Alegbra 2
In systems of equations, how many solutions do parallel lines have?
0 solutions
Inconsistent
0 solutions
row 0 of pascals triangle
1
row 1 of Pascal's triangle
1 1
row 2 of Pascal's triangle
1 2 1
(7 + x ) + y = 7 + ( x + y )
Associative
Arithmetic Mean
Average of 2 numbers the difference between n-1 and n+1 divided by 2 and added to n-1
Standard form
Ax + By = C
Classify a polynomial w degree of X^0
Constant
6 + ( -6 ) = 0
Inverse property of addition
Classify a polynomial w degree of X^1
Linear
Slope formula
M = (y2 - y1) / (x2 - x1)
Absolute Value Graph what's your transformation for the 2nd "2" in 2|x+3|+2
Shift up vertically by 2
formula to find sum of geometric series
Sn = (a(1) (1-r^n) ) / (1-r) n = terms (a(n))
Which formula do you need to know for word problems
Sn = (a(1) (1-r^n) / (1-r)
Arithmetic Series Formula
Sn = (n/2) (a1 + an)
How do you solve an absolute value equation
Split into 2 inequalities Solve each equation Check each solution in the original equation
what is the sigma symbol used to indicate
Summation Notation indicates adding
If 5 = 4 + 1, then 4 + 1 = 5
Transitive property
Explicit Formula for Arithmetic Sequence
an = a1 + (n-1)(d)
Recursive Formula for Arithmetic Sequence
an = an-1 + d a1 =
How does a polynomial react w/ x-axis when there is a multiplicity of 2
bounce off
Consistent
dependent and independent
If the discriminant is positive,
it has two solutions
X ^m / n - power
m
next steps to solve 3^3x-4 = 7
multiply by log log 3^3x-4 = log 7 3x-4 log 3 = log 7 divide the logs solve for x
How do you find the number on top of sigma?
use the formula to side of sigma... if (x(n) - y)....add y to last term if (x(n) + y)....subtract y from last term then divide that by x
equation if y varies directly as x and inversely as z
y = kx / z
vertex
( h , k )
X ^ m / n in radical form
( n √x ) ^m
(√7x) ^3 in x ^m / n
((7x) ^1 / 2) ^3
domain for rational expressions
(- infinity , vertical asymptote) U ( vertical asymptote , vertical asymptote) U ( vertical asymptote , infinity )
y intercept is (x , 0) or (0 , x)
(0 , x)
√(7x)^3 in x ^m / n form
(7x) ^3 / 2
steps to put vertex to standard
1. 4 (x - 1)^2 + 5 2. 4 { (x-1)(x-1) } + 5 3. 4 (x^2 - 2x + 1) + 5 4. 4x^2 - 8x + 4 + 5 5. 4x^2 - 8x + 9
steps to put standard to vertex
1. identify a b & c 2. find x-coordinate of vertex (h=-b/2a) 3. plug value in where x is 4. y = k 5. plug in h value and k value into vertex form
Dependent
2 - infinite solutions
after X method.. to simplify
3x^2 + 4x - 6x - 8 | 3x^2 + 4x | | - 6x - 8 | GCF x ( 3x + 4 ) - 2 ( 3x + 4 ) ( x - 2 ) ( 3x + 4 )
index of ⁵√x
5
√5 • ⁵√5
5 ^1 / 2 • 5 ^1 / 5 5 ^5 / 10 • 5 ^2 / 10 5 ^7 / 10 10√5 ^7
√7x^3 in x ^m / n form
7 ^1 / 2 X ^3 / 2
sum of perfect cubes a ^3 + b ^3
(a + b)(a ^2 - ab + b^2)
difference of perfect squares a ^2 - b ^2
(a - b)(a + b)
difference of perfect cubes a ^3 - b ^3
(a - b)(a ^2 + ab + b ^2)
Exponential Formula for Geometric Sequences
(a)n = (a)1 • r^n-1
Recursive Formula for Geometric Sequences
(a)n = (a)n-1 • r
x intercept is (x , 0) or (0 , x)
(x , 0)
If systems of equations is linear, how many solutions?
