the rate of such growth is proportional to the amount of the quantity present. This means that if P(t) is the amount present at time t, then P(t) = P0ert. r is called the growth constant.

      6. If a function f is continuous on an interval and if f(x) is not 0 at any input x in that interval then f(x) is greater than 0 at every x in that interval or f(x) is less than 0 at every x in that interval.

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      Trigonometry Rules:


      1.
      • The Pythagorean Identities:
        Sin2(x) + Cos2(x) = 1.
        Tan2(x) + 1 = Sec2(x)

      • The Sine Addition formula: Sin(x+y) = SinxCosy + SinyCosx

      • The Sine Subtraction formula: Sin(x-y) = SinxCosy - SinyCosx

      • The Cosine Addition formula: Cos(x+y) = CosxCosy - SinxSiny

      • The Cosine Subtraction formula: Cos(x-y) = CosxCosy + SinxSiny

      • The Sine Double Angle formula: Sin2x = 2SinxCosx

      • The Cosine Double Angle formulae: Cos2x = 1 - 2Sin2x
        = 2Cos2x - 1
        = Cos2x - Sin2x


      2. The Pythagorean Theorem. In the right triangle: Hypotenuse2 = Adjacent2 + Opposite2


      3. Right Angle Trigonometry:

      Sin(A) = Opposite to A
      Hypotenuse
      Cos(A) = Adjacent to A
      Hypotenuse
      Tan(A) = Opposite to A
      Adjacent to A
      Cot(A) = Adjacent to A
      Opposite to A
      Csc(A) = Hypotenuse
      Opposite to A
      Sec(A) = Hypotenuse
      Adjacent to A
      4. The Law of Cosines: In the triangle:
      a2 = b2 + c2 - 2·b·c·Cosa
      Notice that if the angle is 90 degrees, the Law of Cosines becomes the Pythagorean Theorem.
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