Download f(x)

Algebra Review
Polynomial Manipulation
Combine like terms, multiply, FOIL, factor, etc.
Rational Expressions
To multiply rational expressions (fractions), multiply the
numerators, and multiply the denominators.
To divide rational expressions, invert the second expression
and then follow the rules for multiplication.
If possible, it may be helpful to factor the numerators and
denominators before multiplying.
To add or subtract rational expressions, find the least
common denominator, rewrite all terms with the LCD as
the new denominator, then combine like terms.
Rational Equations
To solve rational equations, multiply both sides of the
equation by the LCD of both sides of the equation, and
then solve.
Be sure to check your answers by substituting them back
into the original equation, in case your solution causes
the original expression to become undefined (a zero in
the denominator).
Properties of Exponents and Radicals
a 1
0
a
a a a
x
y
a 
y
x
x
x y
 a xy
a
a
   x
b
b
x
x
1
 x
a
ax
x y
a
y
a
ab  x  a x b x
a  b
x
 a x  bx
Properties of Exponents and Radicals
1
a a
n
a
m

n
2
 a
n
m
a
m
a a
1
n
n
a
a
 n
b
b
n
n
ab  a b
n
n
a b  ab
2
2
n
Equations Involving Radicals
If only one square root is present, isolate it on one side of
the equal sign, square both sides and solve.
If two square roots are present, put one on each side of
the equal sign, square both sides and solve.
When solving equations containing radicals,
extraneous solutions are often introduced, which
means you must check your answer in the original
equation.
Systems of Linear Equations
To solve, use either substitution or the additionsubtraction method.
Quadratic Equations
First put zero on one side of the equal sign and
everything else on the other side.
Then use reverse FOIL
(
)(
)=0
or, use the Quadratic Formula
Quadratic Formula
If
ax  bx  c  0
x
2
 b
b 2  4ac
2a
Inequalities
Use the same methods as in solving equations, with the
exception that if you multiply or divide both sides of the
inequality by a negative number, it reverses the order of
the inequality.
Absolute Value Equations
If
x  a,
xa
or
x  a
Absolute Value Inequalities
Two cases:
<
or
>
x  a 
x a

a  x  a
x  a
or
xa
Geometry
Areas
Rectangle:
A  bh
h
b
Triangle:
Circle:
A  1 bh
2
A r
h
b
2
r
Geometry
Perimeters
Rectangle:
P  2b  2h
h
b
Circle:
C  2 r
r
Geometry
Pythagorean Theorem
z
x
y
x y z
2
2
2
Geometry
Boxes
h
Surface Area:
A  2lw  2lh  2hw
Volume:
V  lwh
w
l
Geometry
Cylinders
r
h
Surface Area:
A  2 r 2  2 r h
Volume:
V   r 2h
Word Problems
Read the problem carefully.
Draw a picture if possible.
Set up a variable or variables, usually for the value
you’re asked to find.
Read again and write an equation.
Solve.
Equations of Lines
Standard Form:
ax  by  c
Slope Intercept Form: y  mx  b
where m is the slope and b is the y-intercept
Slopes of Lines
Given the points (x1, y1) and (x2, y2) on a line, its slope
y2  y1
is m 
x2  x1
Slopes of Lines
If the slope is positive the line is increasing:
If the slope is negative the line is decreasing:
Slopes of Lines
If the slope is zero the line is horizontal:
If the slope is undefined the line is vertical:
Slopes of Lines
If two lines are parallel their slopes are equal.
m1  m2
If two lines are perpendicular their slopes are negative
reciprocals.
1
m1  
m2
Graphing Lines
The x-intercept is the point where the line crosses
the x-axis, found by setting y = 0 and solving for x.
The y-intercept is the point where the line crosses
the y-axis, found by setting x = 0.
Distance and Midpoint Formulas
The distance d between the points (x1, y1) and (x2, y2) is
given by the distance formula:
•
d
 x2  x1  2   y2  y1  2
(x1, y1)
•
•
The coordinates of the point halfway between the points
(x1, y1) and (x2, y2) is given by the midpoint formula:
 x1  x2 y1  y2 
,


2 
 2
(x2, y2)
Conics
Parabolas
Equations of parabolas are quadratic in x or y
y  ax 2
x  ay
2
2


