Download Integrated Algebra Review Sheet

1
Integrated Algebra Review Sheet
Merritt Brown, Andy Janovsky and Millie Rivera-Perez
This packet contains most of the information you will need to know in order to solve problems on the
Integrated Algebra Regents Exam
Equations and Inequalities
1. Evaluate expressions and formulas containing powers and parentheses and the four basic arithmetic
operations.
2. Solve linear equations and inequalities and represent solution sets graphically.
3. Solve quadratic equations both algebraically and graphically.
4. Solve systems of linear equations and inequalities algebraically and graphically.
5. Solve systems consisting of a linear equation and a quadratic equation algebraically and graphically.
1
2
Triangles Area  bh
Right
a 2  b2  c2
1 right angle
Perimeter
leg1  leg 2  hypotenuse 2
2
2
P= a + b + c
Quadrilaterals
Name
Properties
Parallelogram Two pairs of opposite sides
Two pairs of  opposite sides
Rectangle
Parallelogram with 4 right
angles
Rhombus
Parallelogram with 4  sides.
Square
Trapezoid
4  sides and 4 right angles
Exactly one pair of sides.
Isosceles
Trapezoid
Trapezoid with exactly one pair
of  sides.
Areas of irregular figures “ break them up”
Surface area of a rectangular prism= 2lw  2lh  2wh
Area and Perimeter
Formula
A=bh
A=lw
P = 2l + 2w
1
A  d1d 2
2
A  s2
P = 4s
1
A  h(b1  b2 )
2
Surface area of a cube= 6s 2
2
Surface area of a cylinder= 2 r  2 rh
2
Volume of a rectangular prism= length  width  height
Volume of a cylinder =  r 2 h
Volume of a sphere =
4 3
r
3
Regular Polygon – perimeter = sum of all sides
Properties
Additive Identity : a + 0 = a
Multiplicative Identity a 1  a
Additive Inverses (opposites): 8 + (-8) = 0
Multiplicative Inverses (reciprocals)
Commutative Addition: a + b = b + a
Commutative Multiplication ab=ba
Associative Addition (a + b) + c = a + (b + c)
Associative Multiplication (ab)c=a(bc)
a b
 1
b a
Distributive Multiplication over Addition: a( b + c) = ab + ac
Circles
C  2 R
C D
D  2R
A   R2
Proportion and Variation
If
a c

b d
then ad  bc The product of the means is equal to the product of the extremes.
If y varies directly as x then
x1 x2

y1 y2
and y  kx (k is a number ) . Line through (0,0).
longest side small
If two triangles are similar then
longest side big
shortest side small

