Analyze Geometric Sequences and Series
|
|
- Adelia Stewart
- 5 years ago
- Views:
Transcription
1 23 a4, 2A2A; P4A, P4B TEKS Analyze Geometric Sequences and Series Before You studied arithmetic sequences and series Now You will study geometric sequences and series Why? So you can solve problems about sports tournaments, as in Ex 58 Key Vocabulary geometric sequence common ratio geometric series In a geometric sequence, the ratio of any term to the previous term is constant This constant ratio is called the common ratio and is denoted by r E XAMPLE Identify geometric sequences Tell whether the sequence is geometric a 4, 0, 8, 28, 40, b 625, 25, 25, 5,, Solution To decide whether a sequence is geometric, find the ratios of consecutive terms a a 2 } 5 } a 3 } } 5 } a 4 } } 5 } a 5 } } 5 } 40 5 } 0 a 4 2 a2 0 5 a3 8 9 a b c The ratios are different, so the sequence is not geometric a 2 } a 5 25 } } 5 a 3 } a } 25 5 } 5 a 4 } a3 5 5 } 25 5 } 5 a 5 } a4 5 } 5 c Each ratio is } 5, so the sequence is geometric GUIDED PRACTICE for Example Tell whether the sequence is geometric Explain why or why not 8, 27, 9, 3,, 2, 2, 6, 24, 20, 3 24, 8, 26, 32, 264, KEY CONCEPT For Your Notebook Rule for a Geometric Sequence Algebra Example The nth term of a geometric sequence with first term a and common ratio r is given by: r n 2 The nth term of a geometric sequence with a first term of 3 and common ratio 2 is given by: 5 3(2) n 2 80 Chapter 2 Sequences and Series
2 E XAMPLE 2 Write a rule for the nth term Write a rule for the nth term of the sequence Then find a 7 a 4, 20, 00, 500, b 52, 276, 38, 29, Solution a The sequence is geometric with first term a 5 4 and common ratio r 5 } So, a rule for the nth term is: 4 AVOID ERRORS In the general rule for a geometric sequence, note that the exponent is n 2, not n r n 2 Write general rule 5 4(5) n 2 Substitute 4 for a and 5 for r The 7th term is a 7 5 4(5) ,500 b The sequence is geometric with first term a 5 52 and common ratio r 5} } So, a rule for the nth term is: 52 2 r n 2 Write general rule } 2 2 n 2 Substitute 52 for a and 2 } 2 for r The 7th term is a } } 8 E XAMPLE 3 Write a rule given a term and common ratio One term of a geometric sequence is a The common ratio is r 5 2 a Write a rule for the nth term b Graph the sequence Solution a Use the general rule to find the first term r n 2 Write general rule a 4 r 4 2 Substitute 4 for n 2 (2) 3 Substitute 2 for a 4 and 2 for r 5 Solve for a So, a rule for the nth term is: r n 2 Write general rule 5 5(2) n 2 Substitute 5 for a and 2 for r b Create a table of values for the sequence The graph of the first 6 terms of the sequence is shown Notice that the points lie on an exponential curve This is true for any geometric sequence with r > 0 n n at classzonecom 23 Analyze Geometric Sequences and Series 8
3 E XAMPLE 4 Write a rule given two terms Two terms of a geometric sequence are a and a Find a rule for the nth term Solution STEP Write a system of equations using r n 2 and substituting 3 for n (Equation ) and then 6 for n (Equation 2) a 3 r r 2 Equation a 6 r r 5 Equation 2 STEP 2 Solve the system 248 } r 2 Solve Equation for a } r 2 (r 5 ) Substitute for a in Equation r 3 Simplify 24 5 r Solve for r 248 (24) 2 Substitute for r in Equation 23 Solve for a STEP 3 Find a rule for r n 2 Write general rule 523(24) n 2 Substitute for a and r GUIDED PRACTICE for Examples 2, 3, and 4 Write a rule for the nth term of the geometric sequence Then find a 8 4 3, 5, 75, 375, 5 a , r a 2 522, a GEOMETRIC SERIES The expression formed by adding the terms of a geometric sequence is called a geometric series The sum of the first n terms of a geometric series is denoted by S n You can develop a rule for S n as follows S n a r a r 2 a r 3 a r n 2 2rS n 5 2 a r 2 a r 2 2 a r a r n 2 2 a r n S n ( 2 r) a r n So, S n ( 2 r) ( 2 r n ) If r Þ, you can divide each side of this equation by 2 r to obtain the following rule for S n KEY CONCEPT For Your Notebook The Sum of a Finite Geometric Series The sum of the first n terms of a geometric series with common ratio r Þ is: S n 2 rn } 2 r 2 82 Chapter 2 Sequences and Series
4 E XAMPLE 5 Find the sum of a geometric series Find the sum of the geometric series a 5 4(3) r Identify first term 4(3) i 2 Identify common ratio S 6 2 r6 } 2 r 2 Write rule for S } Substitute 4 for a and 3 for r 5 86,093,440 Simplify c The sum of the series is 86,093,440 E XAMPLE 6 Use a geometric sequence and series in real life MOVIE REVENUE In 990, the total box office revenue at US movie theaters was about $502 billion From 990 through 2003, the total box office revenue increased by about 59% per year a Write a rule for the total box office revenue (in billions of dollars) in terms of the year Let n 5 represent 990 b What was the total box office revenue at US movie theaters for the entire period ? Solution a Because the total box office revenue increased by the same percent each year, the total revenues from year to year form a geometric sequence Use a and r to write a rule for the sequence 5 502(059) n 2 Write a rule for b There are 4 years in the period , so find S 4 S 4 2 r4 } 2 r (059)4 } ø 05 c The total movie box office revenue for the period was about $05 billion GUIDED PRACTICE for Examples 5 and 6 7 Find the sum of the geometric series 8 6(22) i 2 8 MOVIE REVENUE Use the rule in part (a) of Example 6 to estimate the total box office revenue at US movie theaters in Analyze Geometric Sequences and Series 83
