Download Freeman High A-Team 2013

Algebra I SOL Topics and Formulas
Equations, Inequalities, Expressions
One Step Equations Addition/Subtraction:
x  3  10 (Perform the inverse “opposite” operations on each side of the equations)
-3 -3
x= 7
x  3  10
**When you see addition, you subtract the value on both sides of the equal sign.
** When you see subtraction, you add the value on both sides of the equal sign.
+3 +3
x = -7
One Step Equations Multiplication/Division:
3x  15
3  3
**When you see a number next to the variable, it indicates multiplication. Therefore, you
are going to divide each side by the leading coefficient (the number in front of the variable)
x = -5
(4)
x
 3 (4)
4
** When you see a number in the denominator that indicates division. Therefore, you are going to
multiply each side by the value in the denominator.
X = -12
Multi-Step Equations:
1.
2.
Add or subtract using the inverse operations on both sides of the equal sign.
If you see a number in the denominator then multiple both sides of the equal sign by the value in the
denominator.
3. If you see a number next to a variable, divide each side of the equal sign by the coefficient.
**Notice that you may not need to perform every step in every problem.
2x  4  15
+4 + 4
2x = 19
2
2
x
19
2
3x
 6
5
3x
 6 (5)
(5)
5
3x  30
3
3
x  10
In calculator
Go to Menu
Equation
Solver
Type in the equation exactly how you see it ( include ( ) if needed)
Hit exe TWICE!!
Inequalities:
Use the same steps as solving equations. The only difference if when you divide or multiply by a negative,
you switch the symbols
2 x  14
2 x 14

