Algebra 2/Trig H

Size: px

Start display at page:

Download "Algebra 2/Trig H"

Alannah Baker
5 years ago
Views:

1 Welcome to Algebra 2/Trig H Welcome to Algebra 2/Trigonometry Honors! We are excited that you will be embarking on a journey to expand your understanding of mathematics and its concepts, tools, and techniques. This class is designed to challenge your understanding of mathematics, expand your critical thinking skills, and prepare you for success in Calculus 1 Honors. About Your Summer Assignment This summer, you will have the opportunity to sharpen your skills from previous classes. Due to the large number of topics we must study to prepare for calculus, we cannot review concepts that you have studied in Algebra 1 and Geometry, and we will immediately begin our study of Algebra 2 when you return in September. To complete your summer assignment: READ ( yes, READ!) each section to make sure you understand the concepts. STUDY the PowerPoint that further reviews the topics from the book. COMPLETE ALL assigned problems (see reverse.) You are expected to correctly complete all problems, not just try them. This assignment is DUE the first day of classes (Thursday, September 6.) You will have one full class period to ask questions about this assignment. Any student who does not fully complete the summer assignment will be moved to a different course. You will have TEST on this material during the first week of classes.

2 ASSIGNMENT: Textbook : Algebra 2 Common Core by Charles, et al. (2012) Your textbook is available for purchase on MBS. Complete the following exercises. Unless specified in the book, all problems should be completed WITHOUT a calculator: Lesson 1-2 (Properties of Real Numbers) #10-48 evens, odds, evens Lesson 1-3 (Algebraic Expressions) #10-62 evens Lesson 1-4 (Solving Equations) #10-60 evens (skip 26, 52) Lesson 1-5 (Solving Inequalities) #10-50 evens, evens Lesson 1-6 (Absolute Value Equations & Inequalities) #10-66 multiples of 3 Lesson 2-1 (Relations and Functions) #8-30 evens Lesson 2-3 (Linear Functions & Slope-Intercept Form) #8-61 multiples of 3 Lesson 2-4 (More About Linear Equations) #10-51 evens Lesson 3-1 (Solving Systems Using Tables & Graphs) #7-37 evens Lesson 3-2 (Solving Systems Algebraically) #10-52 multiples of 3 Lesson 3-3 (Systems of Inequalities) #8-19 evens Lesson 4-4 (Factoring Quadratic Expressions) #14-69 multiples of 3 Lesson 4-5 (Quadratic Equations) #9-17 evens Lesson 4-6 (Completing the Square) #19-45 multiples of 3 Lesson 4-7 (The Quadratic Formula) #11-22 evens Concept Byte for Lesson 6-1 (Properties of Exponents) #1-18 multiples of 3

3 Summer Math Assignment ALGEBRA 2/TRIGONOMETRY HONORS

4 Real Numbers

5 What are real numbers? A real number is any number that you can graph on a number line. Most of the numbers that you have worked with in Algebra 1 and Geometry are examples of real numbers. The symbol for real numbers is:

6 Classifying Real Numbers

7 Example Consider the following set of numbers. 1 3, 0,,.95,, 8, 16 2 List the numbers in the set that are: a. Natural Numbers b. Whole Numbers c. Integers d. Rational Numbers e. Irrational Numbers f. Real numbers

8 Order the following numbers from least to greatest. Graph them on a number line.

9 Properties of Real Numbers

10 Algebraic Expressions

11 An algebraic expression is a mathematical sentence that may contain numbers, variables, and different operations. Expressions do NOT have an equal sign (=). These are equations!