5 23 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p WS for Exs 9, 49, and 59 5 TAKS PRACTICE AND REASONING Exs 27, 54, 55, 59, 63, and 64 5 MULTIPLE REPRESENTATIONS Ex 6 VOCABULARY Copy and complete: The constant ratio of consecutive terms in a geometric sequence is called the? 2 WRITING How can you determine whether a sequence is geometric? EXAMPLE on p 80 for Exs 3 4 IDENTIFYING GEOMETRIC SEQUENCES Tell whether the sequence is geometric Explain why or why not 3, 4, 8, 6, 32, 4 4, 6, 64, 256, 024, 5 26, 36, 6,, } 6, 6 } 3, 2 } 3, 4 } 3, 8 } 3, 6 } 3, 7 },, } 3, 2, } 5, 8 2}, } 3, 2} 3, }, 2} 3, , 5, 25, 25, 0625, 0 23, 26, 2, 24, 248, 24, 2, 236, 08, 2324, 2 02, 06, 8, 54, 62, 3 25, 0, 20, 40, 80, 4 075, 5, 225, 3, 375, EXAMPLE 2 on p 8 for Exs 5 27 WRITING RULES Write a rule for the nth term of the geometric sequence Then find a 7 5, 24, 6, 264, 6 6, 8, 54, 62, 7 4, 24, 44, 864, 8 7, 235, 75, 2875, 9 2, } 3, } 9, } 27, , 2} 6, } 2, 2} 24, , 2,, 05, , 06, 22, 24, 23 22, 208, 2032, 2028, 24 7, 242, 252, 252, 25 5, 24, 392, 20976, 26 20, 80, 270, 405, 27 TAKS REASONING What is a rule for the nth term of the geometric sequence 5, 20, 80, 320,? A 5 5(2) n 2 B 5 5(4) n 2 C 5 5(24) n 2 D 5 5(22) n 2 EXAMPLE 3 on p 8 for Exs WRITING RULES Write a rule for the nth term of the geometric sequence Then graph the first six terms of the sequence 28 a 5 5, r a 522, r a 2 5 6, r a 2 5 5, r 5 } 2 32 a 5 5, r 5 } 8 33 a 4 522, r 52 } 4 34 a , r a 2 5 8, r a , r 5 5 ERROR ANALYSIS Describe and correct the error in writing the rule for the nth term of the geometric sequence for which a 5 3 and r r n 38 5 ra n 2 5 3(2) n 5 2(3) n 2 84 Chapter 2 Sequences and Series
6 EXAMPLE 4 on p 82 for Exs WRITING RULES Write a rule for the nth term of the geometric sequence that has the two given terms 39 a 5 3, a a 5, a a 52 } 4, a a 3 5 0, a a , a a , a a , a a } 4, a } 6 47 a 4 5 6, a } 8 EXAMPLE 5 on p 83 for Exs FINDING SUMS Find the sum of the geometric series (2) i } 4 2 i (4) i } 2 2 i i i } 2 2 i (24) i 54 TAKS REASONING What is the sum of the geometric series 9 2(3) i 2? A 9,680 B 9,68 C 9,682 D 9, TAKS REASONING Write a geometric series with 5 terms such that the sum of the series is 00 ( Hint: Choose a value of r and then find a ) 56 CHALLENGE Using the rule for the sum of a finite geometric series, write each polynomial as a rational expression a x x 2 x 3 x 4 b 3x 6x 3 2x 5 24x 7 PROBLEM SOLVING EXAMPLE 6 on p 83 for Exs SKYDIVING In a skydiving formation with R rings, each ring after the first has twice as many skydivers as the preceding ring The formation for R 5 2 is shown a Let be the number of skydivers in the nth ring Find a rule for b Find the total number of skydivers if there are R 5 4 rings Second ring First ring 58 SOCCER A regional soccer tournament has 64 participating teams In the first round of the tournament, 32 games are played In each successive round, the number of games played decreases by one half a Find a rule for the number of games played in the nth round For what values of n does your rule make sense? b Find the total number of games played in the regional soccer tournament 23 Analyze Geometric Sequences and Series 85
7 59 TAKS REASONING Abinary search technique used on a computer involves jumping to the middle of an ordered list of data (such as an alphabetical list of names) and deciding whether the item being searched for is there If not, the computer decides whether the item comes before or after the middle Half of the list is ignored on the next pass, and the computer jumps to the middle of the remaining list This is repeated until the item is found a Find a rule for the number of items remaining after the nth pass through an ordered list of 024 items b In the worst case, the item to be found is the only one left in the list after n passes through the list What is the worst-case value of n for a binary search of a list with 024 items? Explain 60 FRACTALS The Sierpinski carpet is a fractal created using squares The process involves removing smaller squares from larger squares First, divide a large square into nine congruent squares Remove the center square Repeat these steps for each smaller square, as shown below Assume that each side of the initial square is one unit long Stage Stage 2 Stage 3 a Let be the number of squares removed at the nth stage Find a rule for Then find the total number of squares removed through stage 8 b Let b n be the remaining area of the original square after the nth stage Find a rule for b n Then find the remaining area of the original square after stage 2 6 MULTIPLE REPRESENTATIONS Two companies, company A and company B, offer the same starting salary of $20,000 per year Company A gives a raise of $000 each year Company B gives a raise of 4% each year a Writing Rules Write rules giving the salaries and b n in the nth year at companies A and B, respectively Tell whether the sequence represented by each rule is arithmetic, geometric, or neither b Drawing Graphs Graph each sequence in the same coordinate plane c Finding Sums For each company, find the sum of wages earned during the first 20 years of employment d Using Technology Use a graphing calculator or spreadsheet to find after how many years the total amount earned at company B is greater than the total amount earned at company A 86 5 WORKED-OUT SOLUTIONS on p WS 5 TAKS PRACTICE AND REASONING 5 MULTIPLE REPRESENTATIONS