 2 x 2
x  7
We divided by a
negative….flip
the sign!
Verbal Expressions
Order of Operations PEMDAS
Two less than a number
x-2
Remember these key words:
ADD
more than
increased by
total of
sum
SUBTRACT
less than (switch the numbers)
how much more?
difference
how much farther?
is – equals
is less than → <
is greater than → >
DIVIDE
shared equally
quotient
MULTIPLY
product
twice
of (like half of)
at least → ≥
no more than → ≤
at most → ≤
no less than → ≥
“times the sum of” or “times the difference of” – you need parentheses
So “Five less than three times the sum of a number and seven” means 3(n+7) - 5
Example: Omar wants to have a party. He will pay $300 for a DJ and $10 per person for food. If his bill was
$650, how many people are coming to the party?
Omar has to pay $300 for the DJ no matter what, that is the thing that doesn’t change. His cost for food changes
according to how many people he invites. $650 is his total, so this is the equation:
300 + 10x = 650
You can use solver to get x = 35 people.
SOL 3: Properties
Distributive Property: Multiply a number outside of a parenthesis to every term inside.
6x – 12 = 6(x – 2)
3(n – 7) = 3(n) + 3(-7)
a(x + b) = ax + ab
Identity Property of Multiplication – if you multiply any number by 1 you get the original number
2 x 1  2 x
Identity Property of Addition – if you add zero to any number you get the original number
2x  0  2x
Commutative Property :the numbers can change places (or order)
3+5=5+3
ab = ba
(ab) + 3 = (ba) + 3
(7a + 9 ) +5a = 5a + (9 + 7a)
There is no commutative property of division or subtraction
Reflexive - the exact same thing on both sides of the equal sign
a b  a b
Symmetric – uses the key words “if, then”
If a  b , then b  a
Associative Property: move the parentheses around , the numbers have to stay in exactly the same order.
3x + (5x + 9) = (3x + 5x) + 9
7(xy) = (7x) y
Again – there is no associative property of subtraction or division
Transitive Property of Equality – if two number both equal the same thing then they equal each other (it
always says if…and…then)
If a  b , and b  c , then a  c
Addition Property of Equality – if you add the same thing to both sides of an equation, the two sides are still equal)
Subtraction Property of Equality – if you subtract the same thing from both sides of an equation, the two sides are
still equal)
Multiplication Property of Equality – if you multiply both sides of an equation by the same thing, the two sides are
still equal)
Division Property of Equality – if you divide both sides of an equation by the same thing, the two sides are still
equal)
Inverse Property – any number plus its opposite = 0 ( 4 + (-4) = 0)
any number times its inverse =1 (2 * ½ = 1)
Function, Patterns
Function: x does not repeat (domain is all different numbers)
Graphs use vertical line test (can only cross at one point)
The coordinate plane is made up of an x-axis and an y-axis. To find a certain point on the coordinate plane, you need
an x and a y coordinate that are listed together like this (x, y). The x and y axes are always labeled on the SOL.
A relation is any set of ordered pairs. The domain is the set of all x values or first coordinates and the range is the
set of all y values or second coordinates. A relation is a function if all of the x-values are different. See examples
below.
EX1: {(3, -4), (3, 5), (4, 6)}
not a function – the x’s repeat
EX2: {(7,1), (8,1), (9,1)}
function
EX5:
EX3
1
2
3
EX6:
1
5
5
6
12
18
3
3
5
– Not a function,
EX4
4
4
10
4
5
6
2
5
Functions
8
9
10
Not a Function
D:{3,5}
The x values are repeating
R: {4, 10}
Not a function, the x is repeating
f(x)=abs (x+2)
x=3
9
y
9
y
x^2/9+y^2/25=1
9
8
8
8
7
7
7
6
6
6
5
5
5
4
4
4
3
3
2
2
1
-9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1
x
1 2 3 4 5 6 7 8 9
-9 -8 -7 -6 -5 -4 -3 -2 -1
-1
y
3
2
1
x
1 2 3 4 5 6 7 8 9
-9 -8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-2
-2
-3
-3
-3
-4
-4
-4
-5
-5
-5
-6
-6
-6
-7
-7
-7
-8
-8
-8
-9
-9
-9
function
not a function – use the vertical line test
D = { all real numbers}
R = { all real numbers greater than 0}
x
1 2 3 4 5 6 7 8 9
not a function – use the vertical line test
Slope, X, Y, Intercepts
y=mx+b
m=slope
Ax + By = C standard form
Slope=
y1  y2
x1  x2
m=
b=y-intercept
A
B
Horizontal line (HOY) slope = 0 equation starts with y
Ex. y=4
Vertical line (VUX) slope = undefined equations starts with x
Ex. x=4
Slope, Graph of a line
Types of slope
Positive
y
x
Negative
Zero
y
y
x
Undefined
y
x
x
Graphing a Line
From slope intercept form (y=mx+b):
3 y  2 x  6
1. Get y by itself (add/subtract first if x and y are on both sides)
3
2. Divide everything by the number in front of y
y
m=
2
3
3 3
2
x2
3
b= -2
3.
Identify m (slope) and b (y-intercept)
4.
Start at b on your y-axis make a dot
5.
Now use your slope to make at least two more dots.
y
x
y
x
rise
run
6. We will go down 2 lines and over to the right 3 make
another point
7. Repeat step 6 from your new point.
y
8. Connect your points. Place arrows at the end of your line.
x
From Standard Form Ax  By  C (x and y intercepts):
3 x  4 y  12
3x  12
3 x 12

3
3
x4
4 y  12
4 y 12

4
4
y 3
1. Get x and y on the same side of the equal signs.
2. To solve for x, cover up the y and rewrite the
equation.
3. Solve for x
4. Your point on your graph will be at (4,0) or 4 on the
x-axis
5. To solve for y, cover up the x and rewrite the
equation.
6. Solve for y
7. Your point on your graph will be at (0,3) or 3 on the
y-axis
y
8.
Plot the values of x and y on your graph.
9.
Connect the dots to make a line, place arrows at the end.
x
y
x
Write Linear Equations
Y=mx+b slope intercept
m is your slope (
rise
)
run
b is the y-intercept
Ax+By=C Standard Form
A, B, and C cannot be fractions
A cannot be negative
The GCF between A,B, C has to be 1
From standard form to slope intercept form:
3 x  y  5
-3x
-3x
y  3x  5
1. Move x to the other side (add/subtract)
2. If y has a number in front of it divide everything by that number
3. Reduce all fractions
From slope (m) and y-intercept (b):
m=2
b=10
y=mx+b
y=2x+10
1. Start with y=mx+b
2. Plug in m
Plug in b
From a graph:
y-intercept (b)
b=2
m=
rise 4
=  4
run 1
Rise
Run
1.
Find b (where it crosses
the y-axis)
2.
Find m (slope) (from two
points start at the bottom point,
count up and over to the next
point.
3.
Start with y=mx+b
4.
Plug in m
5.
Plug in b
y  4 x  2
From slope (m) and a point:
m =3
( -4 , 2)
y  mx  b
2  3(4)  b
2  12  b
1. Start with y=mx+b
2. Plug in m, x, and y
3. Solve for b (add or subtract the number on the same side of b to both sides)
+12 +12
14= b
y  mx  b
y  3 x  14
From 2 points:
(-2, 2), (2, 10)
4. Now use y=mx+b
5. Plug in m and b
y2  y1 10  2 8

