LESSON #1  BASIC ALGEBRAIC PROPERTIES COMMON CORE ALGEBRA II


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1 1 LESSON #1  BASIC ALGEBRAIC PROPERTIES COMMON CORE ALGEBRA II Mathematics has developed a language all to itself in order to clarif concepts and remove ambiguit from the analsis of problems. To achieve this, though, we have to agree on basic definitions so that we can all speak this same language. So, we start our course in Algebra II with some basic review of concepts that ou saw in Algebra I. These definitions and properties will be used throughout the ear. SOME BASIC DEFINITIONS Variable: A quantit that is represented b a letter or smbol that is unknown, unspecified, or can change within the contet of a problem. Terms: A single number or combination of numbers and variables using eclusivel multiplication or division. This definition will epand when we introduce higherlevel functions. Epression: A combination of terms using addition and subtraction. Equation: Two epressions that are set equal to each other. To solve an equation, find the value(s) of the given variable(s). If a, b, and REAL NUMBER PROPERTIES c are an real numbers then the following properties are alwas true: 1. The Commutative Properties of Addition and Multiplication: a b b a and ab b a. The Associative Properties of Addition and Multiplication: a b c a b c and abc ab c. The Distributive Propert of Multiplication and Division Over Addition and Subtraction: ca b ca c b and a b a b c c c
2 Eercise #1: The procedure for simplifing the linear epression the real number propert that justifies each step is shown below. State Eercise #: Simplif each of the following epressions b combining like terms. Be careful to onl combine terms that have the same variables and powers. (a) (b) (c) 4 9 (d) Eponents, at their most basic, represent repeated multiplication. The wa the combine, or don't combine, is dictated b this simple premise. Eercise #: The following four steps are given to find the product of the monomials (1) and 4. (a) For steps (1) through (), write the real number propert that justifies each manipulation. () () (b) Eplain wh the final eponent on the variable is 7. (4) 8 7
3 Students (and teachers) can forget the basic properties used in simplifing the product of two monomials because we tend to pick up on the pattern of multipling the numerical coefficients and adding the powers without thinking about the commutative and associative properties that justif our manipulations. Eercise #4: Find the product of each of the following monomials. 6 4 (a) 5 (b) 6 (c) (d) 4 Remember, monomials (or terms) can have more than one variable, just as long as the are all combined using multiplication and division onl. Multipling monomials that contain more than one variable still just involves application of eponent laws and repeated use of the associative and commutative properties. Eercise #5: Find each of the following products, which involve monomials of multiple variables. Carefull consider what ou are doing before appling patterns. 5 (a) 4 5 (b) (c) One of the ke skills we will need this ear will be factoring epressions, especiall factoring out a common factor. To build some skills with this, consider the following problem. Eercise #6: Fill in the missing blank in each of the following equations involving a product such that the equation is then an identit. (An identit is an equation that is alwas true for an value of the variable(s).) 5 8 (a) 6 (b) 1 4 (c) 0 4 The final skill we will review in this lesson is using the distributive propert of multiplication (and division) over addition (and subtraction). Eercise #7: Use the distributive propert to multipl the following monomials and polnomials. (a) 5 (b) 5 6 (c) 7 (d) (e) 4 4
4 4 Now, to build our wa up to factoring in later units, let's make sure we can fill in missing portions of products. Eercise #8: Similar to Eercise #4, fill in the missing portion of each product so that the equation is an identit. (a) (b) (c) (d) 4 9
5 5 LESSON #1  BASIC ALGEBRAIC PROPERTIES COMMON CORE ALGEBRA II HOMEWORK FLUENCY 1. Simplif each of the following epressions b combining like terms. Be careful to onl combine terms that have the same variables and powers. (a) (b) (c) 8 5 (d) Find each of the following products of monomials. 4 (a) 10 5 (b) (c) 4 8 (d) (e) 4t 15t (f) (g) 1 (h) Fill in the missing portion of each product to make the equation an identit. 6 (a) 18 (b) (c) (d) 4 (e) (f)
6 6 4. Use the distributive propert to write each of the following products as polnomials. (a) 45 (b) 510 (c) (d) 10 8 (e) (f) 8 5 (g) (h) 16t t t (i) 1 5. Fill in the missing part of each product in order to make the equation into an identit. (a) (b) 8 10 (c) 18t 45t 5 9 t (d) (e) (f) REASONING Another ver important eponent propert occurs when we have a monomial with an eponent then raised to et another power. See if ou can come up with a general pattern. 6. Write each of the following out as etended products and then simplif. The first is done as an eample. (a) 6 (b) 5 (c) 4 4 (d) a b 7. So, what is the pattern? For positive integers a and b:?
