The
following is a list of information that each student must transfer (hand
written) to
flashcards. The front and back of the flashcards should each be
written in a different color.
The
front of each card should also have a small number in the upper right hand
corner
identifying
which chapter it belongs to.
Front |
Back |
Chapter 1 |
|
a letter used to
represent an unknown number |
variable |
to rewrite an
expression in its simplest form [solve] |
simplify |
to replace
variables with numbers and then simplify |
evaluate |
The ____________
says that an expression may be replaced by another expression that has the
same value. |
substitution
principle |
words that mean
addition |
sum, plus, and,
increased, more than |
words that mean
subtraction |
difference, minus,
decreased, less than, remainder |
words that mean
multiplication |
product, times,
of, by |
words that mean
division |
quotient, divided,
ratio, parts of |
When translating
less than |
reverse the order |
translate: a number is six
less than twice another number |
x = 2y 6 |
When translating
__________, __________, and __________ you
probably use ( ). ex.
translate: twice the sum of a and b. |
the sum of __ and
__, the quantity, which is 2(a+b) |
In a word problem
the verb (usually is) represents _______ |
= |
The order of
operations used to simplify an expression is _______________ |
G grouping (),
[], 1+2 E exponents 3 M multiplication D division A addition S subtraction |
Represents two
things that are equal to one another [problem with an = sign] |
Equation |
An equation with
one or more variables |
Open Sentence |
Any value of a
variable that turns an open sentence into a true statement [solution to an
equation] |
Root |
One or more terms
connected by plus or minus sign. [problem with out an = sign] (Ex. 3 + a, 4y -
z) |
Expression |
The given set of
numbers that a variable may represent. [input values] Written with the
symbol ______ |
Domain Є |
The set of
corresponding positive and negative numbers and zero (Ex.
, -2, -1, 0,
1, 2,
) |
Integers |
The entire
collection of integers and positive and
negative fractions |
Rational numbers |
Numbers that
cannot be expressed as the ratio of two integers |
Irrational numbers |
The set of
rational and irrational numbers |
Real numbers |
The representation
of real numbers as points on a line |
Number line (or number scale) |
The distance
between a number and zero on the number line |
Absolute value |
Symbol used to represent
the absolute value of a number, n |
|n| |
If one number is
greater than another |
Then it is higher
or further to the right on a number line |
The value of a
number |
a numbers
distance and direction from zero |
The absolute value
is |
Absolutely
positive! |
Chapter 2 |
|
Commutative
Property |
the order in which
you add or multiply real numbers does not affect the result. a + b = b + a ab = ba (for all real numbers a,b) |
Associative
Property |
if you are only
adding or multiplying real numbers the grouping of the numbers does not
affect the result (a + b) + c = a +
(b + c) and (ab)c = a(bc) (for all real numbers a,b,c) |
________________
sometimes makes adding or multiplying groups of numbers much easier. ex. 4*17*25*10 = ______ |
Associative
property 17,000 |
Distributive
Property |
a(b + c) = ab +
ac (for all real numbers a,b,c) |
We use the
distributive property for two reasons: |
1. when we get
stuck simplifying with GEMDAS [to destroy parenthesis] 2. to simplify
addition and multiplication. |
Use the
distributive property to multiply
3*6.3 |
3*6.3 = 3(6 + 0.3) =
18 + 0.9 = 18.9 |
Use the
distributive property to solve 75*17 + 25*17 |
17(75 + 25) 17(100) = 1,700 |
If equals are +, -, *, / to equals |
The results are equal |
Either a single number or letter or the product (or quotient) of several numbers or letters. [Things added together] ex. 7, 5ax,
2(a+b), 3yz/2. |
Term |
What happens when
you divide a number by zero? (Ex. 5/0, y/0, or
3/x if x = 0) |
Undefined (meaningless) |
Expressions that
are equal to the same quantity are |
Equal |
To add numbers
with the same sign |
add the numbers
and keep the sign |
To add numbers