1 solution
Independent
1 solution
if the graph is shaded below the line, what is the inequality symbol?
< , less than
if the graph is shaded above the line, what is the inequality symbol?
> , greater than
when you see "continuously" in a word problem, you should use what equation?
A =Pe^rt
Fundamental Theorem of Algebra
A polynomial will have as many roots as its degree
what is this fifth step of solving an absolute value equation, after you have made your t-table
Choose x-values, 2 above and 2 below the middle x - value
Classify a polynomial w degree of X^3
Cubic
( 15 x + 5 ) = 5 ( 3x + 1 )
Distributive
Relation
Domain number appears twice
How to do rational root theorem
First set up p / q chart Calc: put equation in regular screen use numbers in p / q 1 , STO> , X , if = 0 , a root use number that equaled zero and divide synthetic division use polynomial you found and plug into quadratic formula get solutions
What is the first step in solving an absolute value equation
Get the equation into the form of y =
(a + b)( 1 ) = (a + b)
Identity property of multiplication
what is the fourth step of solving an absolute value equation, after you have found the vertex
Make a t-table. Put the vertex in the middle
Absolute Value Graph what's your transformation for |x+3| in 2|x+3|+2
Move to the left 3 points
what is the first step in solving for b when A = 1/2h (a + b)
Multiply by 2 2A = h (a + b)
How do you find the next term in a Geometric Sequence?
Multiply the previous term by the common ratio (r)
Function
Only one number of each domain
what is the third step of solving an absolute value equation, after you have solved for x. what have you just found?
Plug the Middle x-value into the original equation to find y. The point (x,y) that you have just found is the vertex
Classify a polynomial w degree of X^2
Quadratic
Classify a polynomial w degree of X^4
Quartic
Classify a polynomial w degree of X^5
Quintic
( 3 + 4 ) + 7 = ( 3 + 4 ) + 7
Reflexive
Formula for infinite geometric and arithmetic series
S (infinity) = (a(1)) / (1-r)
what is the difference between a series and a sequence
Sequence - an ordered list, commas Series - sum of a list, plus signs
what is the second step of solving an absolute value equation, after you have got the equation into y =
Set what's inside the absolute value sign equal to zero. Solve for x.
when is the no horizontal asymptote?
X^3 +5 / x^2 - 1
can you multiply and divide radical expressions if the index is the same?
Yes
Simplify b = (2A - ha) / h
b = (2A/h) - a
Discriminate
b^2 - 4ac
logb a = x can also be written as
b^x = a
What are you finding in a Arithmetic Sequence?
d= common difference
how to write equation from graph
degree is one more number than how many turning points find roots factors from roots multiply factors together
what is the next step in solving for b when 2A = h (a + b)
distribute h 2A = ha + hb
what is the next step in solving for b when 2A - ha = hb
divide by h b = (2A - ha) / h
how to find horizontal asymptote
divide the highest degree number
quadratic parent function
f (x) = x^2
write polynomial with roots x = 3, multiplicity 3
f(x) = (x - 3)^3
vertex form
f(x)=a(x-h)^2+k
standard form for quadratics
f(x)=ax^2+bx+c
how to find the vertical asymptote
factor the denominator and set equal to zero
How do you find the arithmetic mean
find the average... subtract right number from left divide the difference by 2 add difference to left number
what is the sixth step of solving an absolute value equation, after you have chosen x-values
find the y values to complete the table
to find min or max
find the y-value of ( x , y ) (but put in x= )
steps to solve e^x/5 +4 = 7
get e^x/5 alone e^x/5 = 3 get rid of e ( • ln) (ln) (e^x/5) = (ln)3 x/5 = ln3 solve for x
steps to solve ln(x-1 / 3) = 4
get rid of ln by multiplying by e (e) ln(x-1 / 3) = 4 (e) 4 (e) becomes e^4 (x-1 / 3) = e^4 solve for x
to find zeros..