7 7 LESSON #  MULTIPLYING POLYNOMIALS COMMON CORE ALGEBRA II Polnomials are epressions that are mainl combinations of terms with both addition and subtraction that can have onl constants and positive integer powers. Eercise #1: Use the distributive propert first and then combine each of the following linear epressions into a single, equivalent binomial epression. (a) (b) Eercise #: Which of the following is equivalent to the epression (1) 8 () ? () 5 1 (4) 10 1 In this lesson, we will look at using the distributive propert more than once to multipl polnomials b themselves. Let's start b looking at the product of binomials. Eercise #: Simplif each of the following products. ()( 5) ( )( )
8 8 Eercise #4: Find the product of the binomial 4 with the trinomial 5. An identit is an equation that is alwas true. To prove an equation is true for all real numbers, one or both sides must be manipulated until the same epression is on both sides. Eercise #5: Prove the each equation is an identit. (a) (b) a b ( a b)( a ab b ) (c)
9 Eercise #6: The product of three binomials, just like the product of two, can be found with repeated applications of the distributive propert. (a) Find the product: (b) For what three values of will the cubic polnomial that ou found in part (a) have a value of zero? What famous law is this known as? (c) Test one of the three values ou found in (b) to verif that it is a zero of the cubic polnomial. Eercise #7: Simplif ( )
10 10 LESSON #  MULTIPLYING POLYNOMIALS COMMON CORE ALGEBRA II HOMEWORK FLUENCY 1. Multipl the following binomials and epress each product as an equivalent trinomial. (a) 5 8 (b) 7 (c) 5 (d) 4 10 (e) 15 4 (f) 9. Find each of the following products in equivalent form. (a) 5 (b) (c) 5
11 11. Prove the each equation is an identit. (a) (b) a b a a b ab b (c) 1 1 REASONING 4. Epression 8 4. (a) For what values of will this epression be equal to zero? Show how ou arrived at our answer. (b) Write this product as an equivalent trinomial. (c) Show that this trinomial is also equal to zero at the larger value of from part (a).
12 1 LESSON #  SOLVING LINEAR EQUATIONS COMMON CORE ALGEBRA II We will learn man new equation solving techniques in Algebra II, but the most basic of all equations are those where the variable, sa, is onl raised to the first power. These are known as linear equations. You need to have good fluenc with solving these equations in order to be successful in the beginning portions of Algebra II. Let's start with some practice. Eercise #1: Solve each of the following linear equations for the value of. (a) 5 6 (b) (c) 8 6 (d) When finding intersection of two lines from both Algebra I and Geometr, ou first set the linear equations equal to each other. Eercise #: Find the intersection point of the two lines whose equations are shown below. Be sure to find both the and coordinates. 6 4 and
13 Strange things can sometimes happen when solving linear (and other) equations. Sometimes we get no solutions at all, in which case the equation is known as inconsistent. Other times, an value of will solve the equation, in which case it is known as an identit. Eercise #: Tr to solve the following equation. State whether the equation is an identit or inconsistent. Eplain Eercise #4: An identit is an equation that is true for all values of the substitution variable. Tring to solve them can lead to confusing situations. Consider the equation: 6 1 (a) Test the values of 5 and in this equation. Show that the are both solutions. (b) Attempt to solve the equation until ou are sure this is an identit. Eercise #5: Which of the following equations are identities, which are inconsistent, and which are neither? (a) (b) (c) 8 7 (d) 1 1
14 14 The following equations have two variables, and. The process is the same when solving for a variable in one of these equations using inverse operations to isolate the desired variable in terms of another variable. Eercise #6: Solve each of the following equations for in terms of. (a) 7 (b) 5 9 The following two equations are a preview of a more difficult situation where ou will be solving for in terms of later in the ear. We will not be testing this skill this unit, but it will be helpful to have seen the process so it is familiar when ou have to appl it. Eercise #7: Solve each of the following equations for in terms of. (a) 6 5 (b) 1
15 15 LESSON #  SOLVING LINEAR EQUATIONS COMMON CORE ALGEBRA II HOMEWORK FLUENCY 1. Solve each of the following linear equations. If the equation is inconsistent, state so. If the equation is an identit, also state so. Reduce an noninteger answers to fractions in simplest form. (a) 75 5 (b) 7 5 (c) (d) (d) 1 9 (e) (g) 5 (Cross multipl to begin) 6 18 (f) (h) (i) 18 7
16 16. Solve for in terms of. (a) 4 1 (b) 9 0 (c) (d) 6 6 APPLICATIONS. When finding the intersection of two lines from both Algebra I and Geometr, ou first "set the linear equations equal" to each other. Find the intersection point of the two lines whose equations are shown below. Be sure to find both the and coordinates. 5 1 and 11 REASONING 4. Eplain wh ou cannot find the intersection points of the two lines shown below. Give both an algebraic reason and a graphical reason. 4 1 and 4 10
