Download Interval Notation (P

Linear Equations (1.2)
Solve linear equations by:
(1) removing parentheses
(2) combining like terms
(3) adding or subtracting the same
quantity from both sides
(4) multiplying or dividing the same
quantity from both sides
Solve linear equations with fractions by
first multiplying by the least common
denominator (LCD).
Identify an equation as one of the following:
(1) an identity, if every real number is a
solution
(2) conditional, if it has at least one real
solution
(3) inconsistent, if no real number is a
solution
Absolute Value Equations (1.6)
Solve absolute value equations by:
(1) isolating the absolute value expression
(2) solving the equation without the bars
(3) multiplying the right side by 1 and
solving the equation without the bars
(4) checking all solutions
Interval Notation (1.7)
Set-Builder
Interval
Graph
a  x b
(a , b )
a
()
b
a  x b
[a , b ]
a
[]
b
x a
(a,  )
a
(
x b
(, b ]
]

(,  )

b
Linear Inequalities (1.7)
Reverse the direction of the inequality
symbol when multiplying or dividing by
a negative number.
Absolute Value Inequalities (1.7)
Use the intersection symbol  (and) with
 or  , and use the union symbol  (or)
with  or  .
Models and Applications (1.3)
Solve a formula for a variable by:
(1) factoring the expression with the
variable, if necessary
(2) adding or subtracting the same
quantity from both sides
(3) multiplying or dividing the same
quantity from both sides
Complex Numbers (1.4)
The imaginary unit i is given by:
i   1 or i 2  1
A complex number is a number of the
form a  bi , where a and b are constants.
The complex conjugate of a  bi is
given by: a  bi .
Add or subtract by adding or subtracting
the real and imaginary parts, respectively.
Multiply using the distributive property
or the FOIL method, replacing i 2 with  1.
Divide by multiplying the numerator
and denominator by the complex
conjugate of the denominator.
Quadratic Equations (1.5)
Solve quadratic equations by factoring by:
(1) rewriting in the general form
(2) factoring completely using an
appropriate method
(3) using the zero product principle
(4) solving the resulting equations
Solve quadratic equations by the square
root property by:
(1) isolating a perfect square
(2) taking the square root of both sides
(3) solving the resulting equations
The Quadratic Formula (1.5)
Solve quadratic equations of the form
ax 2  bx  c  0 using the formula:
 b  b 2  4ac
x 
2a
The discriminant d is given by:
d  b 2  4ac
If d  0 , the equation has two unique
real solutions. If d  0 , the equation
has one real solution. If d  0 , the
equation has two imaginary solutions.
Polynomial Equations (1.6)
Solve polynomial equations by factoring
by the same steps used in solving
quadratic equations by factoring.
Radical Equations (1.6)
Solve radical equations by:
(1) isolating the radical expression
(2) squaring both sides
(3) solving the resulting equation
(4) checking all solutions
Rational Equations (1.2)
Solve rational equations by:
(1) multiplying all terms by the LCD
(2) solving the resulting equation
(3) excluding all solutions that result
in a zero denominator
Distance and Midpoint Formulas (2.8)
The distance d between the points
(x1, y1 ) and (x 2 , y 2 ) is given by:
d  (x 2  x1 )2  (y 2  y1 )2 .
The midpoint M of the line segment
joining the points (x1, y1 ) and (x 2 , y 2 )
is given by:
 x  x 2 y1  y 2 
M  1
,
.
2
2


Circles (2.8)
The standard form of the equation
of a circle with center (h , k ) and
radius r is given by:
(x  h )2  (y  k )2  r 2
The general form of the equation
of a circle is given by:
x 2  y 2  Dx  Ey  F  0
Find the general form of the equation
of a circle from its standard form by:
(1) completing the square for the x
and y variables, adding the same
number to both sides
(2) factoring the x and y variables into
perfect squares
Lines (2.3, 2.4)
The slope m of a line through the
points (x1, y1 ) and (x 2 , y 2 ) is given by:
m
y 2  y1 y rise


