1. Chapters 1 and 2
Algebra 2

2. Real Numbers Natural Numbers Whole Numbers Integers Rational Numbers
Irrational Numbers
Imaginary Numbers

3. The opposite or additive inverse of any number a is –a
The sum of opposites is 0
The reciprocal or multiplicative inverse of any number a is 1/a
The product of reciprocals is 1

4. Properties of Real Numbers
Commutative
Associative
Identity
Inverse
Distributive

5. Absolute Value
The absolute value of a number is its distance from zero on the number line
| -4| = 4
| 0 | = 0
| -1 ∙ (-2) | = |2| = 2

6. Evaluating Algebraic Expressions
When you substitute numbers for the variables in an expression and follow the order of operations you evaluate the expression
evaluate a – 2b + ab for a = 3 and b = -1
a – 2b + ab = 3 – 2(-1) + 3(-1)
= 3 – (-2) + (-3)
= – 3
= 2

7. Combining Like Terms
A term is a number, variable or the product of a number and one or more variables.
The coefficient is the numerical factor in a term.
Like terms have the same variables raised to the same powers.
Combine like terms by adding coefficients

8. Try these – in your notebooks
Evaluate 7x – 3xy for x = -2, y = 5 16
Evaluate (k-18)2 -4k for k = 6
120
Combine Like Terms
2x2 + 5x – 4x2 + x – x2
-3x2 + 6x
-2(r + s) – (2r + 2s)
-4r – 4s

9. Practice:
p15 (1-45)odd
Please check you answers in the back of your book when you are done

10. 1.3 & 1.4 Solving Equations and Inequalities EQ: What are the steps to solving linear equations and inequalities?
Warm Up: Solve these problems in your notebook. (Left hand side)
Simplify each expression
5x – 9x – 3
2y + 7x + y – 1
10h + 12g – 8h – 4g
( x + y ) – ( x – y )
- (3 – c) – 4(c – 1)

11. 1.3 & 1.4 Solving Equations and Inequalities EQ: What are the steps to solving linear equations and inequalities?
5x – 9x – 3 = -4x - 3
2y + 7x + y – 1 = 7x + 3y - 1
10h + 12g – 8h – 4g = 8g + 2h
( x + y ) – ( x – y ) = 2y
- (3 – c) – 4(c – 1) = -3c + 1

12. Solving Equations – by steps
Distribute
Combine Like Terms
Combine constants
Solve for variable

13. Solving Equations
A number that makes an equation true is the solution to the equation.
Try these:
8z + 12 = 5z – 21
z = -11
6(t – 2)= 2 (9 – 2t)
T = 3

14. Stations:
Pair up - pick an A and a B. You will turn in ONE sheet of paper with all the problems solved.
Begin at the station on your table.
Student A does the A problem explaining each step to Student B
Student B does the B problem explaining each step to Student A
Add your answers together. They should add to the number on the equation paper.
Once they do, you may move to the next station.

15. Solving For a Variable
Solving for a variable means isolating that variable on one side of the equation.
Solve d = rt for t
Solve A = ½ h ( b1 + b2) for h
Try these:
Solve P = 2L + 2W for W
Solve E = ½ mv2 for v

16. Solving Inequalities Solve just like equations.
Reverse the direction of the inequality symbol if you multiply or divide by a negative.
Graph the solution.
Example: (2 – x) ≤ 41

17. Solving Inequalities – Try these
Solve and graph
3x – 6 < 27
X < 11
12 ≤ 2 ( 3n + 1) + 22
N ≥ -2

18. Compound Inequalities
A pair of inequalities joined by and or or
3x – 1 > -28 and 2x + 7 < 19
Try this:
X – 1 < 3 or x + 3 > 8
2x > x + 6 and x – 7 < 2

19. Exit Pass: Solve these equations and inequalities on a sheet of paper
Exit Pass: Solve these equations and inequalities on a sheet of paper. Place in the Algebra 2 basket on your way out the door.
1. 16x – 15 = -5x + 48
2. 4w – 2(1 - w) = -38
3. -2x < 3 ( x – 5) graph the solution
4. 3x + 4 ≥ 1 and -2x + 7 ≥ 5 graph the solution
Homework:
p21 (1-27) odd
p29 (1-33) odd

20. 1-5 Absolute Value Equations and Inequalities EQ: How do you solve equations with absolute value?
warm up
Solve these equations
5(x-6) = 40
5b = 2(3b-8)
2y + 6y = 15 – 2y + 8
4x + 8 > 20
3a – 2 ≥ a + 6
4(t-1) < 3t + 5 .

21. 1-5 Absolute Value Equations and Inequalities EQ: How do you solve equations with absolute value?
The absolute value of a number is its distance from zero on the number line and distance is non-negative.