shortest side big
Additional Facts
Scientific notation
8.56 104  85,600
8.56 104  .000856
Consecutive Integers: n, n+1, n+2
Consecutive Odd or Even integers: n, n+2, n+4
Two numbers in the ratio 3:5 can be represented as 3x and 5x where x is a number.
Coordinate Geometry
slope 
change in y
change in x
slope 
y2  y1
x2  x1
Parallel lines have the same slope. Example: y = 5x +6 and y=5x – 9.
2
Asmall ssmall
 2
Abig
sbig
3
2
3
If two lines are perpendicular, the product of their slopes is –1. Example: y  x  5 and y   x  7
3
2
6
 2  3 
        1 These are also called opposite reciprocals.
6
 3  2 
The slope of a horizontal line is zero. Example (y = 6)
The slope of a vertical line is undefined. Example (x=5)
The general equation for a straight line is y = mx + b where m is the slope and b is the y-intercept.
The equation of a line in point slope form is y  y1  m( x  x1 ) where m is the slope and  x1 , y1  are the given
coordinates of a point on the line.
Probability
A sample space is all possible outcomes for an event. A tree diagram is one way to list all events in the sample
space.
Biased means UNFAIR!
Theoretical probability is what should happen. Empirical probability is doing the experiment, recording results,
and assigning probabilities.
The counting principle states that if an event can occur x ways and a second event can occur y ways, then both
events can occur in that order xy ways.
Example: The school cafeteria has 4 types of entrees, 6 different drinks, and 3 choices of desserts. How many
different lunches consisting of 1 entrée, 1 drink, and 1 dessert are possible?
 4 63  72 different lunches
The probability of a certain event = 1. The probability of and impossible event is 0
The theoretical probability of an event E…… P( E ) 
number of outcomes in event E
number of outcomes in sample space
The probability of “NOT A” = 1  P  A
P(A or B) = P(A) + P(B)- P(A and B)
If events A and B are independent, then P(A and B)= P(A)P(B)
Examples: On a single roll of a die what is the probability of rolling
1
1 5
Not a six P(not 6)  1  
6
6 6
1 3 1
3
P  3 or odd     (because 3is odd ) 
6 6 6
6
A six P  6  
A 3 or and odd number
When calculating “without replacement” probabilities remember to decrease the denominator by 1
Example: If there are 8 red marbles and 2 two green marbles in a jar. Find the probability of drawing two red
 8  8  64
 8  7  56
marbles? With replacement P Two red      
Without replacement P Two red      
 10  10  100
 10  9  90
4
Permutations are arrangements where order is important, such as medals in a race use n Pr or
Examples: How many ways can 10 runners finish first second and third?
How many arrangements in the word AZALEA?
6! 6 letters 
3! 3 A ' s 
n!
P  720 ways
10 3
 120 arrangements
Statistics
Data can be classified as qualitative, how you feel about something or quantitative, numerical.
The measures of central tendency are the mean, mode, median, and range (both high and low)
The mean (average) of a set of numbers is the sum of the numbers divided by the number of numbers.
The median of a set of numbers is the middle number when the numbers are arranged in order. If the number of
numbers is even, average the two middle numbers.
To find the median term in a data set:
n 1
where n=total frequency, if the quotient is a “.5” then there are two
2
numbers that represent the median.
***If you are asked which is the best measure of central tendency it is the median because the mean can be
corrupted by outliers***
The upper quartile is the median of the scores above the median.
The lower quartile is the median of the scores below the median.
If your grade is the 90th percentile, then your grade is higher than the grades of 90% of the students in the class.
A percentile rank is found by multiplying the percent by the total frequency.
The mode of a set of numbers is the number that has the greatest frequency. A set of numbers can have more
than one mode.
In grouped data with interval length greater than 1 mean, median, and mode cannot be found because you do
not have the original data. You can find the modal interval, and the interval that contains the median.
A scatter plot determines correlation between two independent variables. Correlation may be positive, rising
line of best fit, negative falling line of best fit or no correlation at all, scattered points. Causation is determined
if one variable forces the other to behave that way.
Graphs:
Line graph, Circle graph, Frequency Histogram, Cumulative Frequency Histogram, Scatter Plot, Stem and Leaf
Plot, Box and Whisker Plot
Numbers
Natural numbers are the counting numbers 1, 2, 3, 4, 5…
Whole numbers are the natural numbers plus 0. 0, 1, 2, 3, 4, 5…
Integers consist of natural numbers, zero, and, negatives of natural numbers ... –4, -3, -2, -1, 0, 1, 2, 3 …
The absolute value of an integer is the magnitude of the integer (distance from 0). Examples:
5  5
6  6
 7  7 calculator:
  mathnum1 7 ENTER
5
Sample operations:

5   3  8
 6 5 




5   3  8

 6 5 

30


5   3  2
30

5   3  2
 6  5 



30
 6  5 



30
A prime numbers is a natural number greater than 1 that can only be divided by one and by itself.
A composite number is a number greater than 1 that is not prime. Composite numbers have more than two
factors. Example 6 has factors 1, 2, 3, and 6.
The greatest common factor (gcf) of two numbers is the largest number that divides both numbers.
Order of operations (parentheses, powers, divide and multiply left to right, subtract and add left to right.
Exponents and radicals
A rational number is a number that can be expressed as the ratio of two integers. When expressed in decimal
form rational numbers are either repeating or terminating decimals.
Examples:
4
5
6
7
8
9
5.4
.3333...
9
25
.3
49 .373737...  .37
An irrational number cannot be expressed as a ratio of integers.
Examples:

3
8
5
The set of real numbers consists of all rational and irrational numbers.
Multiply radicals:
 7  5  
35
3 7  4 5   12
35
 5  5   5
Dividing radicals:
18 35
6 7
3 5
Simplify radicals:
50  25 2  25 2  5 2
7 50  7 25 2  7 25 2  7 5 2  35 2
Like
3 56 5 9 5
10 7  7  9 7
Unlike
Combining radicals: 7 12  5 3  7 4 3  5 3 
7 4 3 5 3  7 2 3 5 3 
14 3  5 3 
19 3
6
Negative and Zero Exponents:
1 1
32  2 
3
9
50  1
5 ^ 0Enter
1
61 
6
3^ 2ENTER  Math1 Enter
3
 
