Download Integrated Algebra Regents

Integrated Algebra Regents- Reference Sheet
Properties of Real Numbers:
 Commutative Property of Addition: a + b = b + a
Commutative Property of Multiplication: ab = ba
 Associative Property of Addition: a + (b + c) = (a + b) + c
 Associative Property of Multiplication: (ab)c = a(bc) Distributive Property: a(b + c) = ab + ac
 Additive Identity: a + 0 = a Multiplicative Identity: a(1) = a

Additive Inverse: a + (-a) = 0 Multiplicative Inverse: a(
1
)=1
a
Exponent Rules:
 Addition/Subtraction: When combining like terms do not change exponents.
 Multiplication: Add the exponents of like variables. Division: subtract the exponents of like variables.
 Any base raised to a power of zero = 1. Ex: 20 = 1, x0 = 1
 Power to a Power: Multiply the exponents. Ex: (a2)3 = a6
 Negative Exponents: A negative exponent means to form the reciprocal and then perform the given
power. Ex: 4-2 =
1
1

2
4
16
Lines:





Sets,


Slope-Intercept Form: y = mx + b, m= slope, b = y-int , m 
y2  y 1
x2  x1
Parallel Lines have equal slopes and perpendicular lines have slopes that are negative reciprocals of each
other.
Writing the equation of a line: Find the slope, and then find the y-int by substituting one of the given
points into y = mx + b and solve for b.
Graphing an inequality on a set of axes: > or < use a dashed line, < or > use a solid line
If the inequality is > or > then shade up (above), if it’s < or < then shade down (below).
Graphing an inequality on a number line: use
if < or > and use  if < or >
Interval Notation, and Functions:
A  B (union) = all elements in both sets. Ex: A= {a, b, c}, B= {b, c, d}, A  B = {a, b, c, d}
A  B (intersection) = elements where sets overlap. Ex: A= {a, b, c}, B= {b, c, d}, A  B = {b, c}
A’ Complement (also can be A ) = elements not in the set. Ex: U= {a, b, c, d}, A = {a, d}, A’ = {b, c}
Interval Notation: ( means < or > [ means < or >
Ex: (1, 5] means 1 < x < 5
Ex: x > 5 means (5,  )
Ex: x < 5 means (-  , 5]
 Domain is the x-values of an ordered pair and the range is the y-values of an ordered pair.
Ex: {(1, 3), (5, 6), (3, 4)} Domain = {1, 3, 5} and Range = {3, 4, 6}
 A relation is a function if no x-values are repeated. The example above is a function.
 To see if a graph is a function use the vertical line test. If the vertical line passes through the graph
only once, then it is a function.
 Types of Func: Constant y = 2, Linear y = 2x, Exponential y = 2x, Quadratic y = x2 Abs Value y = |2x|
 Exponential Growth: A = P(1 + r)t, Exponential Decay: A = P(1- r)t, where A is the final amount, P is
the initial amount, r is the rate, and t is the time.
Geometry:
 Area Formulas: Rectangle = lw, Parallelogram and Rhombus = bh, Square = s2, Triangle = ½(bh),


Trapezoid =
1
h(b1  b2 ) , Circle =  r2
2



Perimeter: Add the dimensions of all of the sides of the given figure. For a circle use C=  d
Area of shaded region = Area of larger figure-Area of smaller figure
Number of sides to a polygon: pentagon-5, hexagon-6, heptagon-7, octagon-8, nonagon-9, decagon-10

Volume Formulas: Cube = s3, Rectangular Prism = lwh, Cylinder =


Surface Area Formulas: Rectangular Prism = 2lw + 2hw + 2lh, Cylinder = 2 r  2 rh
Pythagorean Theorem: a2 + b2 = c2 (only applies to a right triangle)
 r 2h
2
is( part )
%

of ( whole) 100
measured  actual
Relative Error:
actual
Percents:
Percent Inc/Dec =
original  new
x100
original
measured  actual
Percent Error:
x100
actual
Radicals:
 Simplifying: Find the largest perfect square that divides evenly into the radicand.
 Adding/Subtracting Radicals: Get a common radicand by simplifying each radical first and then combine
the coefficients and keep the common radicand.
 Multiplying/Dividing: Perform the given operation and then simplify.
Trigonometry: SOHCAHTOA
 Sine
= opp/hyp, Cosine
= adj/hyp, Tangent
= opp/adj (Set up correct ratio and cross multiply to
find a missing side. To find an angle, press the second key then the trig function and then the ratio.)
a
c
x
b
sin x = a/c
cos x = b/c
tan x = a/b
a x
c
b
sin x = b/c
cos x = a/c
tan x = b/a
x
x
x
Angle of Elevation: from
horizontal line of sight up
Angle of Depression: from
horizontal line of sight down
Quadratics and Algebraic Fractions:
x
st
 When factoring look, look for a GCF 1 ! Then D2PS and then a trinomial.
Ex: GCF: 2x – 8 = 2(x – 4)
D2PS: 9x2 – 25 = (3x – 5)(3x + 5)
TRI: x2 – 9x + 14 = (x – 7)(x – 2)
 When graphing a parabola, the axis of symmetry is the line that runs through the middle of the parabola
(in graphing calculator it’s the x-value of the turning point-it must be written as x = #), the roots are
where the graph crosses the x-axis (written in braces { }), and the turning point (vertex) is the point
where the parabola changes direction (point in middle of table of values-must be written in parentheses).
 A fraction is undefined when the denominator = 0.
 Simplifying Algebraic Fractions: Factor 1st then reduce.
 Multiplying Algebraic Fractions: Factor each numerator and denominator and then reduce. Do not reduce
horizontally.
 Dividing Algebraic Fractions: Multiply by the reciprocal of the 2nd fraction.
 Adding/Subtracting Fractions: Get the common denominator 1 st! Then combine the numerators and keep
the common denominator.
 Solving Fractional Equations: Get the common denominator of all fractions and then rewrite the
numerators and solve the resulting equation for the given variable.
Probablity and Stats:
 Complement is the probability of the event not occurring. P(A’) = 1 – P(A)
 P(A and B) = P(A)  P(B),
P(A or B) = P(A) + P(B) – P(A and B)
 Counting Principle: Multiply all possibilities.
 Permutations: order matters –to find how many ways to arrange letters in a word, don’t forget to
eliminate the duplicates. Ex: LILLY to arrange all 5 letters


5!
3!
Measures of Central Tendency: Mean-average, Median-middle (numbers have to be in order), Mode-most
often (there can be more than one mode or no mode)
To Draw a Box and Whisker you need 5 pieces of data: Min, Max, median (2 nd Q), 1stQ, and 3rd Q.
Ex: 1, 4, 5, 7, 8, 9
Min = 1, Max = 9, 1st Q = 4, 2nd Q =
57
= 6, 3rd Q = 8
2
Outliers: values that are far away from the rest of the data. Ex: 1, 57, 65, 80, 95
1 is the outlier.