17 17 LESSON #4  INTRODUCTION TO FUNCTIONS COMMON CORE ALGEBRA II Most higher level mathematics is built upon the concept of a function. Like most of the important concepts in mathematics, the definition of a function is simple to the point of being easil overlooked. Make sure to know the following definition: DEFINITION: A function is an rule that assigns eactl one output value (value) for each input value (value). These rules can be epressed in different was, the most common being equations, graphs, and tables of values. We call the input variable independent and output variable dependent. Eercise #1: An internet music service offers a plan whereb users pa a flat monthl fee of $5 and can then download songs for 10 cents each. (a) What are the independent and dependent variables in this scenario? Independent: Dependent: (b) Fill in the table below for a variet of independent values: Number of downloads, Amount Charged, (c) Let the number of downloads be represented b the variable and the amount charged be represented b the variable, write an equation that models as a function of (d) Based on the equation ou found in part (c), produce a graph of this function for all values of on the interval Use a calculator TABLE to generate additional coordinate pairs to the ones ou found in part (b). Amount Charged, Number of Downloads, 0 40
18 18 Eercise #: One of the following graphs shows a relationship where is a function of and one does not. (a) Draw the vertical line whose equation is on both graphs. (b) Give all output values for each graph at an input of. Relationship A: Relationship B: (c) Eplain which of these relationships is a function and wh. Relationship A Relationship B Eercise #: The graph of the function (a) State this function s intercept. 4 1 is shown below. (b) Between what two consecutive integers does the larger  intercept lie? (c) Draw the horizontal line on this graph. (d) Using these two graphs, find all values of that solve the equation below: 41 (e) Verif that these values of are solutions b using STORE on our graphing calculator.
19 19 A function f ONETOONE FUNCTIONS is called onetoone if different inputs give different outputs. Note, the relation must be a function first to be onetoone. Eercise #4: Of the four tables below, one represents a relationship where is a onetoone function of. Determine which it is and eplain wh the others are not. (1) () () (4) Eercise #5: Consider the following four graphs which show a relationship between the variables and. (1) () () (4) (a) Circle the two graphs above that are functions. Eplain how ou know the are functions. (b) Of the two graphs ou circled, which is onetoone? Eplain how ou can tell from its graph.
20 0 THE HORIZONTAL LINE TEST If an given horizontal line passes through the graph of a function at most one time, then that function is onetoone. This test works because horizontal lines represent constant values; hence, if a horizontal line intersects a graph more than once, an output has been repeated. Eercise #6: Which of the following represents the graph of a onetoone function? (1) () () (4) Eercise #7: The distance that a number,, lies from the number 5 on a onedimensional number line is given b the function D 5 D is not a onetoone function.. Show b eample that
21 1 LESSON #4  INTRODUCTION TO FUNCTIONS COMMON CORE ALGEBRA II HOMEWORK FLUENCY 1. Determine for each of the following graphed relationships whether is a function of using the Vertical Line Test. (a) (b) (c) (d) (e) (f). What are the outputs for an input of 5 given functions defined b the following formulas: (a) 4 (b) 50 (c)
22 . Which of the following graphs illustrates a onetoone relationship? (1) () () (4) 4. Which of the following graphs does not represent that of a onetoone function? (1) () () (4) 5. In which of the following formulas is the variable a onetoone function of the variable? (Hint tr generating some values either in our head or using TABLES on our calculator.) (1) () () (4) 5 6. Which of the following tables illustrates a relationship in which is a onetoone function of? (1) () () (4)
23 APPLICATIONS 7. Evin is walking home from the museum. She starts 8 blocks from home and walks blocks each minute. Evin s distance from home is a function of the number of minutes she has been walking. (a) Which variable is independent and which variable is dependent in this scenario? (b) Fill in the table below for a variet of time values. Time, t, in minutes Distance from home, D, in blocks (c) Determine an equation relating the distance, D, that Evin is from home as a function of the number of minute, t, that she has been walking. (d) Determine the number of minutes, t, that it takes for Evin to reach home. REASONING 8. In one of the following tables, the variable is a function of the variable. Eplain which relationship is a function and wh the other is not Relationship # Relationship #
24 LESSON #5  LESSON FUNCTION NOTATION COMMON CORE ALGEBRA II Functions are fundamental tools that convert inputs, values of the independent variable, to outputs, values of the dependent variable. There is a special notation that is commonl used to show this conversion process. The first eercise will illustrate this notation in the contet of formulas. Eercise #1: Evaluate each of the following given the function definitions and input values. 4 (a) f 5 (b) g 4 (c) h f g h f g 0 h Although this notation could be confused with multiplication, the contet will make it clear that it is not. The idea of function notation is summarized below. FUNCTION NOTATION Recall that function rules commonl come in one of three forms: (1) equations (as in Eercise #1), () graphs, and () tables. The net few eercises will illustrate function notation with these three forms. Eercise #: Boiling water at 1 degrees Fahrenheit is left in a room that is at 65 degrees Fahrenheit and begins to cool. Temperature readings are taken each hour and are given in the table below. In this scenario, the temperature, T, is a function of the number of hours, h. f Output Rule Input h (hours) T h F (a) Evaluate T and T 6. (b) For what value of h is 76 T h? (c) Between what two consecutive hours will Th 100?