with different signs |
subtract the
numbers and keep the sign of the larger number. |
Rules for
Multiplication: For any real
number a a*1 = _____, a*0 = _____, a(-1) = _____ If two numbers
have the same sign, their product is If two numbers
have different signs their product is |
a, 0,
-a positive negative |
A negative times a
negative = |
a positive |
If you multiply an
even number of negatives the answer will be _________ |
positive |
If you multiply an
odd number of negatives the answer will be _________ |
negative |
The reciprocal of
3/4 is ________ |
-4/3 |
Any real number
divided by itself is _____ |
1 |
Fill in the
blanks: a) 1 + ____ =
0 b) 2 + ____ = 0 c) 3/4 + ____ =
0 d) 1(____) = 1 e) 2(____) =
1 f) 3/4(____) = 1 |
a) 1 b) 2 c) Ύ d) 1 e) ½ f) 4/3 |
dividing by 2 is
the same as multiplying by _____ |
½ |
Rules for
division: If two numbers
have the same (different) sign, their quotient is __________ (___________) |
positive
(negative) |
Chapter 3 |
|
equation |
a mathematical way
to represent a balanced system. |
operations that
undo each other |
inverse operations |
To solve an
equation for a specific variable we need to _______________________________ |
get the variable
alone on one side of the equal sign. |
Steps for solving
equations |
1) Rewrite the
equation and simplify each side. 2) Write the
inverse operation needed on both sides to get the variable alone and draw a
line underneath. 3) Perform the
inverse operation for each side of the equation always keeping things lined
up. 4) Go back to step
2) as needed(reverse GEMDAS) 5) Check |
Identity |
An equation that
is true for all values of the variable [something = itself] |
To solve
membership problems, write an equation with the cost of the 1st
plan _________________ |
equal to the cost
of the 2nd plan. |
To solve cost,
income, value problems we usually need 2 equations: |
1) # (number of items) 2) $ (cost/value of items) |
Ch. 4 |
|
bn = The b is called
the ____________ The n is called
the ____________ |
b*b*b*b
(b times
itself n times) (if b is any real
number and n is any + integer) base exponent |
(2y)3 =
_______________ 2y3 =
_______________ |
(2y)(2y)(2y) 2*y*y*y |
One of the numbers or letters multiplied together to make up a term. [Things multiplied together] |
factors |
Any factor (or group of factors) is called the _________ of the product of the remaining factors. [The number in front of the
variable(s)] |
coefficient |
In the term 3xy2, -3 is the _____________ of xy2 |
numerical
coefficient |
An expression that
is either a numeral, a variable, or the product of a numeral and one or more
viariables. [A polynomial with
one term.] |
monomial |
A monomial that is
just a numeral such as 14 |
constant |
A sum of monomials ex. x2
4xy + y2 5 |
polynomial |
A polynomial with
two terms ex. 2x 9 or 2ab
+ b2 |
binomial |
A polynomial with
three terms ex. x2
4x 5 or a2 + 3ab 4b2 |
trinomial |
Terms that have
the exact same variable parts. You can only add
_______________ |
like terms |
__________ is the
process of combining the like terms in an expression. (Show by making _____) |
Simplifying Vs |
The number of
times that the variable occurs as a factor in the monomial. (What is the
exponent?) |
degree of a
variable in a monomial |
degree of a
monomial |
the sum of the
degrees of its variables (add the exponents
of the variables) |
degree of a polynomial |
the greatest of
the degrees of its terms after it has been simplified. |
For the monomial
5x2yz4 what is the degree
of x? what is the degree
of z? what is the degree
of the entire monomial? |
2 4 7 (because 0 + 2 + 1 + 4 = 7) |
To multiply powers
of the same base: keep the base and
_____________________ |
add the exponents |
am*an
= __________ |
am+n |
b2*b*2b6
= ______ |
2b9 |
To find a power of
a power of a base: keep the base and
_____________________ |
multiply the
exponents |
(am)n
= __________ |
amn |
To multiply
polynomials by monomials we just use
________________________ |
the distributive
property. |
To multiply
polynomials by polynomials we just use
________________________ |
the distributive
property more than once.Officially: 1. distribute each
