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Properties
Commutative: Change Order
Reflexive: a=a (yes!)
Associative: Regroup-move ( )
Symmetric: If a=b, then b=a
Distributive: multiply through ( )
Transitive: If a=b and b=c then a=c
Identities: 0 (Add) & 1 (Multiply)
Substitution: replacing one term for two
Inverse: Opposites (add #s to 0 ) & Reciprocals (multiply #s to 1)
Algebra II SOL
Study Sheet
Quadratics
Factoring
Standard : ax2+bx+c
ax2+bx+c
Standard Form:
Put in y= see where the graph crosses the
x axis
2nd calc, Zero, left, right, enter
Roots
Zeros
Solutions
Factors (inverse #s of zeros)
Monomials
Coefficient, Base, Exponent Ex:
3x2
Rules of Exponents
1.
an am =an+m
Multiply: Add
exponents
2.
an /am =an-m Divide: Subtract
exponents
3.
(an)m =an*m
Power of Power:
Multiply exponents
4.
a0 =1
Anything to the 0
a-n =1/an
If an exponent is
power is 1
5.
negative upstairs then it will be positive
in the downstairs.
1. Find a GCF (If there is one)
1. 2. Find a number that multiplies to give you c but adds to give you b.
*c +b
Write in ( x+ ) ( x+ )
Example of Case #1
x 2  16 x  64
= (x-8)(x-8)
Example of Case #2 (you can use slide & divide)
3x 2  x  2
(3x -2)(x+1)
2
Difference of Squares: x  4  ( x  2)( x  2)
Sum of Cubes: a3  b3  (a  b)(a 2  ab  b 2 ) Remember: Same Opposite Always Positive
Difference of Cubes: a3  b3  (a  b)(a 2  ab  b 2 ) SOAP
Solving Quadratic Equations
ax 2 bx  c  0
Remember: Roots = zeros = solutions = x-intercepts
1. Method 1: Solving by factoring
b 2  4ac >0
a. If Factorable, factor the polynomial
b. Set each factor equal to zero and then solve for x
b 2  4ac <0
2. Method 2: Quadratic formula if polynomial is not factorable
2 real roots
2 imaginary
roots
 b  b 2  4ac 2
1 real root
b 2  4ac =0
a. X =
, b  4ac is called the discriminate
(Duplicity)
2a
MATRICES
3. Method 3: Type quadratic equation in calculator y =
a. Graph the equation and where the parabola crosses the x axis is where the solution is
b. Using the table function:
i. 2nd table (2nd graph) : The solution is the x value when y is equal to 0
Absolute Value Equations and Inequalities
|x+2|=7
(x + 2) = 7 or
(x + 2) = - 7
x+2=7
x=5
x + 2 = -7
x = -9
x = -9
or
or
| 2x + 3 | > 6 Put is above 6 and below -6
2x + 3 < 6 [this is the pattern for "greater than"]
2x+3< -6 OR 2x+3 > +6
or x = 5
Remember: Greater - greatOR
| 2x + 3 | < 6 Put it between -6 and 6
–6 < 2x + 3 < 6 [this is the pattern for "less than"]
–6 – 3 < 2x + 3 – 3 < 6 – 3
–9 < 2x < 3
–9
/2 < x < 3/2
Then the solution to | 2x
Between -9/2 and 3/2
● Value is Included
○ Value is Not included
–9
/2 < x < 3/2.
Basic Graphing Concepts
Domain: set of all x values
Range: set of all y values
X intercepts: where the graph
crosses the x axis
Y intercepts: where the graph
crosses the y axis



