Algebra

• Includes numbers we use for counting:
• 1,2,3,4,5,6,7,8,9,...

• Includes natural numbers together with 0:
• 0,1,2,3,4,5,6,7,8,9,...

• includes natural numbers, 0, and the negatives of the natural numbers:
• -5,-4,-3,-2,-1,0,1,2,3,4,5,...

• Are natural number greater than 1 that are divisible by 1 and themselves:
• 2,3,5,7,11,13,17,19,23

• Are natural numbers greater than 1 that are not prime:
• 4,6,8,9,10,12,14,15,16,18,20,21

• Are integers that are exactly divisible by 2:
• -8,-6,-4,-2,0,2,4,6,8,...

• Are integers that are not exactly divisible by 2:
• -9,-7,-5-3-1,1,3,5,7,9,...

• a a is an integer and b is a nonzero integer
• b

x x is a nonterminating, nonrepeating decimal

x x is a terminating decimal, a repeating decimal, or a nontermination nonrepeating decimal

• (-5,8]
• parenthesis indicates that endpoints are not included, brackets indicates that the endpoints are included

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)

is the distance on a number line between 0 and the point with the coordinate.

Result when two numbers are added

Result when one number is subtracted from another number

Result when two numbers are multiplied

• Result when two numbers are divided
• Division by 0 is undefined

The Sum of the values divided by the number of values

• The middle value
• if odd, choose the middle
• if even, find the mean of the middle two values

Value that occurs mors often

• Distance around a figure
• Square P=4s
• Rectangle P=2l+2w
• Triangle P= a+b+c
• Trapezoid P=a+b+c+d

• Distance around a circle
• C=pie * D
• C=2*pie*radius
• (pie is approximately)

• a+b = b+a
• ab = ba

(a+b) + c= a + (b+c)

a(b+c) = ab + ac

x^m xⁿ = x^m+n

• (x^m)ⁿ = x^m*n
• (xy)ⁿ = xⁿ*yⁿ
• (x/y)ⁿ= xⁿ/yⁿ

x⁰=1

• x^-n = 1/xⁿ
• and
• 1/x^-n = xⁿ

x^m/xⁿ = x^m-n

(x/y)^-n = (y/x)ⁿ = yⁿ/xⁿ

A statement indicating that two quantities are equal

Equations have exactly one solution

Equation that is satisfied by every number for which both sides of the the equation are defined. All real numbers

Equation that has no solution

angle whose measure is 90 degree

Angle whose measure is 180 degree

Angle whose measure is greater than 0 degree but less than 90

the sum of two angles equals 90 degrees

the sum of two angles equals 180 degrees

• 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

• When the graph of an equation is a line
• Ax + By = C
• A,B,C are constants
• x,y are variables

Point of a line where the line intersects the y-axis

Point of a line where the line intersects the x-axis

• 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)

the middle point of a line with ends at P(x1,y1) and Q(x2,y2) calculated:

(x1+x2/2 , y1=y2/2)

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

• 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

• 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

Two real numbers a and b if ab=-1

If two nonvertical lines are perpendicular, their slopes are negative reciprocals.

• the point slope equation of a line passing through P(x1,y1) and with the slope M is:
• y-y1=m(x-x1)

• The slope-intercept equation of a line with slope m and y-intercept (0,b) is:
• y=mx+b

• Ax+By=C
• slope = - a/b
• y-inetercept = (0, c/b)

Sets of ordered pairs

the set of all the first components in the relation

The set of all the second components in the relation

Is any set of ordered pairs (a relation) in which each first component determines exactly one second component

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.

The notation y= f(x) denotes that the variable 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.

the graph of the ordered pairs (x,f(x)) that define the function

• a function defined by an equation that can be written in the form:
• f(x)=mx+b
• or
• y=mx+b

• f(x)=x^2 (or y=x^2)
• parabola

f(x)=x^3 (or y=x^3)

• f(x)=lxl (or y=lxl)
• V shaped graph

• 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.

• 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

The graph of y=-f(x) is the graph of f(x) reflected about the x-axis

The process of finding the ordered pair that satisfies both equations in the system

When the system has a solution

When the system has no solution, the solution set is 0 (put line thru the zero)

When the system has infinately many solutions, 2 equations give the same line.

• 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

• 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

• 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