term of the 1st polynomial 2. Simplify
(combine like terms) |
an equation used
to express a rule in concise form |
formula |
Distance =
______________________ |
Rate * Time |
It is customary to
arrange polynomials with |
1. the variables
of a term in alphabetical order. 2. the terms in
descending powers of one of the variables that appears most frequently. |
Ch. 8 |
|
a set of
coordinates that serve to locate a point on a coordinate system |
ordered pair |
Tell whether each
is + or In quadrant I x is _____ and y is ______ In quadrant
II x is _____ and y is ______ In quadrant
III x is _____ and y is ______ In quadrant
IV x is _____ and y is ______ |
+, + -, + -, - +, - |
If an ordered pair
is a solution to an equation it will produce ____________ |
an identity |
Special points at
which a line cuts the axes. |
intercepts |
How do you find
the x intercept |
set y = 0 and
solve the resulting equation for x |
How do you find
the y intercept |
set x = 0 and
solve the resulting equation for y |
Linear equations
in standard form |
ax + by = c a, b, and c are integers. |
The steepness of a
line |
slope |
slope = |
rise
= vertical change = y2 y1 run horizontal change x2 x1 |
When reading from
left to right: lines that go up
have ___________ slope lines that go down
have ___________ slope |
positive negative |
The steeper the
line |
The greater the
absolute value of the slope |
If a line is
straight |
Then its slope is
constant |
The slope of a
horizontal line |
zero |
The slope of a
vertical line |
no slope |
Slope Intercept
Form of a linear equation |
y = mx + b m = slope, b = y intercept (for all
real numbers m and b) |
If the lines have
the same slope |
Then they are
parallel. |
If the lines have
- reciprocal slope |
Then they are perpendicular. |
The three steps to
write the equation of a line in slope intercept form are: |
1. find slope
(m) m = y2 y1 x2
x1 2. find y
intercept (b) substitute x, y, and
m
into y = mx + b and
solve for b 3. write the
equation substitute m and b
into y
= mx + b |
In a function, the
independent variable is ________ technically know
as _________ |
the input the domain |
In a function, the
dependent variable is _________ technically know
as __________ |
the output the range |
Ch. 9 |
|
Two or more
equations in the same variable form |
a system of linear
equations |
To solve a system
of two equations with two variables, you must |
find all ordered
pairs (x, y) that make both equations true. |
We have learned
three methods for solving a system of linear equations, they are |
the graphing
method, the substitution method, and the addition or subtraction method. |
The three possible
solutions to a system of linear equations. |
1. point or
ordered pair - the lines cross 2. no solution
the lines are parallel 3. infinite
solutions the same line (equation) |
The steps of the
Graphing Method are |
1. Solve each
equation for y to get y = mx +b form. 2. Find b, the
y-intercept, on the graph. 3. Use m, the
slope, to graph the line. 4. Write the
solution 5. Check |
The steps of the
Substitution Method |
1. Solve one equation
for one of the variables. 2. Substitute this
expression into the other equation and solve for the other variable. 3. Substitute this
value into the equation in Step 1 to find the value of the first variable. 4. Check |
The steps of the
Addition-or-Subtraction Method |
1. Multiply one or
both equations to get the same or opposite coefficients for one of the
variables. 2. Add or Subtract
the equations to eliminate one variable. 3. Solve the
resulting equation for the remaining variable. 4. Substitute this
value into either original equation to find the value of the first variable. 5. Check. |
Ch. 5 |
|
Things added
together. |
Terms |
Things multiplied
together. |
Factors |
Canceling, any
number divided by itself = ______ |
One |
You may cancel any
common ________ but not common _______. |
factors, terms. |
The largest shared
factors (numeral and literal) |
GCF (Greatest
Common Factor) |
Undistributing [what can each
term be divided by?] (expressing a
number (or algebraic expression) as a product of certain
factors.) |
Factoring |