8 62 CHALLENGE On January of each year, you deposit $2000 in an individual retirement account (IRA) that pays 5% annual interest You make a total of 30 deposits How much money do you have in your IRA immediately after you make your last deposit? MIXED REVIEW FOR TAKS TAKS PRACTICE at classzonecom REVIEW Lesson 32; TAKS Workbook REVIEW Lesson 4; TAKS Workbook 63 TAKS PRACTICE The total cost of carnival tickets for 3 adults and 5 children is $49 The total cost of carnival tickets for 5 adults and 3 children is $55 What is the price, a, of one adult ticket and the price, c, of one child ticket? TAKS Obj 4 A a 5 $5; c 5 $8 B a 5 $725; c 5 $625 C a 5 $8; c 5 $5 D a 5 $0; c 5 $ TAKS PRACTICE What is the relationship between the graphs of y 5 3x 2 and y 5 5x 2? TAKS Obj 5 F G The graph of y 5 5x 2 is a reflection of the graph of y 5 3x 2 in the x-axis The graph of y 5 5x 2 is a 908 rotation of the graph of y 5 3x 2 about the origin H The graph of y 5 5x 2 is narrower than the graph of y 5 3x 2 J The graph of y 5 5x 2 is wider than the graph of y 5 3x 2 QUIZ for Lessons 2 23 Write the next term in the sequence Then write a rule for the nth term (p 794), 3, 5, 7, 2 25, 0, 25, 20, 3 }, 2 }, 3 }, 4 }, , 6, 64, 256, 5 2, 6, 2, 20, 6 9, 36, 8, 44, Find the sum of the series (p 794) 7 4 2i k 5 (k 2 3) 9 6 n 5 2 } n 2 Write a rule for the nth term of the arithmetic or geometric sequence Find a 5, then find the sum of the first 5 terms of the sequence 0, 7, 3, 9, (p 802) } 2, 2, 7 } 2, 5, (p 802) 2 5, 2, 2, 24, 27, (p 802) 3 2, 8, 32, 28, (p 80) 4 2, 4 } 3, 8 } 9, 6 } 27, (p 80) 5 23, 5, 275, 375, (p 80) 6 COLLEGE TUITION In 995, the average tuition at a public college in the United States was $2057 From 995 through 2002, the average tuition at public colleges increased by about 6% per year Write a rule for the average tuition in terms of the year Let n 5 represent 995 What was the average tuition at a public college in 2002? (p 80) EXTRA PRACTICE for Lesson 23, p 02 ONLINE QUIZ at classzonecom 87
9 MIXED REVIEW FOR TEKS TAKS PRACTICE Lessons 2 23 MULTIPLE CHOICE TARGETS A target has red and blue rings that alternate in color The three innermost rings of the target are shown below Which expression is a series that gives the total area A of the target s n blue rings? TEKS a2 classzonecom 4 SEATING ARRANGEMENT At a restaurant, rectangular tables are placed together along their shared edges, as shown in the diagram below How many people can be seated around 8 tables arranged in this way? TEKS a ft The 3 innermost rings of the target A A 5 B A 5 C A 5 D A 5 n n n n (2i 2 )π (4i 2 3)π (4i 2 )π i 2 π 2 SALARY Maria has an annual salary of $45,000 during her first year of employment Her salary increases 35% per year What will Maria s salary be during her 5th year of employment? TEKS a F $49,672 G $5,24 H $5,639 J $53,446 3 CONSTRUCTION A staircase is being built that leads from the ground to an elevated deck The base of the staircase is a concrete slab that is 2 inches tall Each stair is 7 inches tall What is the height of the bottom of the 0th stair? TEKS a A 60 inches B 65 inches C 70 inches D 72 inches F 30 people G 32 people H 34 people J 36 people GRIDDED ANSWER STACKED PILES Pieces of chalk are stacked in a pile Part of the pile is shown below The bottom row has 5 pieces of chalk and the top row has 6 pieces of chalk Each row has one less piece of chalk than the row below it How many pieces of chalk are in the pile? TEKS a 6 RADIOACTIVE DECAY A scientist is studying the radioactive decay of Platinum-97 The scientist starts with a 66 gram sample of Platinum-97 and measures the amount remaining every two hours The recorded amounts (in grams) are 66, 33, 65, 825, After how many hours will the scientist first measure the sample and find that there is less than gram left? TEKS a 88 Chapter 2 Sequences and Series
10 Investigating g Algebra ACTIVITY Use before Lesson Investigating an Infinite Geometric Series TEKS MATERIALS scissors paper a4, a5; P4A QUESTION What is the sum of an infinite geometric series? You can illustrate an infinite geometric series by cutting a piece of paper into smaller and smaller pieces E XPLORE Model an infinite geometric series Start with a rectangular piece of paper Define its area to be square unit STEP Cut paper in half STEP 2 Cut paper again STEP 3 Repeat steps Fold the paper in half and cut along the fold Place one half on a desktop and hold the remaining half Fold the piece of paper you are holding in half and cut along the fold Place one half on the desktop and hold the remaining half Repeat Steps and 2 until you find it too difficult to fold and cut the piece of paper you are holding STEP 4 Find areas The first piece of paper on the desktop has an area of } square unit The second piece has an area of } square unit Write the areas 2 4 of the next three pieces of paper Explain why these areas form a geometric sequence STEP 5 Make a table Copy and complete the table by recording the number of pieces of paper on the desktop and the combined area of the pieces at each step Number of pieces Combined area } 2 }2 } 4 5??? DRAW CONCLUSIONS Use your observations to complete these exercises Based on your table, what number does the combined area of the pieces of paper appear to be approaching? 2 Using the formula for the sum of a finite geometric series, write and simplify a rule for the combined area A n of the pieces of paper after n cuts What happens to A n as n `? Justify your answer mathematically 24 Find Sums of Infinite Geometric Series 89
Find Sums of Infinite Geometric Series
a, AA; PB, PD TEKS Find Sums of Infinite Geometric Series Before You found the sums of finite geometric series Now You will find the sums of infinite geometric series Why? So you can analyze a fractal,
More informationSolve Linear Systems Algebraically
TEKS 3.2 a.5, 2A.3.A, 2A.3.B, 2A.3.C Solve Linear Systems Algebraically Before You solved linear systems graphically. Now You will solve linear systems algebraically. Why? So you can model guitar sales,
More informationEvaluate and Simplify Algebraic Expressions
TEKS 1.2 a.1, a.2, 2A.2.A, A.4.B Evaluate and Simplify Algebraic Expressions Before You studied properties of real numbers. Now You will evaluate and simplify expressions involving real numbers. Why? So
More informationEvaluate Logarithms and Graph Logarithmic Functions
TEKS 7.4 2A.4.C, 2A..A, 2A..B, 2A..C Before Now Evaluate Logarithms and Graph Logarithmic Functions You evaluated and graphed eponential functions. You will evaluate logarithms and graph logarithmic functions.