yk a xh
2
x  h  a y  k 
Vertex at (0, 0)
Vertex at (h, k)
Conics
Parabolas
If the x term is quadratic, (ax2), the parabola is vertical.
If a > 0, it opens up
If a < 0, it opens down
Conics
Parabolas
If the y term is quadratic, (ay2), the parabola is horizontal.
If a > 0, it opens right
If a < 0, it opens left
Conics
Circles and Ellipses
Standard form for an ellipse centered at origin:
b
2
2
x
y
 2 1
2
a
b
a
-a
-b
Conics
Circles and Ellipses
Standard form for a circle centered at origin with radius r:
r
x2  y2  r 2
-r
r
-r
Conics
Hyperbolas
Standard form for hyperbolas centered at origin:
x2 y2
 2 1
2
a
b
-a
a
2
b
2
y
x
 2 1
2
b
a
-b
Conics
If the curve is centered at (h,k), replace the x in the
equation with x-h and replace the y with y-k.
 x  h2   y  k 2
a
2
b
2
1
 x  h2   y  k 2  r 2
 x  h2   y  k 2
a
2
b
2
1
 y  k 2   x  h2
b
2
a
2
1
Functions
Domain and range
The domain of a function f(x) is the set of all
possible x values. (the input values)
The range of a function f(x) is the set of all
possible f(x) values. (the output values)
Functions
Notation and evaluating
If f(x) = 3x + 5,
to find f(2), substitute 2 in for x
f(2) = 3(2) + 5 = 11
f(a) = 3(a) + 5
f(joebob) = 3(joebob) + 5
Functions
Notation and evaluating
Note:
f  x  h  f  x   f h
Functions
Composition of functions
To find f[g(x)], substitute g(x) for x in the f(x) equation
Inverse of a function
The inverse of a function f  x  is denoted as f
1
x
The inverse of a function f(x) “undoes” what f(x) does.
(this means that f  f
1
 x   x and
f
1
 f  x   x )
Functions
Inverse of a function
The domain of f  x   the range of f
1
x
The range of f  x   the domain of f
1
x
(this means that the x and y values are reversed
on the graphs of a function and its inverse.)
Complex Numbers
The imaginary number i is defined as
i  1
so that
i 2  1
Complex numbers are in the form a + bi
where a is called the real part and bi is the imaginary part.
Complex Numbers
If a + bi is a complex number, its complex conjugate is a – bi.
To add or subtract complex numbers, add or subtract the
real parts and add or subtract the imaginary parts.
To multiply two complex numbers, use FOIL, taking
advantage of the fact that i 2  1 to simplify.
To divide two complex numbers, multiply top and
bottom by the complex conjugate of the bottom.
Complex Numbers
Complex solutions to the Quadratic Formula
When using the Quadratic Formula to solve a quadratic
equation, you may obtain a result like  4 , which you
should rewrite as  4  4  1  2i .
In general  a  a i if a is positive.
Polynomial Roots (zeros)
If f(x) is a polynomial of degree n, then f has
precisely n linear factors:
f  x   an  x  c1  x  c2  x  c3 ... x  cn 
where c1, c2, c3,… cn are complex numbers.
This means that c1, c2, c3,… cn are all roots of f(x), so
that f(c1) = f(c2) = f(c3) = … =f(cn) = 0
Note: some of these roots may be repeated.
Polynomial Roots (zeros)
For polynomial equations with real coefficients, any
complex roots will occur in conjugate pairs.
(If a + bi is a root, then a - bi is also a root)
Exponentials and Logarithms
Logs and exponentials are inverse functions, so if
bx  y
then
b0
logb y  x
Exponentials and Logarithms
Properties of logarithms
logb 1  0
logb  xy   logb x  logb y
 x
logb    logb x  logb y
 y
logb x p  p logb x
ln x  loge x
log x  log10 x
Exponentials and Logarithms
Equations
To solve a log equation, rewrite it as an exponential
equation, then solve.
To solve an equation involving exponentials, either put
into form b f  x   b g  x  , which gives f  x   g x  , and
then solve for x, or
take the log of both sides and use the properties of logs to
simplify, then solve.
Sequences and Series
Factorial Notation
If n is a positive number, n factorial is defined as
n !  1 2  3    n  1 n
with 0 !  1
For example,
4 !  4  3  2  1  24
Sequences and Series
An infinite sequence is a list of numbers in a particular
order.
The terms of a sequence are denoted as
a1 , a 2 , a3 , ...an , ...
Sequences and Series
Summation Notation
The sum of the first n terms of a sequence is written as
a1  a 2  a 3  ...  a n 
n
a
k 1
k
Sequences and Series
An infinite series is the sum of the numbers in an
infinite sequence.
a1  a 2  a 3  ...  a n  ... 

a
k 1
k
Sequences and Series
Arithmetic Sequences
A sequence is arithmetic if the difference between
consecutive terms is constant.
a2  a1  a3  a2  a4  a3  ...  d
d is the common difference of the series.
Sequences and Series
Geometric Sequences
A sequence is geometric if the ratio of consecutive terms is
constant.
a 2 a 3 a4


 ...  r
a1 a2 a3
r0
r is the common ratio of the series.
Sequences and Series
Geometric Series
The sum of the terms in an infinite geometric sequence is
called a geometric series.

a  ar  ar  ar  ...  ar  ...   ar k
2
3
n
k 0
If r  1 , the series has the sum
a
S
1 r
Matrices and Determinants
Matrices
An m  n matrix is a rectangular array of numbers with
m rows and n columns.
a11
A 
a 21
a12
a 22
a13 
a 23 
is a 2 3 matrix.
Matrices and Determinants
Matrices
Scalar multiplication of a matrix is performed by
multiplying each element of a matrix by the same
number (scalar).
5a11
5A  
5a 21
5a12
5a 22
5a13 
5a 23 
Matrices and Determinants
Matrices
Matrix addition and subtraction is performed by adding
or subtracting corresponding elements of the two
matrices.
 a11
A  B  a 21
a 31
a12   b11
a 22   b21
a 32  b31
b12   a11  b11
b22   a 21  b21
b32  a 31  b31
a12  b12 
a 22  b22 
a 32  b32 
(note: in order to add or subtract two matrices, they
must be the same size)
Matrices and Determinants
Determinant of a Square Matrix
a b 
The determinant of the 2 2 matrix A  
is

c d 
a b
det  A  A 
c d
a b

c d
 ad  bc