Download algebra 1 study guide units 1

Name _______________________________________________
Block _______
Algebra 1 Study Guide
Unit 1 – Connecting Arithmetic to Algebra
A variable is a letter that stands for a number.
Ex. If x = 5 what is the value for x + 7? 5 + 7 = 12
Exponents are used to show how many times a number or variable is used as a factor.
Ex. x·x·x·x·x·x·x·x = x 7
4·4·4 = 4³
x
Use the y key or the up arrow key ^ for exponents other than 2.
DO equations in PEMDAS order.
Parentheses - grouping symbols (), [] and fraction lines. (Work from the inside out)
Exponents
Multiplication
Division
Addition
Subtraction
***Watch out for parentheses placement***
2·3² = 2·3·3 = 18
BUT, (2·3) ² = 6·6 = 36
A solution of an equation makes the equation TRUE.
A Function has only one “y” value for each “x” value.
A Function can also be identified by the vertical line test.
Each vertical line hits only one point on the line.
Unit 2 – Real Numbers
Real numbers are Positive, negative and zero.
Absolute Value is always positive, except for the absolute value of zero, which is zero!
Addition:
Same sign = add the numbers. 4 + 5 = 9
Different signs = difference of the numbers, sign of the greater absolute value.
4 + (-5) = -1
Subtraction: change second number to the opposite sign, and then add. 4 – (-5) = 4 + 5 = 9
* Any number plus or minus zero = that number* 4 + 0 = 4
4–0=4
Multiplication and Division:
Same signs – Positive product
Different signs – Negative product
Even number of negatives – positive product
Odd number of negative – negative product
*Anything times zero = zero*
Anything times one = that number
Division by ZERO is undefined
Distributive Property: everything in the ( ) is multiplied by what’s outside the ( ).
2(x + 7) = 2x + 14
x (x + 5) = x² + 5x
Like Terms: have the same variable raised to the same power.
3x + 7x = 10x
3x² + 4x² = 7x²
3x² -2x + 6x² + 4x = 9x² + 2x
1
Unit 3 – Probability & Statistics
Proportion: 2 equal ratios.
Cross Product property: the product of the means equals the product of the extremes
2 10
>>>>>>>>>>>> 2 · 25 = 5 · 10 >>>> 50 = 50

5 25
Percent: “ per 100”
is
%

of 100
Probability:
favorable  outcomes
possible  outcomes
Independent Events: do not affect each other
Usually spinners, dice, coins
Dependent Events: what happens with one affects the other event. Look for, “and not replaced”
Usually cards, marbles and socks
Factorial: used to calculate the number of arrangements of a given number of items
Four pictures arranged on the wall = 4! = 4 · 3 · 2 · 1 = 24
Measures of Central Tendency
Mean: add and divide by number of values added
Median: middle number, when the values are put in order, (2 middle numbers, add and divide by 2)
Mode: most frequent number, (there can be no mode, one mode, or multi-modes)
Range: Difference between highest and lowest value
Unit 4 – Linear Equations with One Variable
Undo – SADMEP
“Clean up” any distributive property or like terms first.
Get “x” on one side, get the number on the other side.
Variables on both sides – keep the bigger variable.
Function form: “y” by itself on one side, everything else on the other side
Unit 5 – Graphing Linear Equations
X is the first coordinate, horizontal axis, positive right, and negative left
Y is the second coordinate, vertical axis, positive up, negative down
y  y1
Slope: the tilt of the line: m = 2
x2  x1
Positive: rises left to right
Negative: falls left to right
Zero: horizontal line, y = some number
Undefined: vertical line, x = some number.
Intercepts: where the line crosses the axis
Y intercept, x = zero
X intercept, y = zero
Direct Variation: the line goes through the origin.
2
Unit 6 – Writing Linear Equations
Y = mx + b
Slope-Intercept form m is the slope, b is the y-intercept
Y – y = m (X - x)
Point-Slope form
Ax + By = C
Standard form
x and y are a point on the line.
To write the equation of a line:
Given
Do
Slope and y- intercept replace m with the slope; replace b with the intercept
Slope and a point
replace m with slope, x and y with the given point; find b; write the equation.
Two points
find slope, replace m with slope, use slope and a point to find b, write equation
Unit 7 – Linear Inequalities
In an Inequality: if you multiply or divide by a negative number, you MUST reverse the inequality symbol.
Graphing Inequalities: < or > open circle,  or  filled in circle.
Compound Inequalities
OR: 2 separate inequalities, one < and one >, look like oars.
AND: 1 long extended inequality both symbols < with “x” in the middle, line segment.
Absolute value symbols acts like grouping symbol, UNDO it last
Unit 8 – Systems of Linear Equations & Inequalities
Substitution: Replace one variable with what it equals in terms of the other variable.
Look for an equation that is “y = ______” or “ x = _______”
Solve for the value of the one variable,
Then go back and find the value for the other variable.
Elimination: One variable cancels out, positive in one equation, negative in the other, same number.
Line up the like terms, both variables and constant numbers.
Check to see if one of the variables cancels.
Solve for the value of the other variable.
Go back and find the value of the variable that cancelled.
Elimination with Multiplication: One variable cancels after you multiply one or both equations.
Line up the like terms.
Check to see if one of the variables cancels.
Look for a variable with “1” as a coefficient.
Make one variable in each equation the same number, but opposite signs.
Cancel that variable.
Solve for the other variable.
Go back and find the value of the variable that cancelled.
A System of Equations can have:
One solution -----------------The lines intersect.
No solution ------------------ The lines are parallel and never intersect.
Infinite solutions ----------- The lines are actually the same line.
3
Unit 9 – Exponents and Exponential Functions
To multiply the same base, add the exponents. x3  x 4  x 7
To raise a power to another power, multiply the exponents. ( x3 )4  x12
Raise all the factors in a term to new power. ( x3 y 4 z 5 )3  x9 y12 z15
Any non-zero number to the zero power is 1. 123,9870  1
To change a negative exponent to a positive exponent, move it to the opposite part of the fraction.
x 2 y 4

