1. Flipper
Numbers

2. Numbers Natural Numbers: {1, 2, 3, … {0, 1, 2, 3, … Whole Numbers:
Integers:
{…, −3, −2, −1, 0, 1, 2, 3, …}
Rational Numbers: Fractions, decimals that end, decimals that repeat
3= = = = 1 3
Irrational Numbers: Decimals that never end and never repeat
𝜋 …
Real Numbers: The set of rational numbers and irrational numbers combined

3. Variable and Terms 9 𝑥 2 +4 𝑥 2 +24𝑥𝑦−7 𝑦 2
Variable: A letter or symbol that is used to represent one number or a set of number.
Term: Consist of numbers and variables or a combination of numbers and variables.
Like Terms: Must have the same variable(s) to the same exponent.
Add or subtract coefficients when combining. Exponent
does NOT change.
13 𝑥 2
Like Terms
Variable
Exponent
9 𝑥 2 +4 𝑥 2 +24𝑥𝑦−7 𝑦 2
Term
Coefficient
Variable and Terms
Distributive Property

4. Distributive Property
If a, b and c are real numbers then: 𝑎 𝑏+𝑐 =𝑎𝑏+𝑎𝑐
Examples:
3 𝑥+2 =3𝑥+6
7𝑦 2𝑥+2𝑦 =14𝑥𝑦+14 𝑦 2
Be careful of negative signs outside the parenthesis.
3𝑥−5 𝑥−8 =3𝑥−5𝑥+40=−2𝑥+40
Always SIMPLIFY

5. Order of Operations Please Parenthesis Excuse Exponents My
Step 1 – Evaluate expressions inside grouping symbols
Step 2 – Evaluate Powers
Step 3 – Multiply and divide from left to right
Step 4 – Add and subtract from left to right
Please
Excuse My
Dear
Aunt
Sally
Parenthesis
Exponents
Multiplication
Division
Addition
Subtraction
EX.
24 – (32 + 1) =
24 – (9 + 1) =
24 – 10 = 14
Order of Operations
Equations

6. Equations One Step: Reciprocal: Two Step: Combine Like Terms: x = 4 27 – ( )
x ) = 2 7 – 4
x = –14
Two Step:
Combine Like Terms:
7x – 4x = 21
3x = 21 = 3 3x 21
x = 7

7. Equations (cont.) Distributive Property: Variables on Both Sides
7x + 2(x + 6) = 39
7 – 8x = 4x – 17
7x + 2x + 12 = 39
7 – 8x + 8x = 4x – x
9x + 12 = 39
7 = 12x – 17
9x = 27
24 = 12x
x = 3
No Solution:
2 = x
3x = 3(x + 4)
Identity: ALL REALS
3x = 3x + 12
2x + 10 = 2(x + 5)
3x – 3x = 3x + 12 – 3x
0 = 12
2x + 10 = 2x + 10
Equations (cont.)
Fractions or Decimals

8. Fractions or Decimals
To solve an equation with fractions, multiply through by the LCD to eliminate the fractions.
To solve an equation with decimals, multiply through by either 10, 100, 1000, etc. to eliminate the decimals.
3 4 𝑥− 1 2 = 5 3
0.5𝑥+0.02=−0.2
100(0.5𝑥+0.02=−0.2)
𝑥− 1 2 = 5 3
50𝑥+2=−20
50𝑥=−22
9𝑥−6=20
𝑥=−.44
9𝑥=26
𝑥= 26 9

9. Graph using a Table 𝑦=2𝑥−1 𝑥 𝑦 Graph Using a Table Graph w/Intercepts1 2 3 −3 −1 1 3 5
Draw Table
Pick values
Plug into equation to obtain y-values
Plot points on the coordinate plane
Draw line connecting the points
Graph Using a Table
Graph w/Intercepts

10. Graph w/Intercepts
Use STANDARD FORM
x-intercept
y-intercept

11. Slope Formula RUN Let (x1, y1) = (3, 5) and (x2, y2) = (6, –1). –1 – 5
RISE
RUN
Let (x1, y1) = (3, 5) and
(x2, y2) = (6, –1).
–1 – 5
6 – 3 =
– 6 3 = –2
Horizontal lines have a slope of 0 and vertical lines have an undefined slope.
Slope Formula
Graph Slope-Intercept Form

12. Graph Slope-Intercept Form1 2
Slope = 2
y-intercept = 3
Step 1: Put the equation in 𝑦=
Step 2: Plot the y-intercept on the y-axis
Step 3: From the y-intercept, two options to for slope
Positive slope (+) – Up and to the right
Negative slope (-) – Down and to the right
Step 4: Draw a line through the two points

13. Graph Functions 𝑓(𝑥) Function Notation – f (x) is another name for y
Read “the value of f at x” or “f of x”
f(x) = 37x + 7 x
f(x)
37(0) + 7 = 7 1
37(1) + 7 = 44 2
37(2) + 7 = 81
Graph Functions 𝑓(𝑥)
Point-Slope Form