A fraction bar is
a _____________. When you get stuck with GEMDAS
you must ______________. |
grouping distribute |
(x 3) and (3
x) are __________. To make them
factorable (the same) rewrite (3 x) as _________ |
opposites -1(x 3) |
The first step of
factoring is to ___________ |
take out the GCF
if possible. |
To factor 4 terms |
1. Consider factor
by grouping. 2. Then factor the
common polynomial factors |
To factor 3 terms
of the form ax2 + bx + c |
1. Find 2 factors
of a*c that add to b. 2. Use these
factors to rewrite the trinomial with 2 x terms (4 terms total)* then follow
the steps for 4 terms * if a = 1,
then only 1 step (x )(x ) |
To factor 2 terms |
Is it A2
B2 ? (A + B)(A
B) [the good times and the
bad times] but A2
+ B2 is forever. |
Zero Product
Property: If the product of
factors = 0 ab = 0 |
Then one or more
of the factors = 0 a or b = 0. |
Factoring is often
used to _________ polynomial equations by using
these 3 steps |
solve 1. rewrite in
standard form (ex. ax2 + bx + c = 0) 2. factor
completely 3. set each factor
= 0 and solve. |
Ch.11 & 12
|
|
Radical Rules: √a*b = ___________, √a/b = ____________, you
can only add __________________ |
√a * √b, √a/√b like √ |
Simplifying
Radicals: |
1. No perfect squares inside √ 2. No fractions inside √ 3. No √ in
the denominator |
( )2 and √ are ____________________ they ____________ each other. |
inverse operations undo each other. |
To solve √ equations |
1. get the √ alone 2. ( )2 both sides. |
Now we can solve unfactorable (prime) quadratic equations by ___________________________ |
completing the square. |
Completing the
square will solve ______ quadratic equation using
these 5 steps. |
1. Get into x2 + bx = c form. 2. Add (b/2)2 to both sides. 3. Factor into the perfect square, (x + b/2)2 4. √ both sides 5. Solve for x |
The quadratic
formula =_____________________ We derive it by
____________________________ |
-b ± √ b2 4ac 2a completing the
square. |
Graphing Non-Linear Equations |
|
A quadratic function is one that can be written in the standard form _____________________. When graphed it forms ______________. with a turning point called the ___________. |
y = ax2
+ bx + c, where a
not = 0 a parabola. vertex. |
The vertex has an x-coordinate, Vx, found by ____ that also tells us ___________________________ The vertex has a y-coordinate, Vy, found by _____ |
-b/2a the axis of
symmetry (x = -b/2a) substituting x
back into the function. |
To graph quadratic functions by hand: |
1. Get it in
standard form. 2. Find Vx 3. Make a table of
x & y values, using x-values to the left and right of Vx 4. Plot the
ordered pairs and connect with a smooth curve. |
a,b,c effects on the graph: a b c |
Smile if positive
and frown if negative. |Big number| if
skinny and |small number| if wide. Moves the vertex
side to side and up and down. Moves the vertex
up and down only. |
The solutions of a quadratic equation are the _____ They can be found ___________ or ____________ by identifying the _______________ |
roots algebraically,
graphically x-intercepts |
The discriminant, ________________, tells us _________________________________ |
b2
4ac the number of
solutions (+ = 2, 0 = 1, = none) |
Be able to describe the graphs of: y = x2 y = x3 y = x4 y = √x |
Parabola Bipolar parabola
(½ happy, ½ sad) Skinnier parabola Parabola opening
to the right |
Be able to describe the graphs of: y = 2x y = (½)x y = |2x 4| y = 2/(x+1) |
Exponential growth Exponential decay V Reflection |
The exponential growth model |
FV = PV(1 + r)t |
The exponential decay model |
FV = PV(1 - r)t |
Rational Expression |
Algebraic Fraction |
Anything divided by 0 is ______________ |
undefined |
Any x or y values that make the denominator of a rational expression 0 create _________________ The graph of the rational function will _________ but never ________ them. |
asymptote lines. get close intersect |
Ch.6 |
|
If a fraction is in simplest form |
Then the numerator and denominator have no common factor other than 1 and 1 |
To simplify an algebraic fraction To multiply algebraic fractions |
