includes natural numbers, 0, and the negatives of the natural numbers:
-5,-4,-3,-2,-1,0,1,2,3,4,5,...
Prime numbers
Are natural number greater than 1 that are divisible by 1 and themselves:
2,3,5,7,11,13,17,19,23
Composite numbers
Are natural numbers greater than 1 that are not prime:
4,6,8,9,10,12,14,15,16,18,20,21
Even integers
Are integers that are exactly divisible by 2:
-8,-6,-4,-2,0,2,4,6,8,...
Odd integers
Are integers that are not exactly divisible by 2:
-9,-7,-5-3-1,1,3,5,7,9,...
Rational numbers
a a is an integer and b is a nonzero integer
b
Irrational numbers
x x is a nonterminating, nonrepeating decimal
Real numbers
x x is a terminating decimal, a repeating decimal, or a nontermination nonrepeating decimal
Interval notation
(-5,8]
parenthesis indicates that endpoints are not included, brackets indicates that the endpoints are included
Compound Inequalities
Expressions that involve more than one inequality and involves the word (or,and) so it can be written in interval notation with the union symbol (U, upside down for and)
Absolute value
is the distance on a number line between 0 and the point with the coordinate.
Sum
Result when two numbers are added
Difference
Result when one number is subtracted from another number
Product
Result when two numbers are multiplied
Quotient
Result when two numbers are divided
Division by 0 is undefined
Mean
The Sum of the values divided by the number of values
Median
The middle value
if odd, choose the middle
if even, find the mean of the middle two values
Mode
Value that occurs mors often
Perimeter
Distance around a figure
Square P=4s
Rectangle P=2l+2w
Triangle P= a+b+c
Trapezoid P=a+b+c+d
Circumfrence
Distance around a circle
C=pie * D
C=2*pie*radius
(pie is approximately)
Communitive Property
Multiplication and Addition
a+b = b+a
ab = ba
Associative Property
Multiplication and Addition
(a+b) + c= a + (b+c)
Distributive Property
Multiplication over addition
a(b+c) = ab + ac
The product Rule of exponents
x^{m} x^{n} = x^{m+n}
Power Rules of Exponents
(x^{m})^{n} = x^{m*n}
(xy)^{n} = x^{n}*y^{n}
(x/y)^{n}= x^{n}/y^{n}
Zero exponents
x^{0}=1
Negative Exponents
x^{-n} = 1/x^{n}
and
1/x^{-n} = x^{n}
Quotient Rule
x^{m}/x^{n} = x^{m-n}
Fractions to Negative Powers
(x/y)^{-n} = (y/x)^{n} = y^{n}/x^{n}
Equation
A statement indicating that two quantities are equal
Conditonal equations
Equations have exactly one solution
Identiy
Equation that is satisfied by every number for which both sides of the the equation are defined. All real numbers
Contradiction
Equation that has no solution
Right angle
angle whose measure is 90 degree
Straight angle
Angle whose measure is 180 degree
Acute anle
Angle whose measure is greater than 0 degree but less than 90
Complementary angles
the sum of two angles equals 90 degrees
Supplementary anles
the sum of two angles equals 180 degrees
Coordinate System
x-axis-horizontal line
y-axis-vertical line
origin-point where axis cross
coordinate plane-two axis form this
quadrants-coordinate plane divide into four regions
x-coordinate-point on x axis
y-coordinate-point on y axis
ordered pair-when the order of the coordinate is important
Linear equation
general form
When the graph of an equation is a line
Ax + By = C
A,B,C are constants
x,y are variables
Y-intercept
Point of a line where the line intersects the y-axis
X-intercept
Point of a line where the line intersects the x-axis
Horizonatal and Vertical lines
the graph of x=a is a vertical line with x-intercept at (a,0)
the graph of y=b is a horizoneal line with y intercept at (0,b)
Midpoint
the middle point of a line with ends at P(x1,y1) and Q(x2,y2) calculated:
(x1+x2/2 , y1=y2/2)
Slope of the line
Constant rate of change of line passing through points (x1,y1) and (x2,y2) calculated:
m= Change in y/Change in x
m= y2-y1/ x2-x1
m+ rise/run
Slopes of Horizontal and Vertical lines
all horizontal lines (lines with equations of the form y=b) have a slope of O
all vertical lines (lines with equations of the form x=a) have no defined slope
Slope of parallel lines
Nonvertical parallel lines have the same slope, and lines having the same slope are parallel
Since vertical lines are parallel, lines with no defined slop are parallel
Negative reciprocals
Two real numbers a and b if ab=-1
Slopes of perpendicular lines
If two nonvertical lines are perpendicular, their slopes are negative reciprocals.
Point slope form
the point slope equation of a line passing through P(x1,y1) and with the slope M is:
y-y1=m(x-x1)
Slope-Intercept Form
The slope-intercept equation of a line with slope m and y-intercept (0,b) is:
y=mx+b
Slope and Y-intercept from the general form
Ax+By=C
slope = - a/b
y-inetercept = (0, c/b)
Relations
Sets of ordered pairs
Domain of the relation
the set of all the first components in the relation
Range of the relation
The set of all the second components in the relation
Function
Is any set of ordered pairs (a relation) in which each first component determines exactly one second component
Vertical line test
Determines whether the graph of an equation represents a function. If every vertical line that intersects a graph does so exactly once, the graph represents a function.
Function Notation
The notation y= f(x) denotes that the variable is a function of x.
y is a Function of x
An equation, table, or graph in x and y in which each value of x (the input) determines exactly one value of y (the output) is a function of x.
Graph of a function
the graph of the ordered pairs (x,f(x)) that define the function
Linear function
a function defined by an equation that can be written in the form:
f(x)=mx+b
or
y=mx+b
Squaring function
f(x)=x^2 (or y=x^2)
parabola
Cubing function
f(x)=x^3 (or y=x^3)
Absolute value function
f(x)=lxl (or y=lxl)
V shaped graph
Horizontal Translations
If f is a fuction and k is a positive number, then:
the graphe of f(x-k) is identical to the graph of f(x), except that it is translated k units to the right
The graph of f(x+k) is identical to the graph of f(x), except that it is translated k units to the left.
Vertical translations
If f is a function f and k is a positive number, then:
The graph of f(x)+k is identical to the graph of f(x), except that it is translated k units upward
The graph of f(x)-k is identical to the graph f(x), except that it is translated k units downward
Reflection of a Graph
The graph of y=-f(x) is the graph of f(x) reflected about the x-axis
Solving the system
The process of finding the ordered pair that satisfies both equations in the system
Consistent system
When the system has a solution
Inconsistent system
When the system has no solution, the solution set is 0 (put line thru the zero)
Dependent system
When the system has infinately many solutions, 2 equations give the same line.
Substitution method
1. solve one equation for one of its variables
2. substitute the result from Step 1 into the other equation and solve
3. find the value of the other variable by substittuting the value from step 2 into one of the original equations.
4. State the solution
Addition Medthod
1. write both equations in general form
2. if necessary, multiply the terms of one or both equations to cake the coefficients on one of the variables differ only in sign
3. Add the equation and solve.
4. substitute the value in step 3 into either of the original equations and solve
5. state the solution
Solving a system of three linear equations in three variables
1. pick any two equations and elimate a variable
2. pick a different pair and elinate the same variable
3. solve the resulting pair of two equations in two variables
4. find the value of the third variable, substitute the values from step three into one of the original equations with three variables and solve the equation