7
1

7
3
6 ^ 1( ENTER)(math1) ENTER
3/ 7 ^ 1 ENTER  MATH1 ENTER 
When multiplying monomials, multiply coefficients and add exponents.
Example:
 5x  4 x   20 x
3
6
9
When multiplying polynomials, DISTRIBUTE and follow rules for multiplication above.
3x 2  2 x3  7  
3x  2 x   3x   7  
2
Monomial X Polynomial
3
2
6 x5  21x 2
Polynomial X Polynomial (foil)
 x  6  3x  1 
 x  3x    x  1   6  3x    6  1 
3 x 2  1x  18 x  6 
3 x 2  17 x  6
When raising a power to a power, multiply the exponents and raise the base.
Example:  4 x 2   (43 )  x 2   64 x 2 x3  64 x6
3
3
When dividing monomials, divide coefficients and subtract exponents.
Example:
12 x10
 3 x8
2
4x
When dividing a polynomial by monomial, divide each term in the polynomial
9 x3  15 x 2  3x 9 x3 15 x 2 3x

 3x 2  5 x  1
Example:
3x
3x 3x 3x
When adding or subtracting polynomials, combine only like terms and operate only on the coefficients.
Example:
x  3x  6
3x 2  2 x  9
4 x2  x  3
2
7
3a  5
6a 2  5a  1
6a 2  2a  4
Equations and Inequalities
First degree equations: Isolate the variable by using inverse operations
3x  4  14
2.5a  6  a  3
+4= +4
3 x 18

3
3
a  6  a  6
1.5a
9

1.5 1.5
x6
Literal equations:
3
3

x  9  21 4  x  9  21 3x  36  84
4
4

36  36
3 x 120

3
3
a6
Solve for x : ax  b c
b  b
ax c  b

a
a
c b
x
a
x  40
Solve forc :e  mc 2
e mc 2

m
m
e
 c2
m
e
 c2
m
e
c
m
First degree inequalities: Same as equations BUT if you are dividing by a negative sign changes
3x  5  26
5  5
3 x 21

(reverse)
3 3
x  7
 or  solid dot

 Graph of the solution
7
< or > hollow dot
Quadratics: All can be solved by factoring but ax 2  k is quicker by traditional solution.
3 x 2 192

3
3
x2  5x
x 2  24  5 x
x2  64
x  8
x2  5x  0
x 2  5 x  24  0
x  x  5  0
 x  8 x  3  0
x  8, 3
x  0,5
Answers are the roots of the equation, where the graph would cross the x-axis if the equation were graphed.
8
Systems of Equations
Linear
Solve for x and y 3x  2 y  5 2 x  4 y  18
Multiply to make opposites
2 x  4 y  18 = 2 x  4 y  18
Quad-Linear
Solve algebraically for x and y:
y  2 x 2  5x  5 and y  x  5
Substitute value of y into second equation
2 3x  2 y  5 = 6 x  4 y  10
8 x 8

8
8
x  1
2 x2  5x  5  x  5
2 x2  6 x  5  5
5  5
2
2x  6x  0
2
2
x  3x  0
x  x  3  0
x  0 and x  3
Find Y by substitution
3  1  2 y  5
3  2 y  5
+3
=+3
2y 8

2 2
y4
Solution x=-1 and y=4 or  1, 4 
Find Y’s by substitution
yx5
y0  5
y 3  5
y 5
y 8
Solutions  0,5 and  3,8
Exponential Growth and Decay:
Decay: y  ab x where a  0 and 0  b  1 A is the amount at beginning. B is rate of decay which is 100%
minus given rate and X is the time.
Example: Kathy plans to purchase a car that depreciates (loses value) at a rate of 14% per year. If the initial
cost of the car is $21,000, what is the value of the car after 3 years?
Let a= 21000 b= .86 ( 100% - 14%) and x= 3
v  21000 .86   21000  (.86) ^ 3  13357.176 So the car is worth $13,357.18 after 3 years.
3
Growth: The same formula except now B is found by adding the rate to 100%
Example: Cassandra bought an antique dresser for $500. If the value of her dresser increases 6% annually,
what will the value of the dresser be at the end of 3 years? Round your answer to the nearest dollar.
Let a= $500 b= 1.06 (100% + 6%) and x= 3years
3
v  500 1.06   500  1.06  ^ 3  595.508 So to the nearest dollar it is worth $596.00
Special Products and Factoring
Factoring Binomials: Greatest common factor (GCF): Find the largest number that divides into each
coefficient. Take the lowest power of any common variable terms. Divide each term by GCF. Write answer as
the product of GCF  Answertodivision 
Example: Factor 15 x 4  10 x 2 y GCF= 5x 2
15 x 4 10 x 2 y

