Download Algebra 1 Key Concepts

Algebra Key Concepts
Quiz 1
a letter used to represent an unknown number
to rewrite an expression in its simplest form [solve]
to replace variables with numbers and then
simplify
The ____________ says that an expression may be
replaced by another expression that has the same
value.
words that mean addition
words that mean subtraction
words that mean multiplication
words that mean division
When translating “less than”
translate:
a number is six less than twice another number
When translating __________, __________,
and __________ you probably use ( ).
ex. translate: twice the sum of a and b.
In a word problem the verb (usually “is”)
represents _______
The order of operations used to simplify an
expression is _______________
Represents two things that are equal to one another
[problem with an = sign]
An equation with one or more variables
Any value of a variable that turns an open sentence
into a true statement [solution to an equation]
One or more terms connected by plus or minus
sign. [problem with out an = sign]
(Ex. 3 + a, 4y - z)
The given set of numbers that a variable may
represent. [input values]
Written with the symbol ______
The set of corresponding positive and negative
numbers and zero
(Ex. …, -2, -1, 0, 1, 2, …)
The entire collection of integers and
positive and negative fractions
Numbers that cannot be expressed as the
ratio of two integers
The set of rational and irrational numbers
The representation of real numbers as points on a
line
variable
simplify
evaluate
substitution principle
sum, plus, and, increased, more than
difference, minus, decreased, less than, remainder
product, times, of, by
quotient, divided, ratio, parts of
reverse the order
x = 2y – 6
“the sum of __ and __”, “the quantity”,
“which is”
2(a+b)
=
G – grouping (), [], 1+2
E – exponents
3
M – multiplication
D – division
A – addition
S – subtraction
Equation
Open Sentence
Root
Expression
Domain
Є
Integers
Rational numbers
Irrational numbers
Real numbers
Number line (or number scale)
The distance between a number and zero on the
number line
Symbol used to represent the absolute value of a
number, n
If one number is greater than another
The value of a number
The absolute value is
Quiz 2
Commutative Property
Associative Property
________________ sometimes makes adding or
multiplying groups of numbers much easier.
ex. 4*17*25*10 = ______
Distributive Property
We use the distributive property for two reasons:
Use the distributive property to multiply 3*6.3
Use the distributive property to solve
75*17 + 25*17
If equals are +, -, *, / to equals
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.
What happens when you divide a number by zero?
(Ex. 5/0, y/0, or 3/x if x = 0)
Expressions that are equal to the same quantity are
To add numbers with the same sign
To add numbers with different signs
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 negative times a negative =
If you multiply an even number of negatives the
answer will be _________
If you multiply an odd number of negatives the
answer will be _________
Absolute value
|n|
Then it is higher or further to the right on a number
line
a number’s distance and direction from zero
Absolutely positive!
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)
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)
Associative property
17,000
a(b + c) = ab + ac (for all real numbers a,b,c)
1. when we get stuck simplifying with GEMDAS
[to destroy parenthesis]
2. to simplify addition and multiplication.
3*6.3 = 3(6 + 0.3)
= 18 + 0.9 = 18.9
17(75 + 25)
17(100) = 1,700
The results are equal
Term
Undefined (meaningless)
Equal
add the numbers and keep the sign
subtract the numbers and keep the sign of the larger
number.
a, 0, -a
positive
negative
a positive
positive
negative
The reciprocal of –3/4 is ________
Any real number divided by itself is _____
Fill in the blanks:
a) –1 + ____ = 0
b) 2 + ____ = 0
c) –3/4 + ____ = 0
d) –1(____) = 1
e) 2(____) = 1
f) –3/4(____) = 1
dividing by 2 is the same as multiplying by _____
Rules for division:
If two numbers have the same (different) sign, their
quotient is __________ (___________)
Quiz 3
equation
operations that undo each other
To solve an equation for a specific variable we
need to _______________________________
Steps for solving equations
Identity
To solve membership problems, write an equation
with the cost of the 1st plan _________________
To solve cost, income, value problems we usually
need 2 equations:
Exponents and Polynomials
bn =
-4/3
1
a) 1
c) ¾
e) ½
½
b) –2
d) –1
f) –4/3
positive (negative)
a mathematical way to represent a balanced
system.
inverse operations
get the variable alone on one side of the equal sign.