x 2  x1 x run
If a graph crosses the xaxis at the point
(a ,0) , then a is an xintercept.
If a graph crosses the yaxis at the point
(0, b ) , then b is a yintercept.
The point-slope form of a linear equation
passing through the point (x1, y1 ) with
slope m is given by:
y  y1  m(x  x1 )
The slope-intercept form of a linear
equation with slope m and yintercept b
is given by:
y  mx  b
The general form of a linear equation
is given by:
Ax  By  C  0
The slope of a vertical line is undefined,
and the slope of a horizontal line is zero.
The equation of a vertical line is x  a , and
the equation of a horizontal line is y  b .
Parallel lines have equal slopes: m1  m 2 ,
and perpendicular lines have slopes that
are negative reciprocals: m1   1 m 2 .
Function Basics (2.1)
A function is a relationship between
inputs and outputs, such that each
value of the input is assigned to only
one value of the output.
The set of all possible input values is
the domain, and the set of all possible
output values is the range.
If no vertical line intersects the graph
at more than one point, then y is a
function of x.
More Function Features (2.2)
A function is increasing if it rises from
left to right, decreasing if it falls from
left to right, and constant if it is level.
A function has a relative maximum if
the output value is higher than all other
nearby output values, and a relative
minimum if the output value is lower
than all other nearby output values.
A function is even if f (x )  f (x ) , and
odd if f (x )   f (x ) .
Evaluate piecewise functions using
only the piece that is appropriate for
the given input value.
Transformations of Functions (2.5)
Some common functions are:
(1) the constant function: f (x )  c
(2) the identity function: f (x )  x
(3) the absolute value function: f (x )  x
(4) the quadratic function: f (x )  x 2
(5) the square root function: f (x ) 
x
(6) the cubic function: f (x )  x 3
(7) the cube root function: f (x ) 
3
x
The function h (x )  f (x )  c is an
upward shift of f (x ) by c units, and
the function h (x )  f (x )  c is a
downward shift of f (x ) by c units.
The function h (x )  f (x  c ) is a
leftward shift of f (x ) by c units, and
the function h (x )  f (x  c ) is a
rightward shift of f (x ) by c units.
The function h (x )   f (x ) is a
reflection of f (x ) in the x-axis, and
the function h (x )  f (x ) is a
reflection of f (x ) in the y-axis.
The function h (x )  c  f (x ) is a
vertical distortion of f (x ) , and
the function h (x )  f (cx ) is a
horizontal distortion of f (x ) .
Combinations of Functions (2.6)
The sum of two functions is given by:
( f  g )(x )  f (x )  g (x ) .
The difference of two functions is given by:
( f  g )(x )  f (x )  g (x ) .
The product of two functions is given by:
( fg )(x )  f (x )  g (x ) .
The quotient of two functions is given by:
( f g )(x )  f (x ) g(x ) .
The composition of two functions is given by:
( f  g )(x )  f (g (x )) or (g  f )(x )  g ( f (x )) .
Inverses of Functions (2.7)
If f (g (x ))  x and g ( f (x ))  x for all x,
then g is the inverse function of f,
denoted by f 1.
A function f has an inverse function f 1
if no horizontal line intersects the graph
at more than one point.
Find inverse functions algebraically by:
(1) replacing f (x ) with y
(2) interchanging x and y
(3) solving the new equation for y
(4) replacing y with f
1
(x )
Find inverse functions graphically by
reflecting the graph in the line y  x .
Quadratic Functions (3.1)
The standard form of a quadratic
function with vertex (h , k ) and axis
of symmetry x  h is given by:
f (x )  a(x  h )2  k
The general form of a quadratic