1. factor the numerator and denominator (remember to look for opposites) 2. state restrictions (denominator cant = 0) (set each factor in denominator = 0 and solve) 3. cancel any common factors |
To multiply algebraic fractions |
Use the same rules as simplifying. |
To divide algebraic fractions |
Multiply by its reciprocal. |
To find the GCF |
1. find the prime factorization (factor tree) 2. circle and multiply the common factors |
To find the LCM |
multiply the GCF by all the leftovers in the factor trees. |
If we multiply the numerator and denominator of a fraction by the same non zero number or ________ Then we get an ___________ in a _____________. |
adjustment fraction equivalent fraction different form |
To + or - fractions with common denominators: |
1. + or the numerators 2. keep the common denominator. 3. simplify. |
To + or - fractions with different denominators: |
1. rewrite with factored denominators and space for adjustment fractions 2. find the LCD (LCM of the denominators) and write it off to the side. 3. write in the adjustment fractions needed to produce the LCD. 4. follow the rules for + or - fractions with common denominators. |
Ch.7 |
|
A ratio is just a ____________ |
fraction (of 2
quantities measured in the same units) |
I say the ratio is 3:2, you say ________ |
3x, 2x |
A proportion is just ___________ |
equal ratios. (We use them to solve problems containing
four terms that compare similar objects or situations.) |
In a proportion the 1st and 4th terms are the ______ and the 2nd and 3rd terms are the ______. |
extremes means |
To solve a proportion ______________ |
cross multiply (product of the
means = product of the extremes) |
To eliminate fractions from an equation |
Multiply
everything by the LCD |
% increase (decrease) = |
1. amount of
increase (decrease) original amount 2. covert the
decimal to a percent. |
discount (markup/sales tax) = new price = |
original price *
rate of discount (markup/sales tax) original price
discount (or + markup/sales tax) |
commission earned = |
sales * commission
rate |
profit = |
revenues -
expenses |
The equation for direct variation |
y = kx (k is a non-zero constant) |
The equation for inverse variation |
y = k/x (k is a non-zero constant) (x not = 0) |
The law of levers |
(weight)(distance) = (weight)(distance) Left Right |
Mixture problems are just like _______ problems. We need to write 2 equations, 1 for the _________ and 1 for the ___________ |
coin amount value |
Work problems are set up using ______________ and solved with the fact that _________________ |
Rate of work * Time = work Done work done by A + work done by B = 1 job. |
The rate of work is expressed as a ____________ |
reciprocal. |
(a/b)m = |
am/bm (for every positive integer m) |
A nonzero number to the zero power is ____ |
One |
Negative exponents mean ______________ Ex. a-n = ____ |
take the reciprocal 1/an |
To divide powers of the same base: keep the base and _____________________ am/an = __________ |
subtract the exponents am-n |
To find a power of a quotient __________ (a/b)m = __________ |
Find the power of the numerator and denominator then divide. [distribute the power (top & bottom)] am/bm |
Scientific Notation is used to |
write very big or small numbers. |
Numbers in Scientific Notation have the form |
x.xx * 10n one
digit before the decimal, no extra zeros, 10 to nth power |
The exponent, n, tells you |
how many places to move the decimal point |
With Scientific Notation the exponent n is __________ for large numbers n is __________ for small numbers |
positive negative |
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|
Ch.10 |
|
When solving linear inequalities if you multiply or divide by a negative number you must ________ |
Reverse the inequality symbol |
To solve compound inequalities |
solve two separate problems |
and means or means |
overlap all |
To solve absolute value equations and inequalities: |
Get the absolute value alone then Solve two separate problems, a positive and negative version. |
To graph linear inequalities in two variables |
Use slope-intercept form. Use a dashed line if there is no = |
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