MAC1105 College Algebra by VII9jovw

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									                     Formula List for College Algebra – ACADEMIC SYSTEMS


Quadratic Function:
  A quadratic function is one in the form: f  x   ax 2  bx  c
  where a, b, and c are constants and a is not equal zero.

      A quadratic function in vertex form is: f  x   a  x  h   k or f  x   a  x  xv   yv
                                                                                       2                                  2


      (See below for the meaning of the letters h and k or xv and yv .

Zero-Factor principle:
   ab  0 if and only if a  0 or b  0.

Quadratic Formula:
                                                                       b  b 2  4ac
   The solutions of the equation ax  bx  c  0, where a  0, are x 
                                                          2

                                                                              2a
The Discriminant: b  4ac
                     2

   Students need to memorize “the nature of the solutions” as discussed in class.

Complex Numbers:                            1  i or i 2   1

Vertex of a parabola:
         b                     b                                                b
    xv      and yv  f          . Re call the LINE of SYMMERTRY is x               .
         2a                     2a                                                2a
                                                       b  b    b 4ac  b2 
       Also, the vertex is :  h ,k    xv , yv       , f          ,    .
                                                       2a      2a    2a   4a 

Quadratic Equation in Vertex Form:
  The vertex form of the equation ax 2  bx  c  0, where a  0, is :
       y  a  x  x v  2  y v where  x v , y v  is called the vertex.

The Algebra of Functions:
   Sum :  f  g  x   f  x   g  x                                   Difference :   f    g  x   f  x   g  x 
                                                                                         f        f  x
       Pr oduct :          f  g  x   f  x   g  x                  Quotient :    x          where g  x   0
                                                                                        g         g  x

One-to-one Functions:

      The inverse of a function f is also a function if and only if f is one-to-one.




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Composition of Functions:
  Let f  x  and g  x  reprenet two functions. The composition of f and g ,
     written  f g  x  , is defined as  f g  x   f  g  x   . Here, g  x  must be
                                                                    
     in the domain of f  x  . If it is not , then f  g  x   will be undefined .
                                                               

Inverse Functions:
                                                                                                    1
   Suppose the inverse of f is a function, denoted by f                                                  . Then
      f    1
                 y   x if    and only if f  x   y.

Composition of a Function and its Inverse:
  If a function, f  x  has an inverse f 1  x  , then :
     f    1
                   f   x   x for every x in the domain of f , and

     f          f 1   x   x for every x in the domain of f 1.
---------------------------------------------------------------------------------------------
Linear Equation Formulas:
    Standard or General Form: Ax + By = C
                         y y2  y1             f  x 2   f  x 1  f b   f  a  f  x  h   f  x 
    Slope formula: m                also m                                        
                         x x2  x1                  x 2 x1              ba                   h
    Slope y-intercept form: y  mx  b
      Point Slope form: y  y1  m( x  x1 ) or y  m  x  x 1   y 1

Some Quadratic Function Formulas for Chapter 10 are at the beginning of this handout.

Exponents:
                                                                                          am
      1. a m  a n  a m  n                                                        2.      n
                                                                                               a mn , a  0
                                                                                          a
       3.  a m   a m  n                                                         4.  ab   a m b m
                     n                                                                         m


                    m
          a  am                                                                          1
       5.    m , b  0                                                            6.     m
                                                                                                am , a  0
          b  b                                                                          a
                                                                                               m
         a  m bn
                                                                                                             m
                                                                                        a      b
       7.  n  m , a  0, b  0                                                     8.     
          b    a                                                                        b      a
          9. a 0  1, a  0                                                         10. a 1 n  n a , n is an int eger n  2.
                                          m
                                                 a 
                                   1
       11. a m           n
                             a       n
                                            
                                                   m    1 n
                                                                 n
                                                                      am
                                          

Exponential Function:
   f  x   bx , where b and x are real numbers, b  0 and b  1.



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Exponential Formulas:
   Compound Interest: A  P 1  r  n
                                                                r
      Compound Interest with n Compounding Periods: A  P 1   nt ,
                                                                n
               P  principal , r  annual rate, n  number of compoundings per year ,
               t  number of years, A  amount after t years.
      Compound Interest Continuously: A  P e nt

Exponential Equality:
   If b x  b y , then x  y where b  0 and b  1.