12 You can simplify an algebraic expression by combining like terms. Like terms have the same variables raised to the same powers. 5x 7x 12x

13 Example Simplify this Algebraic Expression 2 2 2( x 3 x) (5x 4) 9x

14 Solve the equation: 2(x 1) = 3(x+2)

15 Solve the equation: (2x 1)(x + 5) = (x + 2)(2x + 3)

16 Solving with Fractions

17 Solve for x. x 3 x 1 x

18 Solving a Formula for One of Its Variables

19 Solving Literal Equations A-Prt=P A=Prt+P A=P(rt+1) A P or rt+1 A P= rt+1

20 1 Solve for h in the area formula for a trapezoid. A= ( ) 2 h a b

21 Solving Linear Inequalities Reminder. The rules for solving equations apply EXCEPT when multiplying or dividing BOTH sides by a NEGATIVE value, the inequality symbol reverses. Let s review

24 Solve the inequality 5x 7 3x 1 and graph the solution set.

25 Compound Inequalities

26 Solve the inequality 4 5x 1 11 and graph the solution.

27 Solve and graph the solution set on a number line. 3 x 1 2 y x

There are two numbers that are the same distance from zero.

28 Absolute Value Refresher The absolute value of a number is its distance from zero on the number line. There are two numbers that are the same distance from zero. One is positive and one is negative. So to remove the absolute value from an equation I will have to use the following rule

30 Find the real solutions of the equation x

31 Solve: x NOTE: When solving an absolute value equation be sure to isolate the absolute value before removing the absolute value symbol.

32 Solve: 2 2x

33 These always confused me. Absolute Value Inequalities

34 This one is an and or intersection problem what they have that is the same. The solution to the inequality is intersection both sets.

35 This one is an or or union problem one set or the other set will make the inequality true. Both answers do not have to make it true at the same time.

36 Solve the inequality: x 8 Yay!!! This one doesn t require any extra work!

37 Solve the inequality: 3x 1 7, and graph the soluton set.

38 Solve the inequality: x 5. I wish they could all be this easy!

39 Solve and graph the solution set on a number line. 2x 5 3

40 Solve the inequality: 2x 3 4, and graph the soluton set.

41 AND I GOT IT RIGHT!!

43 Let s Review Relation Domain Range

44 Function A function is a relation where each number in the domain corresponds to exactly one number in the range.

45 Ways to define a relation/function Table Mapping diagram

46 Equation Graph

47 Vertical Line Test If any vertical line intersects a graph in more than one point, the graph does not define y as a function of x. or A graph in a coordinate plane represents a function if and only if no vertical line intersects the graph more than once.

48 Function Notation y 2x 3 y 2x 3 Using function notation becomes f ( x ) 2x 3

49 Some Examples 2 If f ( x ) 3x 5x 4 find f ( 1) 2 If f ( x ) x -6 find f ( a)

50 f x x x f a h 2 If ( ) 6 find ( ) 3 If g( x) find g( h 3) x 2

51 Solving Linear Inequalities Reminder. The rules for solving equations apply EXCEPT when multiplying or dividing BOTH sides by a NEGATIVE value, the inequality symbol reverses. Let s review

54 Solve the inequality 5x 7 3x 1 and graph the solution set.

55 Compound Inequalities

56 Solve the inequality 4 5x 1 11 and graph the solution.

57 Solve and graph the solution set on a number line. 3 x 1 2 y x

58 Linear Functions

59 Find the slope of the line that passes through (-2,5) and (3,-1) change in y m or change in x

60 Positive Slope Notes on Slope Negative Slope Zero Slope No Slope or Slope Undefined

61 Interpreting a real life situation The line graphs the percent of US adults who smoke cigarettes x years after a. Find the slope of the line segment from 1997 to 2007 Percent Adults (0,24.7) y m change in y change in x = 10 (10,19.5) =-.52 X years after 1997 x b. What does this slope represent? The percent of US adult cigarette smokers is decreasing by.52 percent each year. The change is consistent each year.

