ALGEBRA 2 CHAPTER ONE NOTES SECTION 1-1 REAL NUMBERS Objectives: Classify and order real numbers A is a collection of items called.

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1 ALGEBRA 2 CHAPTER ONE NOTES SECTION 1-1 REAL NUMBERS Objectives: Classify and order real numbers A is a collection of items called. A is a set whose elements belong to another set. The, denoted, is a set containing no elements. THE REAL NUMBER SYSTEM Using Venn Diagrams: RATIONAL NUMBERS can be expressed as a quotient (or ) of two integers, where the denominator is not zero. The decimal form of a rational number either IRRATIONAL NUMBERS, such as 2 and, be expressed as a quotient of two integers. Their decimal forms do not. However, you can approximate these numbers using terminating decimals. Order the numbers from least to greatest.

2 Numbers Real Rational Integer Whole Natural Irrational There are many ways to represent sets. For instance, 1. You can use to describe a set. 2. You can also use, in which the elements in a set are listed between braces, A An has a definite, or finite, number of elements. has an unlimited, or infinite number of elements. Using Number Lines to represent: The set of real numbers between 3 and 5, which is also an infinite set, can be represented on a number line or by an inequality. In the symbols [ and ] are used to an endpoint in an interval, and the symbols ( and ) are used to an endpoint from an interval. An interval that extends forever in the positive direction goes to and an interval that extends forever in the negative direction goes to negative Use interval notation to represent the set of numbers. 7 < x 12 All Real Numbers x -2

3 ALGEBRA 2 CHAPTER ONE NOTES SECTION 1-2 PROPERTIES OF REAL NUMBERS Objectives: Identify and use properties of real numbers. WORDS Additive Identity Property NUMBERS ALGEBRA NUMBERS WORDS Multiplicative Identity Property NUMBERS ALGEBRA WORDS Additive Inverse Property NUMBERS ALGEBRA WORDS Multiplicative Inverse Property NUMBERS ALGEBRA The opposite of any number a is The reciprocal of any nonzero number a is Find the additive and multiplicative inverse of each number

4 WORDS Commutative Property NUMBERS ALGEBRA WORDS Associative Property NUMBERS ALGEBRA WORDS Distributive Property NUMBERS ALGEBRA Identify the property demonstrated by each question = (12 ) = (9 12)

5 ALGEBRA 2 CHAPTER ONE NOTES SECTION 1-3 SQUARE ROOTS Objectives: Estimate square roots. Simplify, add, subtract, multiply, and divide square roots. Radical symbol. The number or expression under the radical symbol is called the. The radical symbol indicates only the square root of a number, called the. To indicate both the positive and negative square roots of a number, use the The perfect squares: Square roots of integers that are not perfect squares are. Estimate to the nearest tenth. Properties of Square Roots 1. Product Property 2. Quotient Property Simplify each expression. Adding and Subtracting Radicals.

7 ALGEBRA 2 CHAPTER ONE NOTES SECTION 1-4 ALGEBRAIC EXPRESSIONS Objectives: Simplify and evaluate algebraic expressions. Write an algebraic expression to represent each situation. A. the number of apples in a basket of 12 after n more are added B. the number of days it will take to walk 100 miles if you walk M miles per day C. Lucy s age y years after her 18 th birthday D. the number of seconds in h hours One way to remember the order of operations is by using the mnemonic PEMDAS. Order of Operations 1. Parentheses and grouping symbols. 2. Exponents. 3. Multiply and Divide from left to right. 4. Add and Subtract from left to right. Evaluate the expression for the given values of the variables. 2x xy + 4y for x = 5 and y = 2 q 2 + 4qr r 2 for r = 3 and q = 7 x 2 y xy 2 + 3y for x = 2 and y = 5. have the same variables raised to the same exponents. are like terms that always have the same value.

8 Simplify the expression. 3x 2 + 2x 3y + 4x 2 6k 2 + 7k) + 9jk 2 7jk 3(2x xy + 3y) 11xy A travel agent is selling 100 discount packages. He makes $50 for each Hawaii package and $80 for each Cancún package. Write an algebraic expression to represent each situation. 1. the number of cards in a 52-card deck after n cards are dealt. 2. the number of days in W weeks 3. Evaluate: rq 2 + 3qr + r 2 for r = 5 and q = Simplify: 5h 2 + 4k + 7h 4h 2.

9 ALGEBRA 2 CHAPTER ONE NOTES SECTION 1-5 EXPONENTS Objectives: Simplify expressions involving exponents. Use scientific notation. In an expression of the form a n, a is the, n is the, and the quantity a n is called a. The exponent indicates the number of times that the. Write the expression in expanded form. s 4 (5z) 2 (2x 1) 3 y 2 3h 3 (k + 3) 2 Properties of Exponents 1. Zero Exponents 2. Negative Exponents

10 3. Product Property 4. Quotient Property 5. Power Property 6. Product to a Power 7. Quotient to a Power Simplify the expression. 3 2 ( 5) 5 ( 2a 3 b) 3 (5x 6 ) 3 3z 7 ( 4z 2 ) is a method of writing numbers by using powers of 10. In scientific notation, a number takes a form m 10 n, where and n is an integer. Simplify the expression. Write the answer in scientific notation. ( )( ) ( )( )

ALGEBRA 2 CHAPTER ONE NOTES SECTION 1-6 Relations and Functions Objectives: Identify the domain and range of relations and functions. Determine whether a relation is a function.