 2
x2  x1 2  2 4
y  mx  b
2  2( 2)  b
2  4  b
m
1. Find Slope of the two points
2. Use y=mx+b and pick ONE of your points
3. Plug in m, x, and y
4. Solve for b
+4 +4
6 = b
y  mx  b
y  2x  6
5. Use y=mx+b
6. Plug in m and b
Going from Slope Intercept to Standard Form of a line:
2
x4
3
2
2
 x  x
3
3
y
2
x y 4
3
1. Move x to the other side of the equation (add or subtract)
(x and y need to be on the same side of the equation, but they are not like terms so do NOT add/subtract them together.)
2
x  y  4)
3
2
3( x  y  4 )
3
2. A can’t be negative, so multiply everything by -1
-1(
3. A, B, and C can’t be fractions, if they are multiply everything
by the denominators.
2 x  3 y  12
To get slope from Standard Form:
m
A
B
3x  4 y  15
m
3 3

4 4
Systems of Equations: 2 equations, 2 variables, 2 answers
x  2y  6
Elimination
x  y  9
3 y  15
y5
x  5  9
**now plug y=5 into either equation  x  4
x  4
Substitution:
Solution: (-4,5)
x  y5
2x  7 y  7
2( y  5)  7 y  7
2 y  10  7 y  7
9 y  3
y
1
3
Solution: ( 4
2 1
,
)
3 3
Graphing- where 2 lines intersect is the solution (x,y)
If they do not cross then they are parallel, it is no solution. If you are using substitution or elimination you
will get different number on each side of the equal sign.
Ex. 4=5 (not true, therefore no solution)
If they are the same line, it is infinitely many solutions. If you are using substitution or elimination you will
get the same number on each side of the equal sign.
Ex. 6=6 (true, therefore infinitely many)
On Calculator: Set each formula in standard form Ax  By  C .
Menu
EQUA
F1: Simultaneous
F1: 2 unknowns (x and y)
Type in A, B, and C for each equation
F1: Solve
Monomials, Exponents, Scientific Notation
A. Multiply the coefficients first, then add exponents when multiplying polynomials
(2 x 2 y3 )(5xy 4 )  10 x3 y 7
B. Multiply the coefficients by itself the number of times the exponent indicates, then Multiply
Exponents when raising a polynomial to a power
(3x 2 y 3 )3  27 x6 y 9
C. Reduce (divide) the coefficients, then Subtract Exponents when dividing polynomials
3x 2 y 5 3x 21 y 57
x