X < -9/2
or
x > 3/2
Direct, Inverse, and Joint Variation Equations
Direct Variation: y = kx, where k is a nonzero constant of variation
k
Inverse Variation: y = or xy = k where k  0
x
Joint Variation: y = kxz
Radicals and Radical equations
3
g(f(x)) = g(2x+3) = 2x + 3 + 2 = 2x + 5
+ 3 | >6 is
x> 3/2 OR X < -9/2
Then the solution to | 2x + 3 | < 6 is the interval
Compositions of functions
f(x) = 2x + 3 g(x) = x + 2
g(f(x)) substitute value of f(x) everywhere there is
an x in the g function
>
8x 8 = 2 x 2  3 x 2
3
1) Take the 8 =2
2) Look at variable x - See how
many times 3 goes into 8 and
how much is left over
a

b
x x
3
5
power
root
x  5 x3
Solving Radical Equations
1) Isolate the radical
2) Square both sides
3) Solve for x
4) Check answer
2 x  3  17
2 x  14
( 2 x ) 2  142
2 x  196
x  98
Arithmetic and Geometric Sequences
1) Arithmetic sequences: Add a common
difference to each term to obtain the next
term
Formulas:
an  an 1  d
an  a1  (n  1)d
Remember a n is the nth term
a1 is the first term
n is the number of the terms
d is the common difference
Arithmetic and Geometric Series
2) Geometric Sequences: Multiply the common
ratio to each term to obtain the next term
Formulas:
an  an 1  r
an  a1r n 1
Remember
a n is the nth term
a1 is the first term
n is the number of the terms
r is the common ratio
REMEMBER SERIES = SUM OF TERMS IN
THE SEQUENCE
1 is the lower limit and 6 is the upper limit
Calculator steps:
1)2nd stat  Math  5 sum(enter)
2) 2nd Stat  OPS  5 seq(enter)
3) ( 2^n , x , 1 , 6)
On calculator screen:
Sum(seq(2^x, x , 1 , 6) = 126
Sum of the first six numbers is
126
Math, # 0, Summation, enter values
Scatter Plots, Curve of Best Fit and predictions
Scatter plots: graph of a real-life data
Curve of Best fit: model the data in a scatter plot
L1: Week L2: Weight in grams of radioactive material
Look at graph to check the shape of the graph
Choose a regression model that fits the shape.
Solve: A chemist has a 100-gram sample of radioactive material. He records the amount of
radioactive material every week for 6 weeks and obtains the data.
1. Read question to see what you are solving for and which data set should be x
2. We are solving for weight in grams
3. Turn on stat plots
4. STAT, EDIT,enter L1, L2
5. Stat  Calc  4 0: ExpReg
6. Xlist: L1
7. YList: L2 (skip Freqlist)
8. STORE Vars  y vars  #1  #1  enter
9. Graph by hitting zoom 9 for zoomstat
10. Check window
11. What about Week 10?
12. 2nd, table set, 10, enter,
13. Table screen, next to week 10, 26.961 grams
Combination or Permutation
STATS Algebra 2
Normal Distribution:
 = mean ( average)
 – standard deviation (how far above or below you are from the mean)
Permutation: # of ways you can select some items
out of a group total, order DOES matter
n!
n Pr 
(n  r )!
Example: Selecting a president, vice president, historian and
treasurer out of 15 members.
15
Z- score:
The z-score a particular data point is the measure of how many standard deviations
that point is away from the mean. You can calculate the z-score of any data point using
the data point formula.
x
z 
x  data point   mean   standard deviation

Ex:
The mean salary for an entry level engineer is $32,000 per year with a standard
deviation of $2,500. Find the z-score of an engineer who is making $35000.
35000  32000
z 
 1.2 (1.2 standard deviations from the mean)
2500
Z-score and Probability/percent
You can calculate the probability/percentage of an event occurring by using the z-score.
P4 
15!
(15  4)!
Math, PROB, # 2 nPr, enter values
Combination - # of ways you can select some items out of a group
total, order DOESN’T matter
n!
n Cr 
(n  r )! r !
Example: selecting a committee of 4 members out of
the total of 15 people where everybody has equal standing
15
C4 
15!
(15  4)!4!
Ex: In a large group of guayule plants, the heights of the plants are normally distributed
with the mean of 12 inches and standard deviation of 2 inches. Find the percent of the
plants are shorter than 16 inches.
z
16  12
 2.0
2
Now look at the z-score table to determine what the percentage is
0.9777 -- probability
97.77% -- percentage
Math, PROB, # 2 nCr,enter values