Welcome to Algebra 2/Trig H 2018-2019 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.
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, 51-59 odds, 60-65 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, 59-65 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
Summer Math Assignment ALGEBRA 2/TRIGONOMETRY HONORS
Real Numbers
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:
Classifying Real Numbers
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
Order the following numbers from least to greatest. Graph them on a number line.
Properties of Real Numbers
Algebraic Expressions
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!
You can simplify an algebraic expression by combining like terms. Like terms have the same variables raised to the same powers. 5x 7x 12x
Example Simplify this Algebraic Expression 2 2 2( x 3 x) (5x 4) 9x
Solve the equation: 2(x 1) = 3(x+2)
Solve the equation: (2x 1)(x + 5) = (x + 2)(2x + 3)
Solving with Fractions
Solve for x. x 3 x 1 x 2 5 2 5
Solving a Formula for One of Its Variables
Solving Literal Equations A-Prt=P A=Prt+P A=P(rt+1) A P or rt+1 A P= rt+1
1 Solve for h in the area formula for a trapezoid. A= ( ) 2 h a b
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
Solve the inequality 5x 7 3x 1 and graph the solution set.
Compound Inequalities
Solve the inequality 4 5x 1 11 and graph the solution.
Solve and graph the solution set on a number line. 3 x 1 2 y x
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
Find the real solutions of the equation x - 3 2.
Solve: x 5 9 7 NOTE: When solving an absolute value equation be sure to isolate the absolute value before removing the absolute value symbol.
Solve: 2 2x 7 14 0
These always confused me. Absolute Value Inequalities
This one is an and or intersection problem what they have that is the same. The solution to the inequality is intersection both sets.
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.
Solve the inequality: x 8 Yay!!! This one doesn t require any extra work!
Solve the inequality: 3x 1 7, and graph the soluton set.
Solve the inequality: x 5. I wish they could all be this easy!
Solve and graph the solution set on a number line. 2x 5 3
Solve the inequality: 2x 3 4, and graph the soluton set.
AND I GOT IT RIGHT!!
Let s Review Relation Domain Range
Function A function is a relation where each number in the domain corresponds to exactly one number in the range.
Ways to define a relation/function Table Mapping diagram
Equation Graph
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.
Function Notation y 2x 3 y 2x 3 Using function notation becomes f ( x ) 2x 3
Some Examples 2 If f ( x ) 3x 5x 4 find f ( 1) 2 If f ( x ) x -6 find f ( a)
f x x x f a h 2 If ( ) 6 find ( ) 3 If g( x) find g( h 3) x 2
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
Solve the inequality 5x 7 3x 1 and graph the solution set.
Compound Inequalities
Solve the inequality 4 5x 1 11 and graph the solution.
Solve and graph the solution set on a number line. 3 x 1 2 y x
Linear Functions
Find the slope of the line that passes through (-2,5) and (3,-1) change in y 5 1 6 6 m or change in x 2 3 5 5
Positive Slope Notes on Slope Negative Slope Zero Slope No Slope or Slope Undefined
Interpreting a real life situation The line graphs the percent of US adults who smoke cigarettes x years after 1997. 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 19.5 24.7 10 0-5.2 = 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.
Of course I remember this duh
Intercepts The x-intercept LOOK: (x,0) The y-intercept LOOK: (0,y)
Graph the linear equation 3x + 2y = 6 by finding its intercepts. y (0,2) (3,0) x
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
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 5 2 3 x 4 1 3 2 1 First: Find the slope. 1 2 1 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
Find an equation of the line L, in slope intercept form, containing the points ( 1, 4) and (3, 1). Graph the line L.
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.
How to Solve a System of Equations There are THREE ways to solve a system of equations: 1. Graphing 2. Substitution 3. Elmination
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.
2. Solving by Substitution
Example Solve by the substitution method: 5x+y=9 -x+3y=11
Example Solve by the substitution method: 3x-2y=7 x+3y=6
3. Solving by Elimination
Example Solve the system: 3x 2y 7 6x 4y 9
Classifying Systems of Equations Consistent systems have at least one solution. Inconsistent systems have NO solutions.
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
Classify the system.
Systems of Inequalities
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.)
Example
System of Inequalities The solution to a system of inequalities is where the shading of the two inequalities overlaps.
Quadratic Equation Techniques Ways to solve quadratic equations: Factoring, if possible Rooting, if possible Completing the square Quadratic Formula
2 Solve the equation by factoring: 2x 7x 4 0
Solve the equation: x 3 2x x 2
3 2 Solve the equation: x 3x 4x 12 0 Oh no!...
Find the real solutions of the equation 2 x 2 3 x 2 2 0.
Solve the following problem by the square root property. 2 4x 7 0
Solve the following problem by the square root property. 2 (x-4) 25
I don t like this method!
Completing the Square
Complete the square to solve the following problem. 2 x 10x 3 0
Solve by completing the square. 2 3x 2x 4 0 NOTE: In order to complete the square the leading coefficient must be 1.
Solve the equation using the quadratic formula. 2 2x 4x 5
Properties of Exponents
Example Simplify: 5 3 7x 9x 5x 3 y 4x 5 y 2
Example Simplify: 2 2 7 3 4 8xy3 2x
Example Simplify: 48x y z 2 16xy 5 3 2
Example Simplify: 6 5 0 6 z 5 xx 7 7
Example ( 5) 2 Simplify Exponential Expressions 5 2 3x 2 y 4
Example Simplify Exponential Expressions 4 4 1 7 2
Example b 5 4 Simplify Exponential Expressions 3 2 2 y 4 5
Example 3x 4 3 Simplify Exponential Expressions 6x 3 y 2
Example Simplify Exponential Expressions x y 4 3 2 5x y 2 3 3
Example Simplify Exponential Expressions ( 2 xy) 3 4 2 8 3 8 7x y 2x y 4xy 2xy 3 7 5 4 54yz 2 6z 7 2