 2
7
15 xy
15
5y
You cannot have a negative exponent. If you get a negative exponent, move the variable and
exponent to the other side and make the exponent positive
Ex:
x 8 z 3
z3
=
y2
x8 y 2
Scientific Notation 2.53 x 105 = 253,000
3.06 x 105 = 0.0000306
Polynomials
Add/Subtract like terms
(The same variables with the same exact exponents)
3xy  4 xy 2  2 x 4 y  12 xy 2  13x 4 y 
3xy  16 xy 2  15 x 4 y
Multiply/Distribute, FOIL (First Outer Inner Last)
( x  5)( x  8)  x 2  8x  5x  40  x 2  3x  40
Factoring Binomial/Trinomial Equations
Signs Trinomial= 2 binomials
x 2  7 x  12 = ( x  4)( x  3)
(+ +)=(+)(+)
(- +)= (-)(-)
x 2  7 x  12 = ( x  4)( x  3)
(+ -)=(+)(-)
x 2  x  12 = ( x  4)( x  3)
(- -)= (+)(-)
x 2  x  12 = ( x  4)( x  3)
Different of Squares
x 2  49  ( x  7)( x  7)
check : x 2  7 x  7 x  49  x 2  49
On calculator: Set equation equal to zero: Ax 2  Bx  C  0
Go to menu
Select EQUA
F2: Polynomial
F1: 2nd degree (highest exponent)
Type in the values for A,B, and C (if there is no A, B, or C in your equation type in 0)
F1: Solve
These are your roots, solutions, zeros to get the factors: place in (x___) with the opposite value.
x  1
x  4
The factors will be (x+1)(x-4)
To turn a root into a factor:
Take the opposite sign and
put in (x __) after x
x  3 / 4 
x  2
The factors will be (4x+3)(x-2)
If you have a fraction as a root: the
denominator goes in front of (_x)
and the numerator takes the
opposite sign and goes after x
Radicals
Simplifying Radicals
To simplify, use n ab  n a  n b or break out into prime factors looking for the same repeated factors (2 or
3 or 4 of a kind—depending on the index).
Ex) 27  3 3 3  3 3
Ex2) 3 54  3 3 3 2  33 2
2 of a kind
Comes out of the radical
Simplifying Radicals with variables:
When working with variables, divide the exponent by the index. The amount divided evenly is the
exponent for the variable on the outside. The remainder, if any, is the exponent for the variable on the
inside.
Ex) 25 x3  5 x 2 x  5 x x
Ex2) 3 16a5b11  2 2 2 2 a3 a2 b3 b3 b3 b2  2ab3 3 2a2b2
a pair
Solve Quadratic Equation
Factors
Set trinomial =0
Factor Polynomial like normal
Set each factor (set of parenthesis) equal to zero
Solve for the variable
You should have two answers!
x 2  6 x  18  0
( x  9)( x  3)  0
( x  9)  0
x9
( x  3)  0
x  3
Roots, Zeros,
Solutions
(Where the graph
crosses the x-axis)
Graphs Parabola Answers is where graph crosses the x-axis
y  x 2  x  12
Roots, Zeros,
Solutions
0  x 2  x  12
0  ( x  3)( x  4)
x  3, 4
Using the calculator:
Factors
Set equation equal to zero
Menu
Graph
Type in equation
F6: Draw
F5: GSOLV
F1: Root
Use the side arrows to go from one root to the other
Using the quadratic formula:
b  b2  4ac
2a
solve: x 2  x  12
1  12  4(1)( 12)
2(1)
1  1  48
2
1  49
2
1  7
2
1  7 1  7
,
2
2
6 8
,
2 2
3, 4
Plug A, B, C into your equation. Ax 2  Bx  C  0
X=3 and x= -4
Functional Value
 ( x) =substitute a value in for the variable
 ( x)  x
2
 5 x Plug in value for x  (5)  52  5(5)   (5)  50
f(x) = 3x – 5
f(7) = 3(7) – 5 = 16
f(x) means the same thing as y. It is just the name of the function
f(7) means take out x, put in 7
EX for f(x) = x2 – 1 find the range if the domain is { -2, -1, 0, 1}
this means put all of those numbers in for x and get an answer for each one. Put all of the negative numbers in
parentheses to square them.
Answer {-1, 0, 3} – in order from smallest to biggest. You got 0 twice but you don’t have to list it twice.
Line of Best Fit
 Enter the set of data into a list (or lists) on your graphing calculator.
 Look at the scatterplot graph, decide which model is most reasonable (linear, quadratic, cubic,
logarithmic (LN), or exponential)
 Calculate the appropriate regression formula using the calculator and plugging in a, b, etc.
 Write the equation down.
Example 1:
Given the data below, find the line of best fit. What would be the value of y when x=6?
{(1, 2.1), (3, 3.1), (5, 4.0), (7, 5.2), (9, 5.9)}