11 ALGEBRA 2 CHAPTER ONE NOTES SECTION 1-6 Relations and Functions Objectives: Identify the domain and range of relations and functions. Determine whether a relation is a function. A is a pairing of input values with output values. It can be shown as a set of ordered pairs (x,y), where x is an and y is an. The set of input values for a relation is called the The set of output values is called the Mapping Diagram Give the domain and range for this relation: {(100,5), (120,5), (140,6), (160,6), (180,12)}. Give the domain and range for the relation shown in the graph. A relation in which the first coordinate is never repeated is called a. In a, there is output for each input, so each element of the domain is mapped to exactly one element in the range. Determine whether each relation is a function. A. from the items in a store to their prices on a certain date B. from types of fruits to their colors

: If a vertical line passes through on the

whether the relation is a If not, identify

12 : If a vertical line passes through on the graph of a relation, the relation is a function. Use the vertical-line test to determine whether the relation is a function. If not, identify two points a vertical line would pass through. Use the vertical-line test to determine whether the relation is a function. If not, identify two points a vertical line would pass through.

This is function notation: ALGEBRA 2 CHAPTER ONE NOTES SECTION 1-7 Function Notation Objectives: Write functions using function notation. Evaluate and graph functions.

13 This is function notation: ALGEBRA 2 CHAPTER ONE NOTES SECTION 1-7 Function Notation Objectives: Write functions using function notation. Evaluate and graph functions. It is read: Some sets of ordered pairs can be described by using an equation. When the set of ordered pairs described by an equation satisfies the definition of a function, the equation can be written in. f(x) is f times x or f multiplied by x. f(x) means So f(1) represents the value of f at x =1 For each function, evaluate ƒ(0), ƒ(.5), and ƒ(-2). ƒ(x) = 8 + 4x For each function, evaluate ƒ(0), ƒ(.5), and ƒ( 2). For each function, evaluate ƒ(0), ƒ(.5),and ƒ( 2). ƒ(x) = x 2 4x The output ƒ(x) of a function is called the it depends on the input value of the function. The input x is called the. because

14 Graph the function. {(0, 4), (1, 5), (2, 6), (3, 7), (4, 8)} A function whose graph is made up of points is called a function. Graph the function f(x) = 3x 1. Graph the function A carnival charges a $5 entrance fee and $2 per ride. Write a function to represents the cost. What is the cost if 12 rides were ridden? A local photo shop will develop and print the photos from a disposable camera for $0.27 per print. Write a function describing the cost.

15 ALGEBRA 2 CHAPTER ONE NOTES SECTION 1-8 Translations and Transformations Objectives: Perform Translations and Transformations of Functions Perform the given translation on the point ( 3, 4). Give the coordinates of the translated point. 5 units right 2 units left and 2 units down A is a transformation that flips a figure across a line called the. Each reflected point is the from the line of reflection, but on the side of the line. Reflections You can transform a function by its ordered pairs. When a function is translated or reflected, the original graph and the graph of the transformation are because the size and shape of the graphs are the same. Use a table to perform each transformation of y=f(x). Use the same coordinate plane as the original function. translation 2 units up reflection across x-axis

Stretches and compressions are not congruent to the original graph.

16 If you pull the points away from the y-axis, you would create a horizontal of the graph. If you push the points towards the y-axis, you would create a horizontal. Stretches and compressions are not congruent to the original graph. Use a table to perform a horizontal stretch of the function y = f(x) by a factor of 3. Graph the function and the transformation on the same coordinate plane. Use a table to perform a vertical stretch of y = f(x) by a factor of 2. Graph the transformed function on the same coordinate plane as the original figure.

17 ALGEBRA 2 CHAPTER ONE NOTES SECTION 1-9 Intro to Parent Functions Objectives: Identify parent functions from graphs and equations. Use parent functions to model real-world data and make estimates for unknown values. Similar to the way that numbers are classified into sets based on common characteristics, functions can be classified into The is the simplest function with the defining characteristics of the family. Functions in the same family are transformations of their parent function. The Constant Function: f(x) = c The Linear Function: f(x) = x Domain: Domain: Range: Range: The Quadratic Function: f(x) = x 2 The Cubic Function: f(x) = x 3 Domain: Domain: Range: Range The Square Root Function: f(x) = x 3 Domain: Range:

18 Identify the parent function for g from its function rule. g(x) = x g(x) = x 3 2 g(x) = ( x) 2 Graph the data from this set of ordered pairs. Describe the parent function and the transformation that best approximates the data set. x y {( 2, 12), ( 1, 3), (0, 0), (1, 3), (2, 12)}

Summary for a n = b b number of real roots when n is even number of real roots when n is odd

Summary for a n = b b number of real roots when n is even number of real roots when n is odd Day 15 7.1 Roots and Radical Expressions Warm Up Write each number as a square of a number. For example: 25 = 5 2. 1. 64 2. 0.09 3. Write each expression as a square of an expression. For example: 4. x