25 5 Eercise #: Given the function h, when does h ( ) 8? Show our table to support our work. Eercise #4: Given the function f 5 7, when does f( )? Find the value of algebraicall. Check our answer with a table. Eercise #5: Given the functions h and does h( ) j( )? 1 j( ) 14, for what positive value of Eercise #6: The function based on this graph. (a) Evaluate f 1, f 1, and f 5. f is defined b the graph shown below. Answer the following questions (b) Evaluate f 0. What special feature on a graph does f 0 alwas correspond to? (c) What values of solve the equation f 0. What special features on a graph does the set of values that solve f 0 correspond to? (d) Between what two consecutive integers does the largest solution to f lie?
26 6 Eercise #7: For a function the graph of g? g it is known that g 7. Which of the following points must lie on (1) 7, () 0, 7 (), 7 (4), 0 Eercise #8: A ball is shot from an aircannon at an angle of 45 with the horizon. It travels along a path 1 given b the equation hd d d, where h represents the ball s height above the ground and d 50 represents the distance the ball has traveled horizontall. Using our calculator to generate a table of values, graph this function for all values of d on the interval 0 d 50. Look at the table to properl scale the ais. What is the maimum height that the ball reaches? At what value of d does it reach this height?
27 7 FLUENCY LESSON #5  FUNCTION NOTATION COMMON CORE ALGEBRA II HOMEWORK 1. Without using our calculator, evaluate each of the following given the function definitions and input values. (a) f 7 (b) g (c) h 5 f 4 g h 41 f g h 14. Using STORE on our calculator, evaluate each of the following more comple functions. (a) f (b) g 5 (c) h 01. f 5 g 4 h f 0 g h 0. Based on the graph of the function g (a) Evaluate g, g 0, g and g 7. shown below, answer the following questions. (b) What values of solve the equation g 0 (c) Graph the horizontal line on the grid above and label. (d) How man values of solve the equation g?
28 8 APPLICATIONS 4. Ian invested $500 in an investment vehicle that is guaranteed to earn 4% interest compounded earl. The amount of mone, A, in his account as a function of the number of ears, t, since creating the account is given b the equation At t. (a) Evaluate A0 and A 10. (b) What do the two values that ou found in part (a) represent? (c) Using tables on our calculator, determine, to the nearest whole ear, the value of t that solves the equation At Justif our answer with numerical evidence. (d) What does the value of t that ou found in part (b) represent about Ian s investment? 5. Phsics students drop a ball from the top of a 50 foot high building and model its height as a function of time with the equation ht 50 16t. (a) Graph the function for all values of t on the interval 0t. Show our table to support our answer. (Hint: Set tbl to 0.1) (b) Determine, to the nearest tenth of a second, when the ball hits the ground.
29 9 LESSON #6  THE DOMAIN AND RANGE OF A FUNCTION COMMON CORE ALGEBRA II Because functions convert values of inputs into value of outputs, it is natural to talk about the sets that represent these inputs and outputs. The set of inputs that result in an output is called the domain of the function. The set of outputs is called the range. Eercise #1: Consider the function that has as its inputs the months of the ear and as its outputs the number of das in each month. In this case, the number of das is a function of the month of the ear. Assume this function is restricted to nonleap ears. (a) Write, in roster form, the set that represents this function s domain. (b) Write, in roster form, the set that represents this function s range. Eercise #: State the range of the function f n n 1 if its domain is the set and range in the mapping diagram below. Domain of f f n 1,, 5. Show the domain Range of f Eercise #: The function g is completel defined b the graph shown below. Answer the following questions based on this graph. (a) Determine the minimum and maimum values represented on this graph. (b) Determine the minimum and maimum values represented on this graph. (c) State the domain and range of this function using set builder notation.