22. Absolute Value Equations
Usually have two solutions
| 2y – 4 | = 12 means
2y – 4 = 12 or 2y – 4 = -12
Isolate the absolute value
Rewrite as two equations
Solve both equations
Be sure to check your answers – they may not always work.

23. Try these | 3x + 2 | = 7 X = 5/3, -3 3|4w – 1| - 5 = 10 W = -13/5, 5
X = 1, /5 is an extraneous solution

24. 1-5 Absolute Value Inequalities
| 3x + 6 | ≥ rewrite the equation as:
3x + 6 ≥ 12 or 3x + 6 ≤ -12
Note: The inequality symbol changes direction for the negative solution
Solve |2x – 3| ˃ 7, graph the solution

25. 1-5 Absolute Value Inequalities
First isolate the absolute value expression
3|2x + 6| -9 ˂ 15

26. 1-5 Absolute Value Inequalities
Exit Pass:
| x + 3 | = 9
|3x – 6| - 7 = 14
|6 – 5x| = -18
2 | x + 3 | ≥ 10
| 2x + 4 | - 6 < 0

27. homework
p every other odd, except 29
(1,5,9,13,17,… etc)

28. Warm up: Complete a 2 minute quick write in your notebook about how to solve absolute value equations and inequalities.

29. There will be a test next Tuesday/Wednesday on solving linear equations and inequalities, including absolute value problems.
There will be basic probability questions.

30. 1-6 Probability EQ: How do you calculate experimental and theoretical probability?
Probability measures how likely it is for an event to occur.
Expressed as a percent- 0% to 100% or
as a number 0 to 1
The probability of an impossible event is 0%
The probability of a certain event is 100%

31. When you gather data from observations you can calculate an experimental probability.

32. The set of all possible outcomes is called the sample space
You can calculate theoretical probability as a ratio of outcomes.

35. Carnival Fish!
Homework:
page 42 (7-21, 25-33)odd
page 45 (51-61) odd

36. Warm Up
Glue the warm up slip into your notebook and complete (page 56)

37. Stations Review
Fold a sheet of binder paper in half lengthwise and width wise so there are four sections on each side.
You will move from station to station completing each set of review problems in a section.
You answers should add together to get the number on the station poster.
Show your work!

38. 2-1 The Coordinate Plane
In an ordered pair ( x,y) the first number is the x coordinate and the second number is the y coordinate
The x-y coordinate plane is divided into four quadrants by the
x and y axes

39. 2-1 Relations and Functions
A relation is a set of pairs of input and output values
The domain is the set of all inputs, or x values of the ordered pairs
The range is the set of all outputs, or y values of the ordered pairs

40. 2-1 Relations and Functions

41. 2-1 Relations and Functions
What is the domain and range of this relation?
Domain {-3, -1, 1}
Range {-4, -2, 1, 3}

42. 2-1 Relations and Functions
What is the domain and range of this relation?
D {-2, -1, 1, 3}
R { -2, 0, 4, 5}

43. 2-1 Relations and Functions
A function is like a machine. Put an input (x) in and get an output (y) out.
A function is a relation in which each element of the domain is matched with exactly one element in the range.

44. 2-1 Relations and Functions

45. 2-1 Relations and Functions
Vertical line test – If a vertical line passes through at least two points on a graph, then the relation is NOT a function

46. 2-1 Relations and Functions
Function notation
Y = 2x can be rewritten as
f(x) = 2x, and read “f of x”
It does not mean f times x
To evaluate the function at x = 3 write
f(3), read “f of 3”

47. Use the function f(a) = 2a + 3
Evaluate the function at:
f(-5)
f(-3)
f(1/2)
f(4)

48. 2-1 Relations and Functions
Homework
p 50 (3-35) odd: Chapter 1 Test

50. 2-2 Linear Equations EQ: How do you graph a line in standard form?
A function whose graph is a line is a linear equation
Because the value of y depends on the value of x, y is called the dependent variable and x is the independent variable
The y intercept is the point where the line crosses the y axis (x = 0)
The x intercept is the point where the line crosses the x axis (y = 0)

51. 2-2 Linear Equations
The standard form of a linear equation is Ax + By = C and is graphed by finding the x and y intercepts
Example: 3x + 2y = 120
Graph 2x + y = 20

52. 2-2 Linear Equations
Slope is the ratio of the vertical change to the horizontal change
Slope = vertical change (rise)
horizontal change (run)
Given two points (x1, y1) and (x2, y2)
Slope = y2 – y1
x2 – x1

53. 2-2 Linear Equations Find the slope between (3,2) and (-9,6)
Change in y = = 4
Change in x = -9 – 3 = -12
Slope = 4/-12 = -1/3
Try these:
( -2, -2) and (4,2)
2/3
(0, -3) and (7, -9)
-6/7