y 4 x 2
To divide the same base, subtract the exponents.
x5
 x2
3
x
x3 2 x 6
Raise both the top and bottom of a fraction to the new power. ( 4 )  8
y
y
Scientific Notation:
Scientific Notation to Decimal Form – move the decimal point the same as the power on “10”
2.3456 x 10 5 = 234560
Decimal Form to Scientific Notation – move the decimal point to leave one digit to the left of it.
234.567 = 2.34567 x 10 2 the power is opposite to direction moved.
The number always looks like: c x 10 n , where 1  c <10 and n is an integer.
Unit 10 – Quadratic Equations and Square Roots
Square Root: what number times itself equals the given number, the opposite of squaring a number
Each positive number has 2 square roots, positive and negative.
The square root of a product = the product of the square roots of the factors.
x 2 y 2  x 2  y 2  xy
The square root of a quotient = the quotient of the square roots of both the top and bottom of the fraction
25
25 5


49
49 7
The Quadratic Formula: For the equation, ax² + bx + c =0,
x
b  b2  4ac
, where a  0 and b 2  4ac  0
2a
Vertical Motion Model: dropped
Height when it stops = h
Initial height = s
Time in motion = t
h = -16t² + s
4
Unit 11 – Polynomials and Factoring
Degree of the polynomial: highest power in the polynomial
0
constant
1
linear
2
quadratic
3
cubic
4
quartic
5
Term of a polynomial: number of terms in a polynomial
1
monomial
2
binomial
3
trinomial
4 or more
polynomial
FOIL: multiplying 2 binomials
( x + 3)( x + 4)
First x · x = x²
Outer x · 4 = 4x
Inner 3 · x = 3x
Last 3 · 4 = 12
x² + 4x + 3x +12 = x² + 7x + 12
* remember: ( x + 2 )² = ( x + 2 )( x + 2)
NOT x² + 2² !!!!!
Zero product property: If 2 factors have a product of zero, then one of the factors is zero.
( x + 2 )( x – 3) = 0 then, x + 2 = 0 or x – 3 = 0 so, x = -2 or x = 3
Factoring x² + bx + c:
If c is positive
Find 2 numbers that multiplied = c and add up to b
Both parentheses get the same sign as is on b
If c is negative
Find 2 numbers that multiplied = c and subtracted = b
The bigger factor gets the same sign as b, the other factor gets the opposite sign
Factoring the Difference of 2 Perfect Squares:
Find the square root of “back” number
Positive sign in one set of parentheses, negative in the other
**All equations must be in x² + bx + c = 0 form before you factor to solve for x**
5
Additional Notes or Reminders
6