More informationAdd, Subtract, and Multiply Polynomials
TEKS 5.3 a.2, 2A.2.A; P.3.A, P.3.B Add, Subtract, and Multiply Polynomials Before You evaluated and graphed polynomial functions. Now You will add, subtract, and multiply polynomials. Why? So you can model
More informationSolve Quadratic Equations by Completing the Square
10.5 Solve Quadratic Equations by Completing the Square Before You solved quadratic equations by finding square roots. Now You will solve quadratic equations by completing the square. Why? So you can solve
More informationSolve Exponential and Logarithmic Equations. You studied exponential and logarithmic functions. You will solve exponential and logarithmic equations.
TEKS 7.6 Solve Exponential and Logarithmic Equations 2A..A, 2A..C, 2A..D, 2A..F Before Now You studied exponential and logarithmic functions. You will solve exponential and logarithmic equations. Why?
More informationGraph Quadratic Functions in Standard Form
TEKS 4. 2A.4.A, 2A.4.B, 2A.6.B, 2A.8.A Graph Quadratic Functions in Standard Form Before You graphed linear functions. Now You will graph quadratic functions. Wh? So ou can model sports revenue, as in
More informationSolve Systems of Linear Equations in Three Variables
TEKS 3.4 a.5, 2A.3.A, 2A.3.B, 2A.3.C Solve Systems of Linear Equations in Three Variables Before You solved systems of equations in two variables. Now You will solve systems of equations in three variables.
More informationA linear inequality in one variable can be written in one of the following forms, where a and b are real numbers and a Þ 0:
TEKS.6 a.2, a.5, A.7.A, A.7.B Solve Linear Inequalities Before You solved linear equations. Now You will solve linear inequalities. Why? So you can describe temperature ranges, as in Ex. 54. Key Vocabulary
More informationPerform Basic Matrix Operations
TEKS 3.5 a.1, a. Perform Basic Matrix Operations Before You performed operations with real numbers. Now You will perform operations with matrices. Why? So you can organize sports data, as in Ex. 34. Key
More information68% 95% 99.7% x x 1 σ. x 1 2σ. x 1 3σ. Find a normal probability
11.3 a.1, 2A.1.B TEKS Use Normal Distributions Before You interpreted probability distributions. Now You will study normal distributions. Why? So you can model animal populations, as in Example 3. Key
More informationYou studied exponential growth and decay functions.
TEKS 7. 2A.4.B, 2A..B, 2A..C, 2A..F Before Use Functions Involving e You studied eponential growth and deca functions. Now You will stud functions involving the natural base e. Wh? So ou can model visibilit
More informationYou evaluated powers. You will simplify expressions involving powers. Consider what happens when you multiply two powers that have the same base:
TEKS.1 a.1, 2A.2.A Before Now Use Properties of Eponents You evaluated powers. You will simplify epressions involving powers. Why? So you can compare the volumes of two stars, as in Eample. Key Vocabulary
More informationModel Direct Variation. You wrote and graphed linear equations. You will write and graph direct variation equations.
2.5 Model Direct Variation a.3, 2A.1.B, TEKS 2A.10.G Before Now You wrote and graphed linear equations. You will write and graph direct variation equations. Why? So you can model animal migration, as in
More informationName Class Date. Geometric Sequences and Series Going Deeper. Writing Rules for a Geometric Sequence
Name Class Date 5-3 Geometric Sequences and Series Going Deeper Essential question: How can you write a rule for a geometric sequence and find the sum of a finite geometric series? In a geometric sequence,
More informationSolve Radical Equations
6.6 Solve Radical Equations TEKS 2A.9.B, 2A.9.C, 2A.9.D, 2A.9.F Before Now You solved polynomial equations. You will solve radical equations. Why? So you can calculate hang time, as in Ex. 60. Key Vocabulary
More informationApply Properties of 1.1 Real Numbers
TEKS Apply Properties of 1.1 Real Numbers a.1, a.6 Before Now You performed operations with real numbers. You will study properties of real numbers. Why? So you can order elevations, as in Ex. 58. Key
More informationSolve Trigonometric Equations. Solve a trigonometric equation
14.4 a.5, a.6, A..A; P.3.D TEKS Before Now Solve Trigonometric Equations You verified trigonometric identities. You will solve trigonometric equations. Why? So you can solve surface area problems, as in
More informationApply Properties of Logarithms. Let b, m, and n be positive numbers such that b Þ 1. m 1 log b. mn 5 log b. m }n 5 log b. log b.