54. Practice
P69 (1-7) odd, (11,13,15)

55. 2-2 Writing Equations Point-Slope form of an equation
y – y1 = m ( x – x1)
Write equation when given a point and slope
Ex: Write in standard form an equation of the line with slope -1/2 through the point (8, -1)

56. 2-2
Try these
Write in slope intercept the equation of the line with slope 2, through the point (4, -2)
Write in slope intercept form the equation of the line with slope 3, through the point (-1, 5)

57. 2-2 Writing an equation given two points. (1,5) and (4, -1)

58. 2-2 Slope Intercept form Y = mx + b M is the slope
B is the y intercept
To find the slope of a line in standard form, solve the equation for y

59. 2-2
Find the slope of 4x + 3y = 7
3x + 2y = 1
3x – 12y = 6

60. 2-2 Parallel lines have the same slope
Perpendicular lines have slopes that are opposite reciprocals of each other
The line perpendicular to y = 3x +7 will have a slope of – 1/3

61. 2-3 Direct Variation EQ: How do you determine if a function is a direct variation?
Warm Up:
find the slope between (3,-5) and (1,2)
write in slope intercept form the equation of the line through (-3,-2) and (1,6)
write in standard form the equation of the line with slope 2/5, through (-1,3)

62. 2-3 Direct Variation EQ: How do you determine if a function is a direct variation?
A linear function y = kx represents direct variation. The slope k is constant.
You can write k = y/x, and y/x is the constant of variation
The rate of change of the function k is constant.
A direct variation function always contains the point (0,0)

63. 2-3 Direct Variation EQ: How do you determine if a function is a direct variation?
What does the graph of a direct variation look like?

64. 2-3 Direct Variation EQ: How do you determine if a function is a direct variation?
Direct Variation from a table. k = y/x
For each table, find y/x for each pair of points.

65. 2-3 Direct Variation Identify direct variation from an equation
Must be able to put equation in the form y = kx
3y = 2x
Y = 2x + 3
Y = x/2
7x + 4y = 10

66. 2-3 Direct Variation EQ: How do you determine if a function is a direct variation?
Direct Variation Activity – Rotate for each task
Group chooses direct variation function. Writes an ordered pair that represents the function on their poster.
Next group determines the constant of variation k for the given point. (k = y/x)
Next group writes the equation for the direct variation in the form y=kx.
Next group constructs a table containing 5 other points that would be on the line.
Next group plots those points and constructs the line through them.
Final group checks all the work and verifies that all parts have been done correctly.

67. 2-3 Direct Variation EQ: How do you determine if a function is a direct variation?
Homework assignment: page 76 (1-45) odd
Chapter 1 make up test on Wednesday during enrichment.

68. 2-3 Direct Variation
Can use direct variation to solve some problems – set up as a proportion
Suppose y varies directly with x, and x = 27 when y = -51. Find x when y = -17.

69. Homework
P 70 (21 -33) odd, (39 – 57) odd
P 76 (1 – 21) odd

70. 2-4 Using Linear Models Both equations represent direct variations
If y = 4 when x = 3, find y when x = 6
If y = 7 when x = 2, find y when x = 8

71. 2-4 Using Linear Models EQ: How do you use linear equations to model real-world situations?
y=mx + b
m = slope which is a rate of change
speed, rate of increase or decrease etc
b = a starting value
beginning height, distance, weight etc
result = (rate of change) ∙ x + (start value)

72. 2-4 Using Linear Models
Jacksonville, FL has an elevation of 12 feet above sea level. A hot air balloon taking off from Jacksonville rises 50 ft/min.
Write an equation to model the balloon’s elevation as a function of time
result = (rate of change) ∙ x + (start value)
Graph the equation
Interpret the intercept at which the graph intersects the vertical axis.

73. Using two points to make a model
A candle is 6 in. tall after burning for 1 hour. After 3 hours it is 5 ½ inches tall.
What is the rate of change? (Slope)
Write an equation in slope intercept form to model the height y of the candle after it has been burning x hours.
What does the y intercept 6 ¼ represent?

74. Using models to make predictions
Using the equation for the candle.
In how many hours will the candle be 4 inches tall?
How tall will the candle be after burning for 11 hours?
When will the candle burn out?

75. whiteboard problems

76. Scatter plot
A scatter plot is a graph that relates two different sets of data by plotting the data as ordered pairs.
You can use a scatter plot to determine a relationship between the data sets.
A trend line is a line that approximates the relationship between the data sets in a scatter plot.

77. Correlation in a scatter plot

78. Draw a trend line that has about the same number of points above and below it
Use the slope and y intercepts to estimate the equation of the line

79. Group work
whiteboard problems

80. page 83 (1-13) all

81. 2-5: Absolute Value Functions and Graphs

82. 2-5: Absolute Value Functions and Graphs

83. 2-5: Absolute Value Functions and Graphs

84. 2-5: Absolute Value Functions and Graphs

85. 2-5: Absolute Value Functions and Graphs

86. 2-5: Absolute Value Functions and Graphs

87. 2-5: Absolute Value Functions and Graphs
Homework: page 92 (1-9, 19-27) odds
Please make your graphs large enough to read!