30 Some functions, defined with graphs or equations, have domains and ranges that stretch out to infinit. Consider the following eercise in which a standard parabola is graphed. Eercise #4: The function f 1 is graphed on the grid below. Which of the following represent its domain and range written in interval notation? 0 (1) Domain:, 4 Range: 4, 6 () Domain:, 4 Range: 4, () Domain:, Range: 4, (4) Domain:, 4 Range: 4, 6 f Eercise #5: Determine if each relation is a function. State the domain and range of each relation in either roster, set builder, or interval notation.
31 For most functions defined b an algebraic formula, the domain consists of the set of all real numbers, given the concise smbol. Sometimes, though, there are restrictions placed on the domain of a function b the structure of its formula. Two basic restrictions will be illustrated in the net few eercises. Eercise #6: The function f (a) Evaluate f 1 and f 6 from the table. (b) Wh does the calculator give an ERROR at 4? 1 has outputs given b the following calculator table. 4 (c) Are there an values ecept 4 that are not in the domain of f? Eplain. Eercise #7: Which of the following values of would not be in the domain of the function 4? Eplain our answer. (1) 0 () () 5 (4) 8 f Error
32 FLUENCY LESSON #6  THE DOMAIN AND RANGE OF A FUNCTION COMMON CORE ALGEBRA II HOMEWORK 1. A function is given b the set of ordered pairs in roster form., 5, 4, 9, 6,1, 8,17. Write its domain and range Domain: Range:. The function h 5 maps the domain given b the set, 1, 0,1, represents the range of h? (1) 0, 6,10,1 () 5, 6, 9 () 5, 6, 7 (4) 1, 4, 5, 6, 9. Which of the following sets. Which of the following values of would not be in the domain of the function defined b f (1) () () (4) 4. Determine an values of that do not lie in the domain of the function f response.?. Justif our Which of the following values of does lie in the domain of the function defined b g 7 (1) 0 () () (4) 5? 6. Which of the following would represent the domain of the function 6? (Hint: Look at the table) (1) : () : () : (4) :
33 7. The function f is completel defined b the graph shown below. (a) Evaluate f 4, f, and f 6. (b) Draw a rectangle that circumscribes (just surrounds) the graph. (c) State the domain and range of this function using interval notation. Domain: Range: 8. Which of the following represents the range of the quadratic function shown in the graph below? (1) 4, (),4 (),4 (4) 4, APPLICATIONS 9. A child starts a pigg bank with $. Each da, the child receives 5 cents at the end of the da and puts it in A d d the bank. If A represents the amount of mone and d stands for the number of das then 0.5 gives the amount of mone in the bank as a function of das (think about this formula). (a) Evaluate A1, A7, and A 0. (b) For what value of d will Ad $ (c) Eplain wh the domain does not contain the value d.5. (d) Eplain wh the range does not include the value A $.10.
34 4 10. Find the Domain and Range of each relation in either interval or set builder notation.
35 5 LESSON #7  KEY FEATURES OF FUNCTIONS DAY 1 COMMON CORE ALGEBRA II The graphs of functions have man ke features whose terminolog we will be using all ear. It is important to master this terminolog, most of which ou learned in Common Core Algebra I. Eercise #1: The function f is shown graphed to the right. Answer the following questions based on this graph. (a) State the intercept of the function. (b) State the intercepts of the function. What is the alternative name that we give the intercepts? is f (c) Over the interval 1 How can ou tell? increasing or decreasing? f (d) Give the interval over which f 0. What is a quick wa of seeing this visuall? (e) State all the coordinates of the relative maimums and relative minimums. Label each. (f) What are the absolute maimum and minimum values of the function? Where do the occur? (g) State the domain and range of interval notation. f using (h) If a second function g is defined b the formula g 1 f, then what is the intercept of g?