14. Point-Slope Form Point Slope Graph in Point-Slope Form 2 3
y + 2 = (x – 3). 2 3

15. 𝐴𝑥+𝐵𝑦=𝐶 Standard Form Integers y – y1 = m(x – x1) y – 1 = –3(x – 1)
Write an equation in standard form of a line whose slope is -3 and goes through the point (1, 1).
y – y1 = m(x – x1)
y – 1 = –3(x – 1)
𝑦−1=−3𝑥+3
𝑦=−3𝑥+4
3𝑥+𝑦=4
Standard Form
Slope-Intercept Form

16. Write an equation of the line that passes through (–2, 5) and (2, –1).
Slope-Intercept Form
Write an equation of the line that passes through (–2, 5) and (2, –1). 3
m =
y2 – y1
x2 – x1 =
–1 – 5
2 – (–2) –6 4 – 2
y = mx + b
5 = – 3 2
(–2)
+ b
2 = b
Substitute – 3 2
for m and 2 for b.
y = – 3 2 x
+ 2

17. Parallel Lines
Parallel Lines – Two lines the will never cross and have the same slope.
Write an equation of the line that passes through (–3, –5) and is parallel to the line y = 3x – 1.
y = mx + b
–5 = 3(–3) + b
4 = b
y = 3x + 4
Parallel Lines
Perpendicular Lines

18. Perpendicular Lines
1.Two lines are perpendicular if and only if the product of their slopes is -1.
2.You can also tell if two lines are perpendicular if their slopes are negative reciprocals of each other.
2 3 ∙− 3 2 =−1
EX:
𝑚=− 1 2
𝑚=2
Opposite reciprocal would be
Write an equation of the line that passes through (4, –5) and is perpendicular to the line y = 2x + 3.
y =
mx + b
y = – x – 3 1 2
–5 =
– (4) + b 1 2
–3 = b

19. Inequalities
<
>
Less than or equal to
Greater than or equal to
Less than
Greater than
Closed Circle
Open Circle
When solving inequalities and you have to multiply or divide by a negative number, reverse the inequality.
< 7 x –6 x –6
> –6 7
x > –42
Inequalities
Compound Inequalities

20. Compound Inequalities
OR Compound Inequality
x < –1 or x  4
AND Compound Inequality
x  –3 and x < 5
= –3  x < 5

21. Absolute Value Equations
Absolute Value: Distance a number is from zero.
NEVER NEGATIVE
x – 3 = 8
x – 3 = or x – 3 = –8
x = or x = –5
Absolute Value Eq.
Absolute Value Ineq.

22. Absolute Value Inequalities
x – 5  7
x – 5  7 or
x – 5  7
x  –2 or
x  12
Remember to flip the inequality for the negative!

23. Graph Systems Special Cases: –x + y = –7 x + 4y = –8 Lines Coincide
Graph both lines on the same coordinate plane.
The ordered pair where the two lines cross is the solution to the system.
–x + y = –7
x + 4y = –8
Special Cases:
Lines Coincide
Lines are Parallel
many solutions
no solution
Graph Systems
Substitution

24. Substitution
Solve one of the equations for one of its variables.
Substitute that expression into the other equation and solve.
Once you find one variable, substitute into either original equation to find the remaining variable.
y = 3x + 2
x + 2y = 11
y = 3x + 2
y = 3(1) + 2
y = 3 + 2
y = 5
x + 2y = 11
x + 2(3x + 2) = 11
7x + 4 = 11
7x = 7
x = 1
(1,5)

25. Elimination
Multiply one or both equations to achieve same coefficient with the same variable.
Add or Subtract the equations to eliminate one of the variables.
Solve the resulting equation for the other variable.
Substitute, in either original equation, to find the value of the eliminated variable.
6x + 5y = 19
6x + 5y = 19
2x + 3y = 5
–6x – 9y = –15
–4y = 4
y = –1
(4,−1)
2𝑥+3 −1 =5
2𝑥=8
𝑥=4
Elimination
Systems of Inequalities

26. Systems of Inequalities
Graph both lines
Solid line ≤ and ≥
Dashed line < and >
Pick a test point
Shade if its true for both inequalities
y > –x – 2
y  3x + 6

27. Exponent Properties Product of Powers Example: Power of a Power
Power of a Product
Exponent Properties
Exponent Properties

28. Exponent Properties
Quotient of Powers
Example:
Power of a Quotient

29. Zero Exponents
Let a be a nonzero number and let n be a positive integer.
A nonzero number to the zero power is 1.
Examples:
(2𝑥𝑦𝑧) 0 =1 =1
Zero Exponents
Negative Exponents

30. Negative Exponents 2𝑥 −2 𝑦 2 = 2 𝑥 2 𝑦 2 𝑥 3 𝑦 −4 𝑧 2 = 𝑥 3 𝑦 4 𝑧 2
To make an exponent positive, write the reciprocal.
2𝑥 −2 𝑦 2 = 2 𝑥 2 𝑦 2
𝑥 3 𝑦 −4 𝑧 2 = 𝑥 3 𝑦 4 𝑧 2

31. EXPONENTIAL GROWTH MODEL
C is the initial amount.
t is the time period.
y = C (1 + r)t
(1 + r) is the growth factor, r is the growth rate.
$250 at 2.5% interest for 6 years
𝑦=250 (1+.025) 6
𝑦=$289.92
Exponential Growth
Exponential Decay

