ࡱ>  ly m`bjbj y{{~TF x4B$ -:".".".|||$^^|{|||".".L2| ".".|Ӡ@;".9Y:H0x+4;;||||||||||x||||||||||||| : Word+-ExampleDefinitionCommon ratioDecay factorExponential decayExponential functionExponential growthExponential regressionArithmetic sequenceGeometric SequenceGrowth factor Exponential interpolationInterest period (time)PolynomialMonomial BinomialTrinomialStandard form of a polynomialAlgebra Tiles Unit 7 Vocabulary Awareness Chart Procedure: 1. Examine the list of words you have written in the first column. 2. Put a + next to each word you know well and for which you can write an accurate example and definition. Your definition and example must relate to the unit of study. 3. Place a  next to any words for which you can write either a definition or an example, but not both. 4. Put a next to words that are new to you. This chart will be used throughout the unit. By the end of the unit, you should have the entire chart completed. Because you will be revising this chart, write in pencil. LINEAR AND EXPONENTIAL FUNCTIONS Make a table of values and graph the following function. Use the domain {-2, -1, 0, 1, 2} y = 3x xy What type of function is this? What is the slope of the line? ____________ Show how the slope is represented in the table. Graph y = 3x using the same domain xy What type of function is this? Find the difference in the y-values for this function as x increases by 1. Is y = 3x a linear function? Why or why not? How do the y-values increase as x increases by 1? Compare the following functions by graphing on the graphing calculator. Use the table on the calculator to complete a table of values for each function. a) y = 2x b) y = 3x c)y = 4x xy xY xy What happens to the graph as the base is increased? Show this using the table of values. Graph y =  EMBED Equation  using the domain {-2, -1, 0, 1, 2} xy What is the difference between the graphs of the functions of y = 3x and y =  EMBED Equation ? In the function y = 3x, as the x values increase, what happens to the y values? In the function, y =  EMBED Equation , as the x values increase, what happens to the y values? y = bx is an example of an ___________________________________ function. Why? y = b-x or  EMBED Equation  is an example of an ________________________________function. Why? LINEAR AND EXPONENTIAL FUNCTIONS Make a table of values and graph the following function. Use the domain {-2, -1, 0, 1, 2}  y = 3x xy-2-6-1-3001326 What type of function is this? linear What is the slope of the line? _3_______ Show how the slope is represented in the table.  Graph y = 3x using the same domain xy-2 EMBED Equation.DSMT4 -1 EMBED Equation.DSMT4 011329 What type of function is this? exponential Find the difference in the y-values for this function as x increases by 1. The differences are not constant, but they increase by the next power of 3. Is y = 3x a linear function? No Why or why not? It does not increase by a constant value. How do the y-values increase as x increases by 1? They are multiplied by 3. Compare the following functions by graphing on the graphing calculator. Use the table on the calculator to complete a table of values for each function. a.) y = 2x b.) y = 3x c) y = 4x xy-20.25-10.5011224 xy-20.11-10.33011329 xy-20.06-10.250114216 What happens to the graph as the base is increased? Show this using the table of values. Graph y =  EMBED Equation  using the domain {-2, -1, 0, 1, 2} xy-29-1301 1.332.11 What is the difference between the graphs of the functions of y = 3x and y =  EMBED Equation ? They are both exponential, but one increases and the other decreases. In the function y = 3x, as the x values increase, what happens to the y values? It increases. In the function, y =  EMBED Equation , as the x values increase, what happens to the y values? It decreases. y = bx is an example of an exponential growth function. Why? The function is exponential because it multiplies at a constant rate. It represents growth because it increases. y = b-x or  EMBED Equation  is an example of an exponential decay function. Why? The function is exponential because it multiplies at a constant rate. It represents decay because it decreases. COMPARE AND CONTRAST MAP     Directions: Place 50 atoms of M&Ms (pieces of candy) in the Ziploc bag. Seal the bag and gently shake for 10 seconds. Gently pour out candy on to the paper plate. Notice the pieces with the print side up. These atoms have "decayed." Count the undecayed pieces (with the print side down) and record the data below. Return the undecayed pieces to the bag. Reseal the bag. Consume the "decayed" atoms. Gently shake the sealed bag for 10 seconds. Continue shaking, counting, and consuming until all the atoms have decayed. Complete the table at each part of the process. Predict the equation that