 5 x 2  3x 2  2 y 
5x2
5x2
Difference of squares (DOS): This binomial does not have a common factor. It factors to two binomials
according to this rule: ( x2  y 2 )   x  y  x  y 
Examples: x  9   x  3 x  3
25x  49   5x  7 5x  7 
2
2
36a  b   6a  b  6a  b 
2
2
Factoring trinomials: Trinomials of the form x 2  bx  c can be factored by answering one question. What
two numbers multiply to “c” and add to “b”.
If “c” is positive signs on the factors are the same:
x2  7 x  12   x  4 x  3
x2  9 x  20   x  4 x  5
If “c” is negative signs on the factors are different, and sign on “b” tells you the sign of the “bigger” number.
a2  a  30   a  6 a  5
a2  3a  28   a  7  a  4 .
When factoring completely, first factor out the GCF then if a binomial remains factor a difference of squares.
If a trinomial remains factor the trinomial accordingly.
Examples: 3ax 2  48a  3a  x 2  16   3a  x  4  x  4  3x 2  15 x  42  3  x 2  5 x  14   3  x  7  x  2 
Vocabulary and Symbols
Words
Symbols
The sum of x and 5
x+5
x increased by 5
5 more than x
The difference between x and 5
x decreased by 5
5 less than x, 5 fewer than x
The product of 5 and x
Multiply x by 5
The quotient of 5 divided by x
5 is greater than x
5 is more than x
5 is less than x
5 is less than or equal to x
5 is at most x
5 is greater than or equal to x
5 is at least x
5 is not equal to x
x-5
5x
5
x
5>x
5<x
5 x
5 x
5 x
Formulas for Some Problems
distance=rate time
rate=
distance
time
time=
distance
rate
Percent change =
amountof change
x100
starting number
Discount  Cost (%)
Sale Pr ice  Cost  Discount
Total cost = unit price x quantity
sales tax = price x tax rate
9
10
Error in Measurement:
Re lativeError 
measure  actual
actual
Percent Error  Re lativeError x100
Trigonometry
tan( A) 
leg opposite A
leg adjacent to A
sin( A) 
leg opposite A
hypotenuse
cos( A) 
leg adjacent to A
hypotenuse
If you are given two sides and asked to calculate an angle, use  tan 1  ,  cos 1  or  sin 1 
Graphs of linear functions and inequalities
The general equation for a straight line is y = mx + b where m is the slope and b is the
y-intercept. When the equation is in general form, (y = mx + b), you can graph it by hand or enter the equation
into the  y   button on the calculator. Next press  table  for a list of x and y coordinates will appear. Choose
at least three points, graph draw the line and label.
Example: 4 x  2 y  8
-4x
= -4x
Slope= 2 up two, right one from the y intercept = (0,-4)
 2 y 4 x 8


or
2
2 2
Press  y   then 2  X , T , , n  - 4  enter  then  table 
Inequalities can graphed in the same manner with three exceptions:
y  2x  4