TEKS 7.5 a.2, 2A.2.A, 2A.11.C Apply Properties of Logarithms Before You evaluated logarithms. Now You will rewrite logarithmic epressions. Why? So you can model the loudness of sounds, as in E. 63. Key
More informationRepresent Relations and Functions
TEKS. a., a., a.5, A..A Represent Relations and Functions Before You solved linear equations. Now You will represent relations and graph linear functions. Wh? So ou can model changes in elevation, as in
More informationSolve Absolute Value Equations and Inequalities
TEKS 1.7 a.1, a.2, a.5, 2A.2.A Solve Absolute Value Equations and Inequalities Before You solved linear equations and inequalities. Now You will solve absolute value equations and inequalities. Why? So
More informationWrite and Apply Exponential and Power Functions
TEKS 7.7 a., 2A..B, 2A..F Write and Apply Exponential and Power Functions Before You wrote linear, quadratic, and other polynomial functions. Now You will write exponential and power functions. Why? So
More informationSolve Radical Equations
6.6 Solve Radical Equations Before You solved polynomial equations. Now You will solve radical equations. Why? So you can calculate hang time, as in Ex. 60. Key Vocabulary radical equation extraneous solution,
More informationMonomial. 5 1 x A sum is not a monomial. 2 A monomial cannot have a. x 21. degree. 2x 3 1 x 2 2 5x Rewrite a polynomial
9.1 Add and Subtract Polynomials Before You added and subtracted integers. Now You will add and subtract polynomials. Why? So you can model trends in recreation, as in Ex. 37. Key Vocabulary monomial degree
More informationWords Algebra Graph. 5 rise } run. } x2 2 x 1. m 5 y 2 2 y 1. slope. Find slope in real life
TEKS 2.2 a.1, a.4, a.5 Find Slope and Rate of Change Before You graphed linear functions. Now You will find slopes of lines and rates of change. Wh? So ou can model growth rates, as in E. 46. Ke Vocabular
More informationGraph and Write Equations of Parabolas
TEKS 9.2 a.5, 2A.5.B, 2A.5.C Graph and Write Equations of Parabolas Before You graphed and wrote equations of parabolas that open up or down. Now You will graph and write equations of parabolas that open
More informationDefine General Angles and Use Radian Measure
1.2 a.1, a.4, a.5; P..E TEKS Define General Angles and Use Radian Measure Before You used acute angles measured in degrees. Now You will use general angles that ma be measured in radians. Wh? So ou can
More informationGraph Simple Rational Functions. is a rational function. The graph of this function when a 5 1 is shown below.
TEKS 8.2 2A.0.A, 2A.0.B, 2A.0.C, 2A.0.F Graph Simple Rational Functions Before You graphed polnomial functions. Now You will graph rational functions. Wh? So ou can find average monthl costs, as in E.
More informationGraph Square Root and Cube Root Functions
TEKS 6.5 2A.4.B, 2A.9.A, 2A.9.B, 2A.9.F Graph Square Root and Cube Root Functions Before You graphed polnomial functions. Now You will graph square root and cube root functions. Wh? So ou can graph the
More information10.7. Interpret the Discriminant. For Your Notebook. x5 2b 6 Ï} b 2 2 4ac E XAMPLE 1. Use the discriminant KEY CONCEPT
10.7 Interpret the Discriminant Before You used the quadratic formula. Now You will use the value of the discriminant. Wh? So ou can solve a problem about gmnastics, as in E. 49. Ke Vocabular discriminant
More informationS.3 Geometric Sequences and Series
68 section S S. Geometric In the previous section, we studied sequences where each term was obtained by adding a constant number to the previous term. In this section, we will take interest in sequences
More information11.3 Solving Linear Systems by Adding or Subtracting
Name Class Date 11.3 Solving Linear Systems by Adding or Subtracting Essential Question: How can you solve a system of linear equations by adding and subtracting? Resource Locker Explore Exploring the
More informationProperties of the Graph of a Quadratic Function. has a vertex with an x-coordinate of 2 b } 2a
0.2 Graph 5 a 2 b c Before You graphed simple quadratic functions. Now You will graph general quadratic functions. Wh? So ou can investigate a cable s height, as in Eample 4. Ke Vocabular minimum value
More informationWrite Quadratic Functions and Models
4.0 A..B, A.6.B, A.6.C, A.8.A TEKS Write Quadratic Functions and Models Before You wrote linear functions and models. Now You will write quadratic functions and models. Wh? So ou can model the cross section
More informationLesson 12: Systems of Linear Equations
Our final lesson involves the study of systems of linear equations. In this lesson, we examine the relationship between two distinct linear equations. Specifically, we are looking for the point where the
More informationMultiplying Polynomials. The rectangle shown at the right has a width of (x + 2) and a height of (2x + 1).