62 Of course I remember this duh

65 Intercepts The x-intercept LOOK: (x,0) The y-intercept LOOK: (0,y)

66 Graph the linear equation 3x + 2y = 6 by finding its intercepts. y (0,2) (3,0) x

69 Two forms for Equations of Lines Point Slope Form For a nonvertical line with slope m that passes through (x 1,y 1 ) the equation is y-y 1 = m(x-x 1 ) Example: slope = -3 point on the line(-1,-2) Y-(-2)= -3(x-(-1)) Y+2= -3(x+1) Slope Intercept Form For a nonvertical line with slope m and y- intercept b the equation is y=mx+b Example: slope =2 y-intercept of 6 Y=2x +6

70 Write the point-slope form of the equation of the line that passes through (-1,2) and (-4,5). Then solve for y. y x y x First: Find the slope Second: Substitue into the point-slope form. y-y m( x x ) 1 1 y 5 1( x 4) Third: Solve for y. y 5 1( x 4) y 5 1x 4 y=-1x+1

72 Find an equation of the line L, in slope intercept form, containing the points ( 1, 4) and (3, 1). Graph the line L.

74 A system of a equations is a set of two or more linear equations. A solution to the system is where the two equations intersect.

75 How to Solve a System of Equations There are THREE ways to solve a system of equations: 1. Graphing 2. Substitution 3. Elmination

76 1. Solving by Graphing To solve a system by graphing: 1. Write in slope-intercept form. 2. Graph. 3. Find the point of intersection, if it exists.

77 2. Solving by Substitution

78 Example Solve by the substitution method: 5x+y=9 -x+3y=11

79 Example Solve by the substitution method: 3x-2y=7 x+3y=6

80 3. Solving by Elimination

81 Example Solve the system: 3x 2y 7 6x 4y 9

82 Classifying Systems of Equations Consistent systems have at least one solution. Inconsistent systems have NO solutions.

83 Classify each system. Solve by the method of your choice. 6x-y=14 2x+3y=18 Solve by the method of your choice. 5x+2y=-1 x+3y=5

84 Classify the system.

85 Systems of Inequalities

86 Graphing a Linear Inequality 1. Write the inequality in slope-intercept form. 2. Graph the line. (Use a dotted line for < or >, solid line for or.) 3. Shade one side of the line. (Shade up for greater than, down for less than.)

87 Example

88 System of Inequalities The solution to a system of inequalities is where the shading of the two inequalities overlaps.

100

101

102 Quadratic Equation Techniques Ways to solve quadratic equations: Factoring, if possible Rooting, if possible Completing the square Quadratic Formula

103 2 Solve the equation by factoring: 2x 7x 4 0

104 Solve the equation: x 3 2x x 2

105 3 2 Solve the equation: x 3x 4x 12 0 Oh no!...

106 Find the real solutions of the equation 2 x 2 3 x

107

108 Solve the following problem by the square root property. 2 4x 7 0

109 Solve the following problem by the square root property. 2 (x-4) 25

110 I don t like this method!

111 Completing the Square

112 Complete the square to solve the following problem. 2 x 10x 3 0

113 Solve by completing the square. 2 3x 2x 4 0 NOTE: In order to complete the square the leading coefficient must be 1.

114

115 Solve the equation using the quadratic formula. 2 2x 4x 5

116 Properties of Exponents

117

118 Example Simplify: 5 3 7x 9x 5x 3 y 4x 5 y 2

119

120 Example Simplify: xy3 2x

121 Example Simplify: 48x y z 2 16xy 5 3 2

122 Example Simplify: z 5 xx 7 7

123

124

125 Example ( 5) 2 Simplify Exponential Expressions 5 2 3x 2 y 4

126 Example Simplify Exponential Expressions

127

128 Example b 5 4 Simplify Exponential Expressions y 4 5

129

130 Example 3x 4 3 Simplify Exponential Expressions 6x 3 y 2

131

132 Example Simplify Exponential Expressions x y x y 2 3 3

133

134 Example Simplify Exponential Expressions ( 2 xy) x y 2x y 4xy 2xy yz 2 6z 7 2

Algebra Summer Review Packet

Name: Algebra Summer Review Packet About Algebra 1: Algebra 1 teaches students to think, reason, and communicate mathematically. Students use variables to determine solutions to real world problems. Skills