Logarithms and Exponents: Conversion Equations
   If b  0 and x  0, then
       y  log b x if and only if x  b y.                        y  ln x if and only if e y  x.

Useful Logarithm Properties:
    log b b  1, because b1  b                                   ln e  1, because e1  e.
        log b 1  0, because b 0  1                              ln1  0, because e 0  1.
        log b b x  x, because b x  b x                          ln e x  x, because e x  e x .
        b log b x  x, for x  0                                  e ln x  x, for x  0.

Other Properties of Logarithms:
   If x, y and b  0, then                                        If x and y  0, then
             a. log b  x y   log b x  log b y                    a. ln  x y   ln x  ln y
                      x                                                    x
             b. log b    log b x  log b y                        b. ln    ln x  ln y
                       y                                                   y
             c. log b  x  k  k log b x                            c. ln  x  k  k ln x

Properties of Natural Logarithms:
      If x and y  0, then
                                                      x
                  a. ln  x y   ln x  ln y   b. ln    ln x  ln y             c. ln  x  k  k ln x
                                                       y

The Natural log and e x :
      ln e x  x, for all x and e ln x  x, for x  0.

Change the base of a logarithm:
                log10 a ln a
      log b a         
                log10 b ln b


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Other Helpful Formulas for College Algebra:
                                            x, if x  0 
      Definition of Absolute Value: x                    
                                            x, if x  0
      Absolute Value Equations and Inequalities:
          a. ax  b  c  c  0  is equivalent to : ax  b  c or ax  b   c
                    b. ax  b  c  c  0  is equivalent to :  c  ax  b  c
                    c. ax  b  c  c  0  is equivalent to : ax  b  c or ax  b   c

             Cube of a Binomial:
                x  y  3 x 3 3x 2 y  3x y 2 y 3                       x  y  3 x 3 3x 2 y  3x y 2 y 3
             Rational Function:
                                                                                P  x
                   A rational function is one of the form f  x  
                                                                                Q  x
                   where P  x  and Q  x  are polynomials and Q  x   0.

             Factorization Formulas:
                The Difference of Two Squares                    A2  B 2  ( A  B)( A  B)
                The Sum of Two Squares                           A2  B 2  prime
                The Difference of Two Cubes                      A3  B3  ( A  B)( A2  AB  B 2 )
                The Sum of Two Cubes                             A3  B3  ( A  B)( A2  AB  B 2 )

                    Trinomial Squares – The Square of a Binomial
                       A2  2 AB  B 2  ( A  B)( A  B)  ( A  B) 2
                       A2  2 AB  B 2  ( A  B)( A  B)  ( A  B)2

             Vertical Asymptotes:
               If Q  a   0, but P  a   0, then the graph of the rational function
                                P  x
                  f  x                  has a vertical asymptote at x  a.
                                Q  x

             Horizontal Asymptotes:
                                  P  x
                Suppose f  x          is a rational function where the deg ree of P  x  is m
                                  Q  x
                   and the deg ree of Q  x  is n.
                           a ) If m  n, the graph of f has a horizontal asymptote at y  0.
                                                                                               a
                           b ) If m  n, the graph of f has a horizontal asymptoteat y  ,
                                                                                               b
                               where a is the lead coefficient of P  x  and b is the lead coefficient of Q  x  .
                           c ) If m  n, the graph of f does not have a horizontal asymptote.
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             Linear Systems of Equations:
                Inconsistent – the system has NO SOLUTIONS (Contradiction)
                Dependent – the system has INFINITELY OR MANY SOLUTIONS (Identity)
                Consistent and Independent – the system has ONE SOLUTION (Conditional)

Linear Regression Analysis:
       Scatterplot:
              STAT, select 1. EDIT (enter in list 1 and list 2)
              Go to Y= and press enter on STATPLOT #1 to turn ON.
              WINDOW (set viewing window) or press Zoom #9
              GRAPH

             Find the Best Line Fit and Linear Regression Line:
                    STAT CALC #4 (finds regression eq.) and press enter once on the screen

             To paste your answer onto Y= and graph line on scatterplot:
                    Go to Y1 = make sure is blank
                    VARS select #5, arrow to EQ, select #1 (pastes eq. in Y1)
                    GRAPH (graphs plot and line)
                    CALC #1 (evaluates for an input)




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