36 Eercise #: Consider the function g 1 8 the domain 4 7. (a) Sketch a graph of the function to the right. defined over 6 (b) State the domain interval over which this function is decreasing. (c) State zeroes of the function on this interval. (d) State the interval over which g 0 g 0 b using the algebraic definition of the function. What point does this correspond to on the graph? (e) Evaluate (f) Are there an relative maimums or minimums on the graph? If so, which and what are their coordinates? You need to be able to think about functions in all of their forms, including equations, graphs, and tables. Tables can be quick to use, but sometimes hard to understand. Eercise #: A continuous function f has a domain of 6 1 with selected values shown below. The function has eactl two zeroes and has eactl two turning points, one at, 4 and one at 9, f (a) State the interval over which f 0. (b) State the interval over which f increasing. is
37 7 LESSON #7  KEY FEATURES OF FUNCTIONS DAY 1 COMMON CORE ALGEBRA II HOMEWORK 1. Is g() a function?. Find g(5). g(). When does g()=? 4. Over what interval(s) is g() increasing? 5. Over what interval(s) is g() decreasing? 6. What are the coordinates of the turning points of g()? State whether each is a relative maimum or minimum. 7. What are the maimum and minimum values of g()? 8. What is the intercept? 9. What are the intercept(s) or zero(s)? 10. What is the domain? 11. What is the range? 1. Over what interval(s) is g ( ) 0? 1. Over what interval(s) is g ( ) 0? COMMON CORE ALGEBRA II, UNIT #1 ESSENTIAL ALGEBRA CONCEPTS
38 8 LESSON #8  KEY FEATURES OF FUNCTIONS DAY COMMON CORE ALGEBRA II Eercise #1: The piecewise linear function f is shown to the right. Answer the following questions based on its graph. (a) Evaluate each of the following based on the graph: (i) f 4 (ii) f (b) State the zeroes of f. (c) Over which of the following intervals is increasing? f alwas (1) 7, () 5,5 (),5 (4) 5, (d) State the coordinates of the relative maimum and the relative minimum of this function. Relative Maimum: Relative Minimum: (e) Over which of the following intervals is the value of the function negative? (1) 7, () 5, (),7 (4) 5, g is defined using the rule g f 5. Evaluate g 0 using (f) A second function this rule. What does this correspond to on the graph of g? (g) A third function h formula g h answer. h is defined b the. What is the value of? Show how ou arrived at our
39 9 Eercise #: For the function g 9 1 do the following. (a) Sketch the graph of g on the aes provided. (b) State the zeroes of g. (c) Over what interval is g decreasing? (d) Over what interval is g 0? (e) State the range of g. Eercise #: Draw a graph of following characteristics. f that matches the Increasing on: 8 4 and 1 5 Decreasing on: 4 1 f 8 5 and zeroes at 6,, and Absolute maimum of 7 and absolute minimum of 5
40 Eercise #4: A continuous function has a domain of 7 10 and has selected values shown in the table below. The function has eactl two zeroes and a relative maimum at 4,1 and a relative minimum at 5, f (a) State the interval on which f is decreasing. (b) State the interval over which f 0.
41 41 LESSON #8  KEY FEATURES OF FUNCTIONS DAY COMMON CORE ALGEBRA II 1. Is h() a function? h(). Find h(4).. When does h() = ? 4. Is h() increasing or decreasing over the interval [4,1]? 5. State an interval where h() is increasing. 6. What are the coordinates of the turning points of h()? State whether each is a relative maimum or minimum. 7. What are the absolute maimum and minimum values of h()? 8. What is the intercept? 9. What are zero(s)? 10. What is the domain? 11. What is the range? 1. Is h() positive or negative over the interval [1,4]? 1. State an interval where h() < A second function f() is defined using the rule f ( ) h( ). Evaluate f(6) using this rule. 15. A third function g() is defined b the formula g( ) 4. What is the value of g(h())?
42 4 16. Draw a graph of m characteristics: that matches the following Increasing on: 4 Decreasing on: 5 and 4 7 m 7 and zeroes at, 0, and 6 Absolute maimum of 9 and absolute minimum of A continuous function, j(), has a domain of 8 6 and has selected values shown in the table below. The function has eactl three zeroes and a relative maimum at,10 and a relative minimum at 5, j (a) State the interval on which j is increasing. (b) State the interval over which j 0.
43 4 LESSON #9  INVERSES OF LINEAR FUNCTIONS COMMON CORE ALGEBRA II The inverse of a function has all the same points as the original function, ecept that the 's and 's have been reversed. Eercise #1: If our function is made up of these points: { (1, 0), (, 5), (0, 4) }. Then the inverse is given b this set of points:. Eercise #: If ou are given a function on a graph, ou can also switch the s and s in the original points to plot the inverse. Use that process to graph the inverses of the two functions below. Which of the inverses is a function? How do ou know? There is a wa to determine if the inverse of a function will also be a function without graphing the inverse. What qualit of the original function determines if the inverse will be a function? Wh does this make sense?