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
An equation that is true for all values of the
variable [something = itself]
equal to the cost of the 2nd plan.
1) # (number of items)
2) $ (cost/value of items)
b*b*b*b … (b times itself “n” times)
(if b is any real number and n is any “+” integer)
The b is called the ____________
base
The n is called the ____________
exponent
3
(2y) = _______________
(2y)(2y)(2y)
2y3 = _______________
2*y*y*y
One of the numbers or letters multiplied together to factors
make up a term.
[Things multiplied together]
Any factor (or group of factors) is called the
_________ of the product of the remaining factors. coefficient
[The number in front of the variable(s)]
In the term –3xy2, -3 is the _____________ of xy2 numerical coefficient
An expression that is either a numeral, a variable,
monomial
or the product of a numeral and one or more
viariables.
[A polynomial with one term.]
A monomial that is just a numeral such as 14
constant
A sum of monomials
polynomial
ex. x2 – 4xy + y2 – 5
A polynomial with two terms
ex. 2x – 9 or 2ab + b2
A polynomial with three terms
ex. x2 – 4x – 5 or a2 + 3ab – 4b2
Terms that have the exact same variable parts.
You can only add _______________
__________ is the process of combining the like
terms in an expression. (Show by making _____)
The number of times that the variable occurs as a
factor in the monomial. (What is the exponent?)
degree of a monomial
degree of a polynomial
For the monomial 5x2yz4
what is the degree of x?
what is the degree of z?
what is the degree of the entire monomial?
To multiply powers of the same base:
keep the base and _____________________
am*an = __________
b2*b*2b6 = ______
To find a power of a power of a base:
keep the base and _____________________
(am)n = __________
To multiply polynomials by monomials we just
use ________________________
To multiply polynomials by polynomials we just
use ________________________
an equation used to express a rule in concise form
Distance = ______________________
It is customary to arrange polynomials with
Graphing Linear Equations
a set of coordinates that serve to locate a point on a
coordinate system
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 ____________
Special points at which a line cuts the axes.
How do you find the x intercept
binomial
trinomial
like terms
Simplifying
“Vs”
degree of a variable in a monomial
the sum of the degrees of its variables
(add the exponents of the variables)
the greatest of the degrees of its terms after it has
been simplified.
2
4
7 (because 0 + 2 + 1 + 4 = 7)
add the exponents
am+n
2b9
multiply the exponents
amn
the distributive property.
the distributive property more than once.Officially:
1. distribute each term of the 1st polynomial
2. Simplify (combine like terms)
formula
Rate * Time
1. the variables of a term in alphabetical order.
2. the terms in descending powers of one of the
variables that appears most frequently.
ordered pair
+, +
-, +
-, +, an identity
intercepts
set y = 0 and solve the resulting equation for x
How do you find the y intercept
Linear equations in standard form
The steepness of a line
slope =
When reading from left to right:
lines that go up have ___________ slope
lines that go down have ___________ slope
The steeper the line
If a line is straight
The slope of a horizontal line
The slope of a vertical line
Slope Intercept Form of a linear equation
set x = 0 and solve the resulting equation for y
ax + by = c
a, b, and c are integers.
slope
rise = vertical change
= y2 – y1
run
horizontal change
x2 – x 1
positive
negative
The greater the absolute value of the slope
Then its slope is constant
zero
no slope
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
1. find slope (m)
m = y2 – y1
slope intercept form are:
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 ________ the input
technically know as _________
the domain
In a function, the dependent variable is _________ the output
technically know as __________
the range
Systems of Linear Equations
Two or more equations in the same variable form
a system of linear equations
To solve a system of two equations with two
find all ordered pairs (x, y) that make both
variables, you must
equations true.