When dividing by a negative the inequality symbol reverses.
The graph of the line is dashed when < or > is being graphed.
The solution is a shaded region, if  or  shade under the line  or  shade over the line
Example y  2 x  6
+2x +2x
y  2x  6
Y intercept ( 0 , -6) and slope= 2 up two right one
draw a solid line and shade under. Mark shaded region solution set S
Graphs of non-linear functions
Parabola: y  ax 2  bx  c
If a>0 the parabola holds water MIN. If a<0, the parabola looks like the path of
a projectile MAX. If a is large the parabola is narrow. If a is small the parabola is wide. If b is increased the
parabola shifts b units to the left, decreased shifts b units to the right. Changes in c result in a change in the yintercept, so the parabola will move up if c is increased and down if c is decreased.
11
b
Equation of axis of symmetry: x 
Use this value of x as the middle value after you press  table  .
2a
To find the vertex algebraically substitute x into equation of parabola to find y.
To find vertex graphically examine table OR 2nd TRACE (3/ 4)
To determine the roots of a parabola examine the graph, see where it crosses the x axis. Roots are not ordered
pairs  x, y  they are a set of the X values where parabola crosses x-axis 1st x, 2 nd x
Absolute Value y   abs  graph has a “V” shape.
Exponential
y  ax
graph has an almost “half parabola” shape, NEVER touches x-axis
Additional Information
A function is a relationship where each “x” corresponds to only one “y”.
A relation is a set of ordered pairs ( x ,y)
The relation  3,7  2,5 1, 1 3, 5 is a function because each x has only one y.
The relation  3,9 2,6 5,11 2,8 is not a function because the 2 has two different y’s 6 and 8.
The graph of a function will intersect a vertical line once.
An algebraic expression is a math phrase made up of operation signs, numbers, and/or variables.
2a 2  7 a  3 is an expression.
An equation is a math sentence which shows that two expressions are equal. 3x  7  2x  21 is an equation.
Sets and set notation:  X /  2  X  3  is read the set of x’s which is greater than -2 and less than or equal
to 3. The set can be expressed in several ways { -1,0,1,2,3} is a list of the integral elements. It can also be
graphed on a number line with a hollow dot on -2 and a solid dot on 3. 1.2  X /  2  X  3  is read 1.2 is an
element of set of x’s which is greater than -2 and less than or equal 3 because 1.2 lies in that interval.
The answer can also be expressed in interval notation as (2,3] .
The universal set ( U )or domain is the set of all elements under consideration for a given example.
A B is the union of sets A and B, which is all the elements in both sets
A
B is the intersection of sets A and B which are the elements in that are in both sets.
A ' is the complement of set A, which are the elements of the universal set that are not in set A.
A set with no elements is called the empty { } or null  set. Use one or the other to express this.
Examples: U  1, 2,3...10 A  1,3, 4,7B  2, 4,6,7,8C  5,9
A
B  1, 2,3, 4,6,7,8 A
B  4,7 A C   B '  1,3,5,9,10
Venn diagrams are pictorial representations of union, intersection, and complements of sets.
Using units to solve problems:
Convert 30 miles per hour to feet per second.
30 miles 1 hour 1 min 5280 ft 30  5280 ft
ft




 44
hour
60 min . 60 sec
mile
3600 sec
sec
Algebraic Fractions
An algebraic fraction is the quotient of two expressions.
3a
7
x 2  16
x4
12
x  16
is undefined when x=4
x4
A fraction is in simplest form when the numerator and denominator have a GCF of 1. To simplify a fraction
divide both numerator and denominator by the GCF, polynomials require factoring.
2
A fraction is undefined when the denominator is equal to zero.
Simplify
3a 2 / 3a a
3a 2
GCF= 3a

12a / 3a 4
12a
Simplify
x 2  16  x  4  x  4   x  4 


  x  4
x4
1
 x  4
In order to add or subtract fractions you must have like denominators
Subtract:
6z 4z 2z


5
5
5
Add
( x  3)(3) (2 x  1)  4  3x  9 8 x  4 11x  5
x  3 2x 1





LCD =12
4
3
4(3)
3 4
12
12
12
When multiplying fractions, simplify first by factoring, “any top with any bottom”, then multiply horizontally.
16 x 14 y 2 2 2 y 4 y
Multiply

 

21y 8 x
3 1
3
Multiply
 x  4  x  4  
3x
3x
3x
x 2  16
 2


x  4 x  7 x  12
 x  4
 x  3 x  4  x  3
When dividing fractions multiply by the reciprocal of the divisor, “KEEP, CHANGE, FLIP”
Divide:
 x  5 x  1  6a  x  1
30a 3
25a 2
30a 3
x2  6 x  5
30a 3





2
2
2
2
x  25 x  6 x  5 x  25
25a
25a 2
5  x  5
 x  5  x  5 
Calculator Suggestions (TI-84)
1. Use the (-) key for negative numbers and put the negative number in parentheses.
Example:
If x= -3 evaluate 5 x 2  4 x  8
5   3  4   3  8 = 41
2
2. Use  MATH1 to change decimals to fraction and  MATH 2 for vice versa.
Add
1 2
9
  1/ 2    2 / 5   .9 math1 enter  
2 5
10
3. Use (^) for powers greater than 2. Example : 53  5 ^ 3
4. Use  MATH  NUM1 for absolute value items and graphs
 MATH  PRB2 for permutations n Pr and  PRB4 for factorials n!
12!  12  math  PRB4 enter   479001600
6 P3  6  math  PRB 2  3 enter   120
5
Use
6
Before performing any trigonometry (SOHCAHTOA) check  MODE  and set  DEGREE 
Looking for side use  SIN  COS TAN  looking for angle use  Sin 1  Cos 1 Tan1 
7
Before graphing any function press STATPLOT  2nd y   and make sure PLOTS are OFF.