32. EXPONENTIAL DECAY MODEL
t is the time period.
C is the initial amount.
y = C (1 – r)t
(1 – r ) is the decay factor, r is the decay rate.
$13,000 car depreciates at a rate of 6% per year. What is the value in 3 years?
𝑦=13000 (1−.06) 3
𝑦=$10,797.59

33. Add/Sub Polynomials Add. (2x3 – 5x2 + x) + (2x2 + x3 – 1)
Subtract. (4x2 – 3x + 5) – (3x2 – x – 8)
4x2 – 3x + 5 – 3x2 + x + 8
(4x2 – 3x2) + (–3x + x) + (5 + 8)
x2 – 2x + 13
Add/Sub Polynomials
Mult. Polynomials

34. Degree (largest exponent)
Polynomials
Degree (largest exponent)
Classify by Degree
0. Constant
1: Linear
2: Quadratic
3: Cubic
4: Quartic
5: 5th degree polynomial
Constant Term
Leading Coefficient
Polynomials do not have variable exponents or negative exponents

35. Multiply a Monomial by a Polynomial
Find the product 2x3(x3 + 3x2 – 2x + 5).
2x3(x3 + 3x2 – 2x + 5)
Write product.
= 2x3(x3) + 2x3(3x2) – 2x3(2x) + 2x3(5)
Distributive property
= 2x6 + 6x5 – 4x4 + 10x3
Product of powers property
Mult. Mono by Poly
Foil/Rectangle

36. FOIL/Rectangle Method𝑥 2 𝑥
𝑥 2 2𝑥 3 3𝑥 6

37. Special Products 𝑥 2 −121 (𝑥−11)(𝑥+11) Example: A) Examples: A) B)
𝑎=𝑥
𝑏=11
(𝑥−11)(𝑥+11)
𝑥 2 −121
Examples: A) B)
Special Products
Factor GCF

38. Factor GCF
In order to factor out the greatest common factor, it has to be common to ALL terms. Look at coefficients first then variables. A.
4x3 – 44x2 + 96x
= 4x(x2– 11x + 24)
Look for GCF first
= 4x(x– 3)(x – 8)
Factor remaining polynomial B.
50h4 – 2h2
= 2h2 (25h2 – 1)
Look for GCF first
= 2h2 (5h – 1)(5h + 1)
Look for special patterns (difference of two squares)

39. Factor Trinomials EX. x2 – 4x + 3 = (x – 3)( x – 1) x2 + 3x + 2 =
ADD
Multiply
Two + #’s
Two - #’s
One + # One - #
One + # One - #
EX. x2 – 4x + 3 =
(x – 3)( x – 1)
x2 + 3x + 2 =
(x + 2)(x + 1)
Factor Trinomials
Factor Special Products

40. Factor Special Products
a. y2 – 16
= (y + 4)(y – 4)
b m2 – 36
= (5m + 6)(5m – 6) c.
4s2 + 4st + t2 d.
9x2 – 12x + 4
= (2s + t)2
= (3x – 2)2

41. Factor by Grouping
Use factor by grouping if there is no common monomial to factor out.
x3 + 3x2 + 5x + 15 = (x3 + 3x2) + (5x + 15) a.
= x2(x + 3) + 5(x + 3)
= (x + 3)(x2 + 5)
y2 + y + yx + x = (y2 + y) + (yx + x) b.
= y(y + 1) + x(y + 1)
= (y + 1)(y + x)
Factor by Grouping
Graph Quadratics

42. Graph Quadratics − 𝑏 2𝑎 1. Find the x-coordinate of the vertex.
2. Make a table of values, using x values to the left and to the right of the vertex.
3. Plot the points and connect them with a smooth curve to form a parabola.
Graph y = 3x2 – 6x + 2.
− − =1 x y

43. Solve Using Square Roots
d > 0
d = 0
d < 0
a. 2x2 = 8
c. b = 5
b. m2 – 18 = – 18
x2 = 4
m2 = 0
b2 = – 7
x = ± 4 = ± 2 
m = 0
No Solution
Solve Using Square Roots
Complete the Square

44. Complete the Square 𝑏 2 2 x2 – 16x = –15 x2 – 16x + 64 = –15 + 64
𝑏 2 2
x2 – 16x = –15
Find new c value = 8 2 =64
x2 – 16x = –
(x – 8)2 = 49
Factor the left side and simplify the right side
x – 8 = ±7
Take the square root of both sides
x = 8 ± 7
Add 8 to both sides
Simplify
𝑥=15, 1

45. Quadratic Formula 2x2 – x – 7 = 0 x = b2 – 4ac + – –b 2a – (–1) – +
*Make sure quadratic is in standard form before identifying a, b and c.
Example:
2x2 – x – 7 = 0
x =
b2 – 4ac + – –b 2a
* Solutions are where the graph crosses over the x-axis.
– (–1) – +
( –1)2 – 4(2)(–7)
2(2) = 4 = + –
The solutions are 2.14 and -1.64
Quadratic Formula