44 44 Eercise #: On the grid below the linear function f ( ) 4 is graphed along with the line. (a) How can ou quickl tell that f( ) is a onetoone function? (b) Graph the inverse of 4 on the same grid. Recall that this is easil done b switching the and coordinates of the original line. (c) What can be said about the graphs of 4 and its inverse with respect to the line? Steps for Finding the Inverse of a Function From an Equation 1. If necessar, substitute for f().. SWITCH X AND Y.. Solve for. 4. If necessar, substitute f 1 ( ) for. 4 (d) Find f 1 ( ) (the inverse of f( )). Eercise #4: Find the inverse of each of the following functions. (a) g( ) 5 (b) 4 1
45 45 Eercise #5: Which of the following represents the equation of the inverse of 5 0? (1) () 1 0 () (4) Eercise #6: Which of the following represents the inverse of the linear function (1) 8 () 8 () 1 (4) 1 8? Eercise #7: What is the intercept of the inverse of (1) 15 () 9 9? 5 () 1 (4) 9 5 Eercise #8: Which of the following linear functions would not have an inverse that is also a function? Eplain how ou made our choice. (1) () () (4) 5 1
46 46 FLUENCY LESSON #9  INVERSES OF LINEAR FUNCTIONS COMMON CORE ALGEBRA II HOMEWORK 1. The graph of a function and its inverse are alwas smmetric across which of the following lines? (1) 0 () () 0 (4) 1. Which of the following represents the inverse of the linear function 4? (1) () 1 8 () 1 8 (4) If the intercept of a linear function is 8, then we know which of the following about its inverse? (1) Its intercept is 8. () Its intercept is 1 8. () Its intercept is 8. (4) Its intercept is Which of the following represents the equation of the inverse of 4 (1) 4 () () 18 (4) ? 5. Find the inverse of each of the following functions. (a) h( ) 5 7 (b) 1 f ( ) 9
47 47 A linear function is graphed below. Answer the following questions based on this graph. (a) Write the equation of this linear function in m b form. (b) Sketch a graph of the inverse of this function on the same grid. (c) Write the equation of the inverse in m b form. (d) What is the intersection point of this line with its inverse? APPLICATIONS 8. A car traveling at a constant speed of 58 miles per hour has a distance of miles from Poughkeepsie, NY, given b the equation 58 4, where represents the time in hours that the car has been traveling. (a) Find the equation of the inverse of this linear a function in form. b (b) Evaluate the function ou found in part (a) for an input of 7. (c) Give a phsical interpretation of the answer ou found in part (b). Consider what the input and output of the inverse represent in order to answer this question. REASONING 9. Given the general linear function m b, find an equation for its inverse in terms of m and b.
48 48 LESSON # 10  AVERAGE RATE OF CHANGE COMMON CORE ALGEBRA II When we model using functions, we are ver often interested in the rate that the output is changing compared to the rate of the input. The average rate of change is an eceptionall important concept in mathematics because it gives us a wa to quantif how fast a function changes on average over a certain domain interval. AVERAGE RATE OF CHANGE For a function over the domain interval a b, the function's average rate of change is calculated b: f change in the output f b f a change in the input b a = 1 1 g 1. Eercise #1: Consider (a) Calculate the average rate of change for g() over the following intervals: (i) (ii) 1 5 Eercise #: Nelson took a summer job, for five weeks, where he received a weekl salar plus tips. His take home pa is recorded in the table shown to the right. What was the average rate of change in his weekl takehome pa over the five weeks of his job? Eplain what this means in the contet of the problem. Week Weekl Salar 1 $60 $65 $7 4 $75 5 $80 Eercise #: What tpe of function would have the same average rate of change over an interval? Eplain our answer.
49 49 Eercise #4: The graph below shows the variation in the average temperature of Earth's surface from , according to one source. During which ears did the temperature variation change the most per unit time, or 1975 to 000? Eplain how ou determined our answer. Eercise #5: The table below shows the average diameter of a pupil in a person s ee as he or she grows older. What is the average rate of change of a person s pupil diameter from age 0 to age 80? Eplain what this value means in the contet of the problem. Eercise #6: An astronaut drops a rock off the edge of a cliff on the Moon. The distance,, in meters, the rock travels after t seconds can be modeled b the function. What is the average speed of the rock between 5 and 10 seconds after it was dropped? Use appropriate units to label our answer. Eercise #7 (Challenge): The table below represents a linear function. Fill in the missing entries
50 50 LESSON #10  AVERAGE RATE OF CHANGE COMMON CORE ALGEBRA II HOMEWORK FLUENCY 1. For the function g given in the table below, calculate the average rate of change for each of the following intervals g (a) 1 (b) 1 6 (c) 9 (d) Eplain how ou can tell from the answers in (a) through (c) that this is not a table that represents a linear function.. Which has a greater average rate of change over the interval 4 f. Provide justification for our answer. function the function g 16 or the
51 51 APPLICATIONS. The table below shows the cost of mailing a postcard in different ears. What was the average rate of change in the cost of a postal stamp from 006 to 01. Eplain our answer in the contet of the problem. 4. An object travels such that its distance, d, awa from its starting point is shown as a function of time, t, in seconds, in the graph below. (a) What is the average rate of change of d over the interval 5t 7? Include proper units in our 14, 79 answer. 11, 64 7,64 (b) The average rate of change of distance over time (what ou found in part (a)) is known as the average speed of an object. Is the average speed of this object greater on the interval 0t 5 or 11t 14? Justif. Distance (feet) 5, 0 Time (seconds) REASONING 5. What makes the average rate of change of a linear function different from that of an other function? What is the special name that we give to the average rate of change of a linear function?