2 
function with vertex   2ba , c  b4a 




and axis of symmetry x   b 2a is
given by:
f (x )  ax2  bx  c
Polynomial Functions (3.2)
The standard form of a polynomial
function is given by:
f (x )  an x n  an 1x n 1    a2x 2  a1x  a0 .
The leading coefficient is an and
the degree is n.
All polynomial functions have smooth,
continuous graphs.
For a polynomial function,
(1) if n is odd and an  0 , then the
graph rises right and falls left,
(2) if n is odd and an  0 , then the
graph rises left and falls right,
(3) if n is even and an  0 , then the
graph rises both right and left, and
(4) if n is even and an  0 , then the
graph falls both right and left.
If c is a zero of odd multiplicity, then the
graph touches the xaxis at (c ,0 ) . If c is
a zero of even multiplicity, then the graph
crosses the xaxis at (c ,0 ) .
Polynomial Division (3.3)
Use long division to divide a polynomial
f (x ) by a divisor d(x ) by:
(1) dividing the first term of f (x ) by the
first term of d(x ) and writing it as a
term of the quotient q (x )
(2) multiplying the term of q (x ) by every
term of d(x )
(3) subtracting the product from f (x )
(4) bringing down the next term of f (x )
(5) repeating the previous steps until the
remainder r (x ) cannot be divided
A polynomial function can be written as:
f (x )  d(x )  q(x)  r (x )
Use synthetic division to divide a polynomial
f (x ) by a divisor x  c by:
(1) writing c in a box and the coefficients
of f (x ) in a row to the right
(2) writing the leading coefficient of f (x )
in the bottom row
(3) multiplying c times the value on the
bottom row and writing it in the middle
row of the next column
(4) adding the top and middle row of the
next column and writing the sum in
the bottom row
(5) repeating steps 3 and 4 until all
columns are filled
(6) writing the quotient and remainder
using descending powers of x
If f (x ) is divided by x  c , then the
remainder is f (c ) . If f (c )  0 , then
x  c is a factor of f (x ) .
Zeros of Polynomial Functions (3.4)
If f (x )  an x n  an 1x n 1    a2x 2  a1x  a0
has integer coefficients, then all rational
zeros must be of the form p q , where p
is a factor of a0 and q is a factor of an.
If f (x ) is a polynomial function of
degree n, then f (x )  0 has exactly
n complex solutions.
The number of positive real zeros of f (x )
is the number of sign changes of f (x ) , or
is less than that by an even number.
The number of negative real zeros of f (x )
is the number of sign changes of f ( x ) , or
is less than that by an even number.
Rational Functions (3.5)
A rational function is given by:
f (x ) 
n (x )
,
d (x )
where n (x ) and d(x ) are polynomial
functions of degree n and d, and have
leading coefficients an and bd.
The vertical intercept is found by
setting x  0 and solving for f (x ) .
The horizontal intercepts are found by
setting the numerator equal to zero and
solving for x.
The line x  a is a vertical asymptote
if f (x )   as x  a .
The line y  b is a horizontal asymptote
if f (x )  b as x   .
The vertical asymptotes are found by
setting the denominator equal to zero
and solving for x.
The horizontal asymptote is:
(1) y  0 , if n  d
(2) y  an bd , if n  d
(3) nonexistent, if n  d
A rational function is even if the
numerator and denominator are
either both even or both odd.
A rational function is odd if the
numerator is even and the
denominator is odd, or vice versa.
Exponential Functions (4.1)
An exponential function is given by:
f (x )  b x , with b  0 .
The common base is 10, and the
natural base is e  2.718 .
For periodic compounding, the
account balance A is given by:
nt
r

A  P 1   ,
 n
where P is the principal, r is the
annual interest rate, n is the
number of compoundings per
year, and t is the number of years.
For continuous compounding, the
balance is given by: A  Pert .
Logarithmic Functions (5.2)
If x  b y , then y  log b x , and y is the
logarithm of x.
A logarithmic function is given by:
f (x )  logb x .
Negative exponents indicate reciprocals,
and fractional exponents indicate radicals.
The common log is written log x ,
and the natural log is written ln x .
Basic Properties of Logarithms (4.2)
1. log b b  1
2. log b 1  0
3. log b b x  x
4. b
log x
b
x
More Properties of Logarithms (4.3)
5. logb xy  logb x  logb y
x
6. log b    log b x  log b y
y 
7. logb x n  n  logb x
8. loga x 
logb x
logb a
Exponential and Logarithmic Equations (4.4)
Solve exponential and logarithmic equations
by one of the following methods:
(1) using logarithms to remove bases
(2) using bases to remove logarithms
(3) using properties of logarithms
Linear Systems (5.1)
Solve systems of linear equations by
substitution by:
(1) solving one of the equations for
one of the variables
(2) substituting that expression into
the other equation
(3) solving the resulting equation
(4) substituting the answer into the
first equation and solving
Solve systems of linear equations by
elimination by:
(1) rewriting both equations in the
form Ax  By  C
(2) multiplying all terms in one or
both equations by a constant
(3) adding the two equations together
to eliminate one of the variables
(4) solving the resulting equation
(5) substituting the answer into one of
the original equations and solving