Page 1 of 6 10.2 Multiplying Polynomials What you should learn GOAL 1 Multiply two polynomials. GOAL 2 Use polynomial multiplication in real-life situations, such as calculating the area of a window in
More information1.2 Inductive Reasoning
1.2 Inductive Reasoning Goal Use inductive reasoning to make conjectures. Key Words conjecture inductive reasoning counterexample Scientists and mathematicians look for patterns and try to draw conclusions
More informationPreCalc 11 Chapter 1 Review Pack v1
Period: Date: PreCalc 11 Chapter 1 Review Pack v1 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Determine the first 4 terms of an arithmetic sequence,
More informationThe Remainder and Factor Theorems
Page 1 of 7 6.5 The Remainder and Factor Theorems What you should learn GOAL 1 Divide polynomials and relate the result to the remainder theorem and the factor theorem. GOAL 2 Use polynomial division in
More informationModel Inverse Variation
. Model Inverse Variation Rational Equations and Functions. Graph Rational Functions.3 Divide Polynomials.4 Simplify Rational Epressions. Multiply and Divide Rational Epressions.6 Add and Subtract Rational
More informationPre-Calc 2nd Semester Review Packet - #2
Pre-Calc 2nd Semester Review Packet - #2 Use the graph to determine the function's domain and range. 1) 2) Find the domain of the rational function. 3) h(x) = x + 8 x2-36 A) {x x -6, x 6, x -8} B) all
More informationConvert the equation to the standard form for an ellipse by completing the square on x and y. 3) 16x y 2-32x - 150y = 0 3)
Math 370 Exam 5 Review Name Graph the ellipse and locate the foci. 1) x 6 + y = 1 1) foci: ( 15, 0) and (- 15, 0) Objective: (9.1) Graph Ellipses Not Centered at the Origin Graph the ellipse. ) (x + )
More informationFor Your Notebook E XAMPLE 1. Factor when b and c are positive KEY CONCEPT. CHECK (x 1 9)(x 1 2) 5 x 2 1 2x 1 9x Factoring x 2 1 bx 1 c
9.5 Factor x2 1 bx 1 c Before You factored out the greatest common monomial factor. Now You will factor trinomials of the form x 2 1 bx 1 c. Why So you can find the dimensions of figures, as in Ex. 61.
More informationEvaluate and Graph Polynomial Functions
5.2 Evaluate and Graph Polynomial Functions Before You evaluated and graphed linear and quadratic functions. Now You will evaluate and graph other polynomial functions. Why? So you can model skateboarding
More informationCumulative Test 1. Evaluate the expression Answers [32 (17 12) 2 ] [(5 + 3)2 31]
Name Date Cumulative Test 1 Evaluate the expression. 1. 7 + 6 3. 4 5 18 3. 4[3 (17 1) ] 4. 3 [(5 + 3) 31] 5. 3(5m 4) when m = 6. 9x 4 when x = 3 Write an algebraic expression, an equation, or an inequality.
More informationIn #8-11, Simplify the expression. Write your answer using only positive exponents. 11) 4
Algebra 1 Summer Review 16-17 Name Directions: Do your best, this is a refresher from what you learned in Algebra 1. The more you review now, the easier it will be for you to jump back into Algebra Chapter
More informationFinal Exam Review - DO NOT WRITE ON THIS
Name: Class: Date: Final Exam Review - DO NOT WRITE ON THIS Short Answer. Use x =,, 0,, to graph the function f( x) = x. Then graph its inverse. Describe the domain and range of the inverse function..
More informationFinal Exam Review. Name: Class: Date: Short Answer
Name: Class: Date: ID: A Final Exam Review Short Answer. Use x, 2, 0,, 2 to graph the function f( x) 2 x. Then graph its inverse. Describe the domain and range of the inverse function. 2. Graph the inverse
More information1. What are the various types of information you can be given to graph a line? 2. What is slope? How is it determined?
Graphing Linear Equations Chapter Questions 1. What are the various types of information you can be given to graph a line? 2. What is slope? How is it determined? 3. Why do we need to be careful about
More informationTennessee Comprehensive Assessment Program TCAP. TNReady Algebra II Part I PRACTICE TEST. Student Name. Teacher Name
Tennessee Comprehensive Assessment Program TCAP TNReady Algebra II Part I PRACTICE TEST Student Name Teacher Name Tennessee Department of Education Algebra II, Part I Directions This booklet contains constructed-response
More informationMath Studio College Algebra
Math 100 - Studio College Algebra Rekha Natarajan Kansas State University November 19, 2014 Systems of Equations Systems of Equations A system of equations consists of Systems of Equations A system of
More informationAlgebra 1. Functions and Modeling Day 2
Algebra 1 Functions and Modeling Day 2 MAFS.912. F-BF.2.3 Which statement BEST describes the graph of f x 6? A. The graph of f(x) is shifted up 6 units. B. The graph of f(x) is shifted left 6 units. C.
More informationApply Exponent Properties Involving Quotients. Notice what happens when you divide powers with the same base. p a p a p a p a a
8. Apply Eponent Properties Involving Quotients Before You used properties of eponents involving products. Now You will use properties of eponents involving quotients. Why? So you can compare magnitudes
More informationSolving and Graphing Linear Inequalities 66.1 Solve Inequalities Using Addition and Subtraction
Solving and Graphing Linear Inequalities 66.1 Solve Inequalities Using Addition and Subtraction 6.2 Solve Inequalities Using Multiplication and Division 6.3 Solve Multi-Step Inequalities 6.4 Solve Compound
More informationGraph and Write Equations of Ellipses. You graphed and wrote equations of parabolas and circles. You will graph and write equations of ellipses.
TEKS 9.4 a.5, A.5.B, A.5.C Before Now Graph and Write Equations of Ellipses You graphed and wrote equations of parabolas and circles. You will graph and write equations of ellipses. Wh? So ou can model
More informationChapter 7 - Exponents and Exponential Functions
Chapter 7 - Exponents and Exponential Functions 7-1: Multiplication Properties of Exponents 7-2: Division Properties of Exponents 7-3: Rational Exponents 7-4: Scientific Notation 7-5: Exponential Functions
More informationUnit 4 Linear Functions
Algebra I: Unit 4 Revised 10/16 Unit 4 Linear Functions Name: 1 P a g e CONTENTS 3.4 Direct Variation 3.5 Arithmetic Sequences 2.3 Consecutive Numbers Unit 4 Assessment #1 (3.4, 3.5, 2.3) 4.1 Graphing
More informationPRECALCULUS GROUP FINAL FIRST SEMESTER Approximate the following 1-3 using: logb 2 0.6, logb 5 0.7, 2. log. 2. log b
PRECALCULUS GROUP FINAL FIRST SEMESTER 008 Approimate the following 1-3 using: log 0.6, log 5 0.7, and log 7 0. 9 1. log = log log 5 =... 5. log 10 3. log 7 4. Find all zeros algeraically ( any comple
More informationFactor and Solve Polynomial Equations. In Chapter 4, you learned how to factor the following types of quadratic expressions.