We have learned three methods for solving a
the graphing method, the substitution method, and
system of linear equations, they are
the addition or subtraction method.
The three possible solutions to a system of linear
1. point or ordered pair - the lines cross
equations.
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.
Factoring
Things added together.
Things multiplied together.
Canceling, any number divided by itself = ______
You may cancel any common ________ but not
common _______.
The largest shared factors (numeral and literal)
Undistributing
[what can each term be divided by?]
(expressing a number (or algebraic expression) as a
product of certain factors.)
A fraction bar is a _____________. When you get
stuck with GEMDAS you must ______________.
(x – 3) and (3 – x) are __________. To make them
factorable (the same) rewrite (3 – x) as _________
The first step of factoring is to ___________
To factor 4 terms
To factor 3 terms of the form ax2 + bx + c
To factor 2 terms
Zero Product Property:
If the product of factors = 0
ab = 0
Factoring is often used to _________ polynomial
equations by using these 3 steps
Solving Nonlinear Equations
Radical Rules:
√a*b = ___________, √a/b = ____________,
you can only add __________________
Simplifying Radicals:
( )2 and √ are ____________________
they ____________ each other.
To solve √ equations
Terms
Factors
One
factors, terms.
GCF (Greatest Common Factor)
Factoring
grouping
distribute
opposites
-1(x – 3)
take out the GCF if possible.
1. Consider factor by grouping.
2. Then factor the common polynomial factors
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
)
2
2
Is it A – B ?
(A + B)(A – B)
[the good times and the bad times]
2
2
but A + B is forever.
Then one or more of the factors = 0
a or b = 0.
solve
1. rewrite in standard form (ex. ax2 + bx + c = 0)
2. factor completely
3. set each factor = 0 and solve.
√a * √b,
√a/√b
like √
1. No perfect squares inside √
2. No fractions inside √
3. No √ in the denominator
inverse operations
undo each other.
1. get the √ alone
2. ( )2 both sides.
Now we can solve unfactorable (prime) quadratic
equations by ___________________________
Completing the square will solve ______ quadratic
equation using these 5 steps.
The quadratic formula =_____________________
We derive it by ____________________________
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 ___________.
The vertex has an x-coordinate, Vx, found by ____
that also tells us ___________________________
The vertex has a y-coordinate, Vy, found by _____
To graph quadratic functions by hand:
a,b,c effects on the graph:
a
b
c
The solutions of a quadratic equation are the _____
They can be found ___________ or ____________
by identifying the _______________
The discriminant, ________________,
tells us _________________________________
Be able to describe the graphs of:
y = x2
y = x3
y = x4
y = √x
Be able to describe the graphs of:
y = 2x
y = (½)x
y = |2x – 4|
y = 2/(x+1)
The exponential growth model
The exponential decay model
Rational Expression
Anything divided by 0 is ______________
Any x or y values that make the denominator of a
completing the square.
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
-b ± √ b2 – 4ac
2a
completing the square.
y = ax2 + bx + c, where a not = 0
a parabola.
vertex.
-b/2a
the axis of symmetry (x = -b/2a)
substituting x back into the function.
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.
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.
roots
algebraically, graphically
x-intercepts
b2 – 4ac
the number of solutions (+ = 2, 0 = 1, – = none)
Parabola
Bipolar parabola (½ happy, ½ sad)
Skinnier parabola
Parabola opening to the right
Exponential growth
Exponential decay
“V”
Reflection
FV = PV(1 + r)t
FV = PV(1 - r)t
Algebraic Fraction
undefined
rational expression 0 create _________________
The graph of the rational function will _________
but never ________ them.
Algebraic Fractions
If a fraction is in simplest form
asymptote lines.
get close
intersect
Then the numerator and denominator have no
common factor other than 1 and –1
To simplify an algebraic fraction
1. factor the numerator and denominator
(remember to look for opposites)
To multiply algebraic fractions
2. state restrictions (denominator can’t = 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 adjustment fraction
fraction by the same non zero number or ________ equivalent fraction
Then we get an ___________ in a _____________. 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.