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:
a b  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 
2a
2
The Discriminant: b  4ac
Students need to memorize “the nature of the solutions” as discussed in class.
2
Complex Numbers:
 1  i or i 2   1
Vertex of a parabola:
b
xv 
and yv  f
2a
b
 b 
.
  . Re call the LINE of SYMMERTRY is x 
2a
 2a 
 b  b    b 4ac  b2 
Also, the vertex is :  h ,k    xv , yv   
, f    
,
.
4a 
 2a    2a
 2a
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 
Pr oduct :
Difference :
f
 g  x   f  x   g  x 
f  x
 f 
Quotient :    x  
where g  x   0
g  x
g
 f  g  x   f  x   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:
Suppose the inverse of f is a function, denoted by f
1
f
 y   x if
1
. Then
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
f
1
f   x   x for every x in the domain of f , and
f 1   x   x for every x in the domain of f 1.
--------------------------------------------------------------------------------------------Linear Equation Formulas:
Standard or General Form: Ax + By = C
f  x 2   f  x 1  f b   f  a  f  x  h   f  x 
y y2  y1
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
 a mn , a  0
n
a
1. a m  a n  a m  n
2.
3.  a m   a m  n
4.  ab   a m b m
n
m
m
am
a
5.    m , b  0
b
b
6.
m
a m bn
7.  n  m , a  0, b  0
b
a
n

a

1
n
m
 

m
a
b
8.     
b
a
10. a 1 n  n a , n is an int eger n  2.
9. a 0  1, a  0
11. a m
1
 am , a  0
m
a
a 
m
1 n

n
am
Exponential Function:
f  x   b x , 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 ent
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 b0  1
ln1  0, because e0  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
b. log b    log b x  log b y
 y
c. log b  x  k  k log b x
x
b. ln    ln x  ln y
 y
c. ln  x  k  k ln x
Properties of Natural Logarithms:
If x and y  0, then
a. ln  x y   ln x  ln y
x
b. ln    ln x  ln y
 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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c. ln  x  k  k ln x
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:
A rational function is one of the form f  x  
P  x
Q  x
where P  x  and Q  x  are polynomials and Q  x   0.
Factorization Formulas:
The Difference of Two Squares
The Sum of Two Squares
The Difference of Two Cubes
The Sum of Two Cubes
A2  B 2  ( A  B)( A  B)
A2  B 2  prime
A3  B3  ( A  B)( A2  AB  B 2 )
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
f  x 
P  x
Q  x
has a vertical asymptote at x  a.
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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