TEKS 5.4 2A.1.A, 2A.2.A; P..A, P..B Factor and Solve Polynomial Equations Before You factored and solved quadratic equations. Now You will factor and solve other polynomial equations. Why? So you can find
More information2.5 Justify a Number Trick
Investigating g Geometry ACTIVITY Use before Lesson 2.5 2.5 Justify a Number Trick MATERIALS paper pencil QUESTION How can you use algebra to justify a number trick? Number tricks can allow you to guess
More informationSolve. Label any contradictions or identities. 1) -4x + 2(3x - 3) = 5-9x. 2) 7x - (3x - 1) = 2. 3) 2x 5 - x 3 = 2 4) 15. 5) -4.2q =
Spring 2011 Name Math 115 Elementary Algebra Review Wednesday, June 1, 2011 All problems must me done on 8.5" x 11" lined paper. Solve. Label any contradictions or identities. 1) -4x + 2(3x - 3) = 5-9x
More informationGraph Linear Inequalities in Two Variables. You solved linear inequalities in one variable. You will graph linear inequalities in two variables.
TEKS.8 a.5 Before Now Graph Linear Inequalities in Two Variables You solved linear inequalities in one variable. You will graph linear inequalities in two variables. Wh? So ou can model data encoding,
More informationGraph and Write Equations of Circles
TEKS 9.3 a.5, A.5.B Graph and Write Equations of Circles Before You graphed and wrote equations of parabolas. Now You will graph and write equations of circles. Wh? So ou can model transmission ranges,
More informationExponential and Radical Functions
Exponential and Radical Functions 11A Exponential Functions 11-1 11- Lab Geometric Sequences Exponential Functions Model Growth and Decay 11- Exponential Growth and Decay 11-4 Linear, Quadratic, and Exponential
More information, 500, 250, 125, , 2, 4, 7, 11, 16, , 3, 9, 27, , 3, 2, 7, , 2 2, 4, 4 2, 8
Warm Up Look for a pattern and predict the next number or expression in the list. 1. 1000, 500, 250, 125, 62.5 2. 1, 2, 4, 7, 11, 16, 22 3. 1, 3, 9, 27, 81 4. 8, 3, 2, 7, -12 5. 2, 2 2, 4, 4 2, 8 6. 7a
More informationModeling with Exponential Functions
CHAPTER Modeling with Exponential Functions A nautilus is a sea creature that lives in a shell. The cross-section of a nautilus s shell, with its spiral of ever-smaller chambers, is a natural example of
More informationHow can you use multiplication or division to solve an equation? ACTIVITY: Finding Missing Dimensions
7.3 Solving Equations Using Multiplication or Division How can you use multiplication or division to solve an equation? 1 ACTIVITY: Finding Missing Dimensions Work with a partner. Describe how you would
More informationChapter 8. Sequences, Series, and Probability. Selected Applications
Chapter 8 Sequences, Series, and Probability 8. Sequences and Series 8.2 Arithmetic Sequences and Partial Sums 8.3 Geometric Sequences and Series 8.4 Mathematical Induction 8.5 The Binomial Theorem 8.6
More informationUnit 12: Systems of Equations
Section 12.1: Systems of Linear Equations Section 12.2: The Substitution Method Section 12.3: The Addition (Elimination) Method Section 12.4: Applications KEY TERMS AND CONCEPTS Look for the following
More informationMATH 080 Final-Exam Review
MATH 080 Final-Exam Review Can you simplify an expression using the order of operations? 1) Simplify 32(11-8) - 18 3 2-3 2) Simplify 5-3 3-3 6 + 3 A) 5 9 B) 19 9 C) - 25 9 D) 25 9 Can you evaluate an algebraic
More information(MATH 1203, 1204, 1204R)
College Algebra (MATH 1203, 1204, 1204R) Departmental Review Problems For all questions that ask for an approximate answer, round to two decimal places (unless otherwise specified). The most closely related
More informationChapter 8 Sequences, Series, and Probability
Chapter 8 Sequences, Series, and Probability Overview 8.1 Sequences and Series 8.2 Arithmetic Sequences and Partial Sums 8.3 Geometric Sequences and Partial Sums 8.5 The Binomial Theorem 8.6 Counting Principles
More informationALGEBRA I EOC REVIEW PACKET Name 16 8, 12
Objective 1.01 ALGEBRA I EOC REVIEW PACKET Name 1. Circle which number is irrational? 49,. Which statement is false? A. a a a = bc b c B. 6 = C. ( n) = n D. ( c d) = c d. Subtract ( + 4) ( 4 + 6). 4. Simplify
More informationSolutions Key Exponential and Radical Functions
CHAPTER 11 Solutions Key Exponential and Radical Functions xzare YOU READY, PAGE 76 1. B; like terms: terms that contain the same variable raised to the same power. F; square root: one of two equal factors
More informationName Class Date. Simplifying Algebraic Expressions Going Deeper. Combining Expressions
Name Class Date 1-5 1 Simplifying Algebraic Expressions Going Deeper Essential question: How do you add, subtract, factor, and multiply algebraic expressions? CC.7.EE.1 EXPLORE Combining Expressions video
More informationGeometric Sequences and Series
12-2 OBJECTIVES Find the nth term and geometric means of a geometric sequence. Find the sum of n terms of a geometric series. Geometric Sequences and Series ACCOUNTING Bertha Blackwell is an accountant
More informationFunction: State whether the following examples are functions. Then state the domain and range. Use interval notation.
Name Period Date MIDTERM REVIEW Algebra 31 1. What is the definition of a function? Functions 2. How can you determine whether a GRAPH is a function? State whether the following examples are functions.