represents this exponential function. Graph the number of undecayed atoms vs. time in your calculator. Find the exponential equation that fits the curve. TrialNumber of undecayed atoms050 Product of powersExpanded productProduct as a single powerExplanation EMBED Equation.3  EMBED Equation.3  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   Quotient of powersExpanded quotientQuotient as a single powerExplanation EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  Power of a powerExpanded powerPower as a single powerExplanation EMBED Equation.3  EMBED Equation.3  EMBED Equation.3   EMBED Equation.3  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   Power of a fractional powerExpanded powerPower as a single powerExplanation EMBED Equation.DSMT4  EMBED Equation.DSMT4 x EMBED Equation.DSMT4  EMBED Equation.DSMT4  Product of powersExpanded productProduct as a single powerExplanation EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 When you multiply 2 xs together times 3 xs that are multiplied together, you get 5 xs multiplied together which is  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.DSMT4 When you multiply 5 xs together times 4 xs that are multiplied together, you get 9 xs multiplied together which is  EMBED Equation.3  EMBED Equation.3  EMBED Equation.DSMT4  EMBED Equation.DSMT4 When you multiply 1 xs times 3 xs that are multiplied together, you get 4 xs multiplied together which is  EMBED Equation.3  EMBED Equation.3  EMBED Equation.DSMT4  EMBED Equation.DSMT4 When you multiply 2 xs together times 4 xs that are multiplied together, you get 6 xs multiplied together which is  EMBED Equation.3  EMBED Equation.3  EMBED Equation.DSMT4 When multiplying variables that have exponents, if the bases are the same, add the exponents. Quotient of powersExpanded quotientQuotient as a single powerExplanation EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  When you have 5 xs that are multiplied together and you divide the product by 2 xs multiplied together, 2 xs will divide, yielding 1 and 3 x s or  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 XWhen you have 4 xs that are multiplied together and you divide the product by 3 xs multiplied together, 3 xs will divide, yielding 1, and 1 x  EMBED Equation.3  EMBED Equation.3  EMBED Equation.DSMT4 When you have 9 xs that are multiplied together and you divide the product by 4 xs multiplied together, 4 xs will divide, yielding 1 and 5 x s or  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.DSMT4 When you have 2 xs that are multiplied together and you divide the product by 5 xs multiplied together, 2 xs will divide yielding 1, and 3 x s in the denominator of the fraction or  EMBED Equation.DSMT4  EMBED Equation.3  EMBED Equation.DSMT4 When dividing variables that have exponents, if the bases are the same, subtract the exponents Power of a powerExpanded powerPower as a single powerExplanation EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 Multiply  EMBED Equation.DSMT4 together three times, add the exponents to get  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.DSMT4 Multiply  EMBED Equation.DSMT4 together four times, add the exponents to get  EMBED Equation.3  EMBED Equation.3  EMBED Equation.DSMT4  EMBED Equation.DSMT4 Multiply  EMBED Equation.DSMT4 together two times, add the exponents to get  EMBED Equation.3  EMBED Equation.3  EMBED Equation.DSMT4  EMBED Equation.DSMT4 Multiply  EMBED Equation.DSMT4 together four times, add the exponents to get  EMBED Equation.3  EMBED Equation.3  EMBED Equation.DSMT4 When taking a power to a power, multiply the exponents. Power of a fractional power Expanded powerPower as a Single powerExplanation EMBED Equation.DSMT4  EMBED Equation.DSMT4 xMultiply  EMBED Equation.DSMT4 two times, add the exponents to get x. EMBED Equation.DSMT4  EMBED Equation.DSMT4 xMultiply  EMBED Equation.DSMT4  three times, add the exponents to get x. EMBED Equation.DSMT4 xMultiply  EMBED Equation.DSMT4  n times, add the exponents to get  EMBED Equation.DSMT4 .  Read the following passages and provide the solutions to the problems below each passage. 