52 5 LESSON #11  USING TABLES ON YOUR CALCULATOR COMMON CORE ALGEBRA II The graphing calculator is an amazing device that can do man things. One function that it is particularl good at is evaluating epressions for different input values. We will be looking at two tools on the calculator toda, the STORE feature and TABLES. First let's look at how to use STORE. Eercise #1: Find the value of each of the following epressions b using the STORE feature on our calculator. (a) 7 for 5 (b) 6 5 for 10 (c) 7 0 for Sometimes the calculator can even tell us useful information even when it has a hard time evaluating an epression. Eercise #: Consider the epression 6. What happens when ou tr to use STORE to evaluate this epression for 5? Evaluate the epression b hand to help eplain what the calculator is tring to tell us. Eercise #: Let's work with the product of two binomials again, specificall (a) Find their product in trinomial form. and 5. (b) Evaluate both the trinomial and the original product for. What do ou notice? (c) Use the STORE command to evaluate the trinomial from (a) for 5. Wh does the value of the trinomial turn out to be this specific value at 5? Eplain.
53 The STORE feature is etremel helpful when ou are tring to determine the value of an epression at one or two input values of. But, if ou want to know an epression's value for multiple inputs, then TABLES are a much better tool. Eercise #4: The epression 16 has an integer zero somewhere on the interval Use a TABLE to find the zero on this interval. Show the table. 5 Table commands can be particularl good at establishing proof that two epression are equivalent. This is particularl helpful when ou've done a number of manipulations and ou want to have confidence that ou've produced an algebraicall equivalent epression. Eercise #5: Consider the more comple algebraic epression shown below: 5 8 (a) This relativel comple epression simplifies into a linear binomial epression. Determine this epression carefull. Show our work below. (b) Set up a table using the original epression and the one ou found in (a) over the interval 0 5. Compare values to determine if ou correctl simplifies the original epression
54 54 Eercise #6: Consider the equation shown below: 5( ) 6( 1) (a) Solve the equation algebraicall. (b) Store our answer, and plug it into both sides of the equation to show that the are equal. (c) Graph both sides of the equation as Y1 and Y. Cop our table including the value of that was the solution to the equation. What do ou notice? Eercise #7: You do not have the skills necessar at this point to solve the equation below. Using the same method as eercise 6c, find the three solutions to this equation. Cop our table below
55 55 LESSON #11  USING TABLES ON YOUR CALCULATOR COMMON CORE ALGEBRA II HOMEWORK FLUENCY 1. Use the STORE feature on our calculator to evaluate each of the following. No work needs to be shown. (a) 7 18 for 8 (b) 5 for (c) for 5 (d) 5 8 for 1 (e) 4 5 for (f) 4 9 for 5. The STORE features is particularl helpful in checking to see if a value is a solution to an equation. Let's see how this works in this problem. Consider the relativel eas linear equation: (a) Solve this equation for (b) Using STORE,determine the value of both the left hand epression, 6, and the right hand epression, 4 9, at the value of ou found in (a). (c) Wh does what ou found in part (b) verif that our solution is correct (or possibl incorrect if ou made a mistake in (a))?. Two of the following values of are solutions to the equation: are and provide a justification for our answer Determine which the 5 6 8
56 56 4. The quadratic epression 8 10 has its smallest value for some integer value of on the interval Set up a TABLE to find the smallest value of the epression and the value of that gives this value. Show our table below. 5. Consider the comple epression (a) Multipl the two sets of binomials and combine like terms in order to write this epression as an equivalent trinomial in standard form. Show our work. (b) Set up a TABLE to verif that our answer in part (a) is equivalent to the original epression. Don't hesitate to point out that it is not equivalent (which means ou either made a mistake in our algebra or in our table set up). Show our table. 6. The product of three binomials is shown below. Write this product as a polnomial in standard form. (Its highest power will be ) Set up a table for the answer ou found in #6 on the interval 5 5. Where does this epression have zeroes?
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