graph in calculator or.. set (x-5) = 0 solve for x
to find h for translating standard to vertex
h = -b / 2(a)
Rule of Signs
how ever many times the signs change, is how many possible solutions
what does the number on top of sigma mean?
how many terms are in the list... a(n)
How do you know if a series converges or diverges?
if r<1 converges if r>1 diverges if r=1 then neither
direct variation
if y varies directly as x y = kx
inverse variation
if y varies inversely as x k = xy
If the discriminant is 0,
it only has one solution.
steps to solve log 3 + log x = 4
log 3x = 4 can also be written as 3x = 10^4 solve for x
log3x = 4 can also be written as
log10 (3x) =4 (3x) = 10^4
X ^ m / n - root / index
n
do you need to solve | x - 3 | > -4 ?
no, all real numbers
do you need to solve | x - 3 | < -4 ?
no, it's no solution
Rational Root Theorem
p / q p = factors of constant q = factors of leading coefficient
What are you finding in a Geometric Sequence?
r= common ratio
Solve by completing the square 3x^2 - 12x + 6
rewrite in x^2 + bx = c divide so a = 1 Add ( b / 2 )^2 to both sides Factor ( ( b / 2 ) ) Square root Plus or Minus!!! Split into two and solve
when you're given a number, and you're finding what nth it is.. how do you find the nth
set the number that you're given equal to your explicit formula and solve for n
Rational Root Theorem w no calc
set up p / q chart rule of signs to possibly eliminate use synthetic division and go through (p / q) list is remainder is zero, you have a factor set polynomial you found = 0 or quadratic formula solutions
to find y intercept
set x equal to 0 (x=0)
to find x intercept
set y equal to 0 (y=0)
Geometric Mean
square root of n-1 and n+1
How does a polynomial react w/ x-axis when there is a multiplicity of 1
straight through
what is the next step in solving for b when 2A = ha + hb
subtract ha 2A - ha = hb
How does a polynomial react w/ x-axis when there is a multiplicity of 3
swerve
If the discriminant is negative,
the answer is no solution.
Arithmetic indicates what
the difference is the same between every number
Absolute Value Graph what's you're transformation for the first 2 in 2|x+3|+2
the the slope, slope of 2
After you have set what is in the | absolute value | sign to 0 and have solved, what is the number that you have come up with?
this is known as the middle x-value
When do you need the recursive formula
to find a term in the past
When do you need the exponential formula
to find the nth term in the future (....)
how do you find the formula to the side of sigma for arithmetic?
use explicit formula and simplify to (x(n) + y)
Remainder Theorem
when f(x) is divided by (x-a) the remainder is f(a) when you divide a polynomial by (x-3) then the number you get after you plug 3 into the polynomial is your remainder
radicand of ⁵√x
x
How do you find the Geometric mean
x , ________ , y \|(x)(y) Multiply left term and right term and and square it
Domain
x - value side to side
Quadratic Formula
x = [ -b ± √(b² - 4ac) ] /2a
axis of symmetry
x = h
End behavior of negative coefficient, even degree
x → ∞ y → -∞ x → -∞ y → -∞
End behavior of negative coefficient, odd degree
x → ∞ y → -∞ x → -∞ y → ∞
End behavior of positive coefficient, odd degree
x → ∞ y → ∞ x → -∞ y → -∞
End behavior of positive coefficient, even degree
x → ∞ y → ∞ x → -∞ y → ∞
is vertical asymptote x= or y=
x=
simplify x⁵ y^2 z^-6 / x^-3 y^5 z^2
x^8 / y^3 z^8
in a word problem for direct/inverse variation what variable is dependent and what variable is independent
y - dependent x - independent
Range
y - values up to down
Slope intercept form
y = mx + b
joint variation
y varies jointly with x and z y = kxz
Point slope form
y-y1=m(x-x1)
is horizontal asymptote x= or y=
y=