1. Two types of number sequences are represented in this activity-arithmetic sequences and geometric sequences. When the terms of a sequence are based on adding a fixed number to each previous term, the pattern is known as an arithmetic sequence. The fixed number is called the common difference. Examples: Look at the pattern of -7, -3, 1, 5. What is the common difference between each of the terms? The common difference is +4. So to find the next term in the sequence, you would add 4 to 5 to obtain 9. Now, determine the common difference for 17, 13, 9, and 5. The common difference is __, and the next term in the sequence is ___. A rule for finding the nth term of an arithmetic sequence can be found by using the following guidelines: Consider the sequence 7, 11, 15, 19,... in which the common difference is +4. Let n = the term number of the sequence. Let A(n) = the value of the nth term of the sequence A(1) = 7; A(2) = 7 + 4 = 11; A(3) = 7 + 2(4) = 15; A(4) = 7 + 3(4) = 19; and A(n) = 7 + (n-1)4. If you were asked to find the 10th term of the sequence, A(10) = 7 + (10 - 1)4 = 43, the 20th term would be ________________________. Thus, the nth term A(n) rule for an arithmetic sequence is A(n) = a + ( n - 1) d where a = the first term, n = the term number, and d = the common difference. Find the common difference of the arithmetic sequence. 2. 9, 13, 17, 21, . . . 3. 5, 5.3, 5.6, 5.9, . . . 4. 2, , , , . . . 5. The common difference in an arithmetic sequence is ____________ a positive number. Find the first, fourth, and tenth terms of the arithmetic sequence described by the given rule. 6.  EMBED Equation.DSMT4  7.  EMBED Equation.DSMT4  8.  EMBED Equation.DSMT4  Geometric Sequences: Two types of number sequences are represented in this activity-arithmetic sequences and geometric sequences. When the terms of a sequence are determined by multiplying a term in the sequence by a fixed number to find the next term, the sequence is called a geometric sequence. Examples: Look at the pattern of 3, 12, 48, 192. What is the common ratio between each of the terms? The common ratio is 4. So to find the next term in the sequence, you would multiply 4 by 192 to obtain 768. Now, determine the common ratio for 80, 20, 5, and 1.25. The common ratio is ___, and the next term in the sequence is _________. A rule for finding the nth term of an geometric sequence can be found by using the following guidelines: Consider the sequence 1, 3, 9, 27,... in which the common ratio is 3. Let n = the term number of the sequence. Let A(n) = the value of the nth term of the sequence. A(1) = 1; A(2) = 1(3) = 3; A(3) = 1(3)(3) = 1(3)2; A(4) = 1(3)(3)(3) = 1(3)3; and A(n) =  EMBED Equation.DSMT4  If you were asked to find the 10th term of the sequence, A(10) = 1(3)9 = 19,683, the 20th term would be ____________________________________. Thus, the nth term A(n) rule for an geometric sequence is  EMBED Equation.DSMT4 where a = the first term, n = the term number, and r = the common ratio. The common difference or the common ratio can be used to determine whether a sequence is arithmetic or geometric. If there is no common difference or ratio, then the sequence is neither geometric nor arithmetic. Examples: 2, 4, 6, 8... the common difference is 2 so the sequence is arithmetic. 2, 4, 8, 16 ... the common ratio is 2 so the sequence is geometric. Find the common ratio of the sequence: 9. 2, 10, 50, 250, . . . 10. 164, 82, 41, 20.5, . . . 11. Find the next three terms of the sequence 3, 9, 27, 81... Determine whether the sequence is arithmetic or geometric. 12. 2, 10, 50, 250, . . . 13. Find the first, fourth, and eighth terms of the sequence.  EMBED Equation.DSMT4  14. Find the next three terms of the sequence. Then, write a rule for the sequence. 648, 216, 72, 24 Read the following passages and provide the solutions to the problems below each passage. 1. Two types of number sequences are represented in this activity-arithmetic sequences and geometric sequences. When the terms of a sequence are based on adding a fixed number to each previous term, the pattern is known as an arithmetic sequence. The fixed number is called the common difference. Examples: Look at the pattern of -7, -3, 1, 5. What is the common difference between each of the terms? The common difference is +4. So to find the next term in the sequence, you would add 4 to 5 to obtain 9. Now, determine the common difference for 17, 13, 9, and 5. The common difference is -4, and the next term in the sequence is 1. A rule for finding the nth term of an arithmetic sequence can be found by using the following guidelines: Consider the sequence 7, 11, 15, 19,... in which the common difference is +4. Let n = the term number of the sequence. Let A(n) = the value of the nth term of the sequence. A(1) = 7; A(2) = 7 + 4 = 11; A(3) = 7 + 2(4) = 15; A(4) = 7 + 3(4) = 19; and A(n) = 7 + (n-1)4. If you were asked to find the 10th term of the sequence, A(10) = 7 + (10 - 1)4 = 43, the 20th term would be A(20) = 7 + (20-1)4 = 83. Thus, the nth term A(n) rule for an arithmetic sequence is A(n) = a + ( n - 1) d where a = the first term, n = the term number, and d = the common difference. Find the common difference of the arithmetic sequence. 2. 9, 13, 17, 21, . . . Answer: +4 3. 5, 5.3, 5.6, 5.9, . . . Answer: +0.3 4. 2, , , , . . . Answer:  EMBED Equation.DSMT4  5. The common difference in an arithmetic sequence is sometimes a positive number. Find the first, fourth, and tenth terms of the arithmetic sequence described by the given rule. 6.  