Plug x into List 1 and y into List 2
Graph the scatterplot.
It should represent the line y  .485 x  1.635
Using the equation for the line of best fit, predict the y value when x = 6:
 Plug 6 in for x.
 Simplify
y  .485(6)  1.635  4.545
On calculator: Menu
STAT
Type all values of X in list 1
Type all values of Y in list 2
F1: GPH1
F1:GPH1
F1:CALC
F2: X
Replace m and b with the data given
Curve Best Fit
Write Equation using y  Ax 2  Bx  C
Example:
The manager of a large bookstore counts the number of books that are sold each hour since the store has opened. For
example, since the store opens at 8:00 A.M., she will record the number of books sold from 8:00-9:00 a.m. as the
number sold in the first hour. The table below shows some of her data.
Hours Since Opening
Number of Books
0
0
4
26
8
38
12
37
16
22
Write an equation for a function to model the data.
Type the Hours in L1 and Books in L2
You will see the following on your calculator:
Quad Reg
y  ax 2  bx  c
a= -.4241071429
b= 8.160714286
c= .0285714286
R 2 =.9999396718
Plug a, b, and c back into the equation. Round to the second decimal place.
y  0.42 x 2  8.16 x  0.03 This is the equation for the curve best fit!
On calculator: Menu
STAT
Type all values of X in list 1
Type all values of Y in list 2
F1: GPH1
F1:GPH1
F1:CALC
F2: x 2
F6:draw
Statistics
Mean- averages (add up all the data and then divide by # of objects)
Mode- Most
Median- middle # put in order smallest to biggest
Range- subtract the lowest value from the highest value
Stem-leaf, Box-whisker read directions, look at graphs
Normal Distribution:
A normal distribution shows data in a symmetrical, bell-shaped curve.
Data is centered around the mean (  )
The standard deviation (  ) tells how each data value in the set differs (deviates) from the mean.
The variance (  2 ) is the squared deviation from the mean of a data set.
On the Calculator:
Menu
Stats
Type data into L1
F2: Calc
F1: Var
x  mean (μ)
σx= standard deviation
Label the normal curve indicating the following: mean and deviation
Example 2:
The test scores on a college algebra test are as follows: 67, 69, 71, 75, 78, 78, 83, 85, 85, 85, 85, 86, 87, 89, 92, 95,
98, 98, 98, 100, 100, 100, 100, 100, 100.
A) What is the mean?
B) What is the Standard deviation?
C) What is the absolute mean deviation?
Z-Score
A “z-score” represents the number of standard deviations away from the mean
 A z-score with a negative value lies below the mean.
 A z-score of 0 lies at the mean
 A z-score with a positive value lies above the mean.
Z-scores are a way to compare different normal distributions,
x
, where  is the mean and  is the standard deviation. “x” is the
To calculate the value of a z-score, z 

number you are seeking.
Example 4:
The mean height of eleventh-grade boys at Franklinton High School was 69.5 inches and the standard deviation of
the data was 3 inches.
The mean height of eleventh-grade girls at Franklinton High School was 64 inches and standard deviation of 2.25
inches.
How many standard deviations away from the mean is a boy who is 65 inches and a girl who is 65 inches tall?
Find the z-score of Boys:
Find the z-score of Girls:
z
z
65  69.5
 1.5
3
65  64
 0.44
2.25
y
Direct Variation
 Equation: y = kx (k is the constant of variation);
 graph is a line thru the origin
 Solve first equation for k
 substitute k into another equation and solve for the unknown variable
x
Example 1: If y varies directly as x and y is 6 when x is 18, find y when x is 24.
y  kx
Set up to solve for k: 6  18k
1
k
3
Then plug k into the formula and find the missing variable
y  kx
1
y  (24)
3
y 8
II Inverse Variation
 Example – the speed of a car and the time it takes to reach the destination
k
 Equation: xy  k or y  ( k is the constant variation)
x
 graph is a hyperbola in opposite quadrants (Quad I & III or Quad II and IV)
 To solve find k and substitute it and remaining numbers into eqn. again.
Example 2:
If y varies inversely as x and y = 10 when x = 2, find y when x = 6.
y
x
xy  k
If (2)(10)  k
20  k
xy  k
6 y  20
Find y now y  20
6
10
y
3