More information11.4 Partial Sums of Arithmetic and Geometric Sequences
Section.4 Partial Sums of Arithmetic and Geometric Sequences 653 Integrated Review SEQUENCES AND SERIES Write the first five terms of each sequence, whose general term is given. 7. a n = n - 3 2. a n =
More informationMATHEMATICS. Perform a series of transformations and/or dilations to a figure. A FAMILY GUIDE FOR STUDENT SUCCESS 17
MATHEMATICS In grade 8, your child will focus on three critical areas. The first is formulating and reasoning about expressions and equations, including modeling an association in bivariate data with a
More informationEssential Question How can you use a rational exponent to represent a power involving a radical?
5.1 nth Roots and Rational Exponents Essential Question How can you use a rational exponent to represent a power involving a radical? Previously, you learned that the nth root of a can be represented as
More informationdate: math analysis 2 chapter 18: curve fitting and models
name: period: date: math analysis 2 mr. mellina chapter 18: curve fitting and models Sections: 18.1 Introduction to Curve Fitting; the Least-Squares Line 18.2 Fitting Exponential Curves 18.3 Fitting Power
More information2.3 Solve: (9 5) 3 (7 + 1) 2 4
7 th Grade Final Exam Study Guide No Calculators ATN Suppose the Rocy Mountains have 72 cm of snow. Warmer weather is melting the snow at a rate of 5.8 cm a day. If the snow continues to melt at this rate,
More informationTwo-Digit Number Times Two-Digit Number
Lesson Two-Digit Number Times Two-Digit Number Common Core State Standards 4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using
More informationLesson 26: Problem Set Sample Solutions
Problem Set Sample Solutions Problems and 2 provide students with more practice converting arithmetic and geometric sequences between explicit and recursive forms. Fluency with geometric sequences is required
More informationACTIVITY: Simplifying Algebraic Expressions
. Algebraic Expressions How can you simplify an algebraic expression? ACTIVITY: Simplifying Algebraic Expressions Work with a partner. a. Evaluate each algebraic expression when x = 0 and when x =. Use
More informationSequences, Series, and Probability
Sequences, Series, and Probability 9. Sequences and Series 9. Arithmetic Sequences and Partial Sums 9.3 Geometric Sequences and Series 9.4 Mathematical Induction 9.5 The Binomial Theorem 9.6 Counting Principles
More informationADVANCED ALGEBRA (and Honors)
ADVANCED ALGEBRA (and Honors) Welcome to Advanced Algebra! Advanced Algebra will be challenging but rewarding!! This full year course requires that everyone work hard and study for the entirety of the
More informationPRACTICE FINAL , FALL What will NOT be on the final
PRACTICE FINAL - 1010-004, FALL 2013 If you are completing this practice final for bonus points, please use separate sheets of paper to do your work and circle your answers. Turn in all work you did to
More informationSequence Not Just Another Glittery Accessory
Lesson.1 Skills Practice Name Date Sequence Not Just Another Glittery Accessory Arithmetic and Geometric Sequences Vocabulary Choose the term from the box that best completes each statement. arithmetic
More information1. Does each pair of formulas described below represent the same sequence? Justify your reasoning.
Lesson Summary To model exponential data as a function of time: Examine the data to see if there appears to be a constant growth or decay factor. Determine a growth factor and a point in time to correspond
More informationMTH 65-Steiner Exam #1 Review: , , 8.6. Non-Calculator sections: (Solving Systems), Chapter 5 (Operations with Polynomials)
Non-Calculator sections: 4.1-4.3 (Solving Systems), Chapter 5 (Operations with Polynomials) The following problems are examples of the types of problems you might see on the non-calculator section of the
More informationThe given pattern continues. Write down the nth term of the sequence {an} suggested by the pattern. 3) 4, 10, 16, 22, 28,... 3)
M60(Precalculus) Evaluate the factorial expression. 9! ) 7!! ch practice test ) Write out the first five terms of the sequence. ) {sn} = (-)n - n + n - ) The given pattern continues. Write down the nth
More informationWriting and Graphing Inequalities
.1 Writing and Graphing Inequalities solutions of an inequality? How can you use a number line to represent 1 ACTIVITY: Understanding Inequality Statements Work with a partner. Read the statement. Circle
More informationSEQUENCES & SERIES. Arithmetic sequences LESSON
LESSON SEQUENCES & SERIES In mathematics you have already had some experience of working with number sequences and number patterns. In grade 11 you learnt about quadratic or second difference sequences.
More informationUsing Properties of Exponents
6.1 Using Properties of Exponents Goals p Use properties of exponents to evaluate and simplify expressions involving powers. p Use exponents and scientific notation to solve real-life problems. VOCABULARY
More informationChapter 4.1 Introduction to Relations
Chapter 4.1 Introduction to Relations The example at the top of page 94 describes a boy playing a computer game. In the game he has to get 3 or more shapes of the same color to be adjacent to each other.
More informationUsing the Laws of Exponents to Simplify Rational Exponents
6. Explain Radicals and Rational Exponents - Notes Main Ideas/ Questions Essential Question: How do you simplify expressions with rational exponents? Notes/Examples What You Will Learn Evaluate and simplify
More informationSeries, Exponential and Logarithmic Functions
Series, Eponential and Logarithmic Functions 4 Unit Overview In this unit, you will study arithmetic and geometric sequences and series and their applications. You will also study eponential functions
More informationLesson 1. Unit 6 Practice Problems. Problem 1. Solution
Unit 6 Practice Problems Lesson 1 Lesson 2 Lesson 3 Lesson 4 Lesson 5 Lesson 6 Lesson 7 Lesson 8 Lesson 9 Lesson 10 Lesson 11 Lesson 12 Lesson 13 Lesson 14 Lesson 15 Lesson 16 Lesson 17 Lesson 18 Lesson
More information