88. Practice Graph these absolute value functions Y = | 3x + 6 |

89. 2 – 6 Families of functions EQ: How do translations affect the graph of a parent function?
A family of functions is made up of functions with common characteristics
A parent function is the simplest function with these characteristics
A translation shifts a graph horizontally, vertically or both. It results in a graph of the same shape and size but possibly in a different position.

90. 2 – 6 Families of functions EQ: How do translations affect the graph of a parent function?
Absolute value functions
y = |x| parent function
y = |x| + k shifts vertex of function k units up (down if negative)
y = | x – h | shifts vertex of function h units to the right (to the left if h is negative)
y = a|x| stretches |x| by a factor of a (slope)
y = -a|x| reflects the graph of |x| over the x axis

91. 2 – 6 Families of functions EQ: How do translations affect the graph of a parent function?
y = a|x – h| + k
what does h do?
what does k do?
what does a do?

92. 2 – 6 Families of functions EQ: How do translations affect the graph of a parent function?
How is each graph different from the parent function y = |x|?
y = |x+1|
y = -|x|
y = | x | - 3
y = | x - 2 | + 4

93. 2 – 6 Families of functions EQ: How do translations affect the graph of a parent function?
homework: page 99 (1-11, 17-19) all
Chapter 2 test on Monday October 1

94. 2 – 6 Families of functions
Graph y = |x|
On the same graph, graph
y = |x| + 3
y = |x| - 2
Describe how adding a constant outside the absolute value affects the graph of the parent function

95. 2 – 6 Families of functions
Explain how a function of the form y= |x| + k is different from the parent function.
A vertical translation moves the graph of the parent function up (or down) k units.
Write the equation for the graph of y = |x| translated 5 units down.
Y = |x| translated 7 units up.

96. 2 – 6 Families of functions
On a new graph, draw the parent function y = |x|
On the same graph, draw y = |x + 2|
On the same graph draw y = | x – 4|
Describe how adding a number inside the absolute value affects the graph of the parent function

97. 2 – 6 Families of functions
For a positive number h, y = | x - h| is a horizontal translation of the parent function to the right h units
Y = |x + h| is a horizontal translation h units to the left.

98. 2 – 6 Families of functions

99. 2 – 6 Families of functions
Graph y = 2 |x|
Graph y = - |x|
Graph y = ½ |x|
How does multiplying a graph by a number larger than one affect the graph?
How does multiplying a graph by a number less than one affect the graph?
How does multiplying by a negative affect the graph?

100. 2 – 6 Families of functions
A vertical stretch multiplies all y values by the same factor greater than one, stretching the graph vertically
A vertical shrink multiplies all y values by a factor less than one, compressing the graph vertically
Multiplying by a negative factor reflects the graph over the x axis

101. 2 – 6 Families of functions
A function is a vertical stretch of y = |x| by 5 – what is the equation?
Reflect the function across the x axis. What is the equation?

102. 2 – 6 Families of functions

103. 2 – 6 Families of functions
Write equations for the graphs obtained by translating y = |x|
10 units right
4 units down
7 units left, 6 units up
Reflection across x axis
Vertical shrink by a factor of 2/3

104. Homework Page 92 (33-43) odd Page 99 (1-13) odd Page 102 (1-10)
Chapter 1 & 2 test next week Tuesday

105. Warm up Graph the following functions y = 2x + 3 y = -1/3x +1

106. 2-7 Two Variable Inequalities
A linear inequality is an inequality in two variables whose graph is a region in the coordinate plane that is bounded by a line.
To graph a linear inequality:
Graph the boundary line
Determine which side of the line contains solutions
Determine if the boundary line is included

107. 2-7 Two Variable Inequalities
A dashed boundary line indicates the line is not part of the solution
A solid boundary line indicates the line is part of the solution
Choose a test point to check if a region makes the inequality true – use (0,0), if it is not on the line
Example: graph y > ½ x - 1

108. 2-7 Two Variable Inequalities
Try this on your whiteboard – graph:
y ≤ 2x + 3
Graph the line y = 2x + 3
Check the test point (0,0)
Is the line part of the solution?

109. 2-7 Two Variable Inequalities
Graph y ˃ -4x + 3

110. 2-7 Two Variable Inequalities
Graph the absolute value inequality
y ≤ | x – 4 | + 5
-y + 3 > | x + 1 |

111. Homework:
Page 106 (1,5,9,11,15,19, 25)
Corrections to quiz – use quiz as study guide
Chapter 2 test on Monday

112. Study Guide Answers a b c d d b a c a c d b