EMBED Equation.DSMT4  Answer: 12, 21, 39 7.  EMBED Equation.DSMT4  Answer: -6,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  8.  EMBED Equation.DSMT4  Answer: -3, -9.6, -22.8 Geometric Sequences: Two types of number sequences are represented in this activity-arithmetic sequences and geometric sequences. When the terms of a sequence are determined by multiplying a term in the sequence by a fixed number to find the next term, the sequence is called a geometric sequence. Examples: Look at the pattern of 3, 12, 48, 192. What is the common ratio between each of the terms? The common ratio is 4. So to find the next term in the sequence, you would multiply 4 by 192 to obtain 768. Now, determine the common ratio for 80, 20, 5, and 1.25. The common ratio is 0.25, and the next term in the sequence is  EMBED Equation.DSMT4  or 0.3125. A rule for finding the nth term of an geometric sequence can be found by using the following guidelines: Consider the sequence 1, 3, 9, 27,... in which the common ratio is 3. Let n = the term number of the sequence. Let A(n) = the value of the nth term of the sequence, A(1) = 1; A(2) = 1(3) = 3; A(3) = 1(3)(3) = 1(3)2; A(4) = 1(3)(3)(3) = 1(3)3; and A(n) =  EMBED Equation.DSMT4 . If you were asked to find the 10th term of the sequence, A(10) = 1(3)9 = 19,683. the 20th term would be  EMBED Equation.DSMT4  Thus, the nth term A(n) rule for an geometric sequence is  EMBED Equation.DSMT4 where a = the first term, n = the term number, and r = the common ratio. The common difference or the common ratio can be used to determine whether a sequence is arithmetic or geometric. If there is no common difference or ratio, then the sequence is neither geometric nor arithmetic. Examples: 2, 4, 6, 8... the common difference is 2 so the sequence is arithmetic. 2, 4, 8, 16 ... the common ratio is 2 so the sequence is geometric. Find the common ratio of the sequence. 9. 2, 10, 50, 250, . . . Answer: -5 10. 164, 82, 41, 20.5, . . . Answer:  EMBED Equation.DSMT4  11. Find the next three terms of the sequence 3, 9, 27, 81... Answer: 243, 729, 2187 Determine whether the sequence is arithmetic or geometric. Geometric 12. 2, 10, 50, 250, . . . Answer: -1250 13. Find the first, fourth, and eighth terms of the sequence.  EMBED Equation.DSMT4  Answer: -2; -16; -256 14. Find the next three terms of the sequence. Then write a rule for the sequence. 648, 216, 72, 24 Answer: 8,  EMBED Equation.DSMT4 ;  EMBED Equation.DSMT4  Word+-ExampleDefinition ParabolaQuadratic Equation in Standard FormZerosCompleting the SquareScatterplotDiscriminantQuadratic FormulaVertexAxis of SymmetryY-interceptX-interceptSolutions to a quadratic equationReal solutionComplex SolutionRoots of a quadraticVertex form of a quadratic equation Procedure: 1. Examine the list of words you have written in the first column. 2. Put a + next to each word you know well and for which you can write an accurate example and definition. Your definition and example must relate to the unit of study. 3. Place a  next to any words for which you can write either a definition or an example, but not both. 4. Put a next to words that are new to you. This chart will be used throughout the unit. By the end of the unit you should have the entire chart completed. Because you will be revising this chart, write in pencil. 1. Graph  EMBED Equation.DSMT4   a. Determine the axis of symmetry: ______ b. Determine the vertex: _______ c. Name the solution(s) to  EMBED Equation.DSMT4 : _________ 2. Graph  EMBED Equation.DSMT4   a. Determine the axis of symmetry: ______ b. Determine the vertex: ______ c. Name the solution(s) to  EMBED Equation.DSMT4 : _________ 3. Graph  EMBED Equation.DSMT4   a. Determine the axis of symmetry: ____ b. Determine the vertex: ______ c. Name the solution(s) to  EMBED Equation.DSMT4 : _________ 4. Graph  EMBED Equation.DSMT4   a. Determine the axis of symmetry: ______ b. Determine the vertex: _________ c. Name the solution(s) to  EMBED Equation.DSMT4 : _________ 1. Graph  EMBED Equation.DSMT4   a. Determine the axis of symmetry: _x = -1 b. Determine the vertex: ( -1.5, -.25 ) c. Name the solution(s) to  EMBED Equation.DSMT4 : (-2, 0) ( -1, 0) 2. Graph  EMBED Equation.DSMT4   a. Determine the axis of symmetry: _x = 2 b. Determine the vertex: ( 2, 0) c. Name the solution(s) to  EMBED Equation.DSMT4 : (2, 0) 3. Graph  EMBED Equation.DSMT4   a. Determine the axis of symmetry: _x = 0 b. Determine the vertex: ( 0, 3 ) c. Name the solution(s)  EMBED Equation.DSMT4 : (No real solution) 4. Graph  EMBED Equation.DSMT4   a. Determine the axis of symmetry: _x = -2.5 b. Determine the vertex: ( -2.5, -.25) c. 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