common core math 3 notes unit 1 day 1 i. systems of linear...

Math 3 Honors Notes – Unit 1 – Modeling with Linear Functions

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Common Core Math 3 Notes – Unit 1 – Day 1 “Systems” I. Systems of Linear Equations Solve each system by graphing.

A. 2 0

2 3

x y

y x

B.

2 4

2

x y

x y

A system of two linear equations in two variables is two equations considered together. To solve a system is to find all the ordered pairs that satisfy both equations.

When solving a system three situations can occur:

1. The lines intersect. There is one solution, the point where they intersect. The system is called a consistent system.

2. The lines are parallel. There is no solution, the lines do not intersect. The system is called an inconsistent system.

3. The lines are the same. There are infinitely many solutions, the lines coincide. The system is called dependent.

Systems of equations can be solved by various methods which will be discussed in this chapter.


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II. Word Problems

1. Travis and his band are planning to record their first CD. The initial start-up cost is $1500 and each CD will cost $4 to produce. They plan to sell their CD’s for $10 each. How many CD’s must the band sell before they make a profit? 2. A service club is selling copies of their holiday cookbook to raise funds for a project. The printer’s set-up charge is $200 and each book costs $2 to print. The cookbooks will sell for $6 each. How many cookbooks must the members sell before they make a profit? III. Substitution Method

To solve a system of equations by the substitution method:

1. Solve one of the equations for one of the variables.(Choose a variable with a coefficient of 1 or -1 if possible.)

2. Substitute this expression into the other equation to produce an equation with only one variable.

3. Solve the equation in Step 2 for the remaining variable. 4. Substitute this solution into the expression obtained in Step 1. 5. Solve for the second variable. 6. Write your solution set as an ordered pair, and check in each equation.

Solve each system by substitution.

A. 2 8

3 2

x y

x y

B.

5 3 2

2 3

x y

x y


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IV. Elimination Method

Steps for elimination-by-addition method:

1. Write each equation in standard form if needed. 2. If necessary, multiply one or both equations by some constant which will make the x or y coefficients opposites. 3. Add the equations from step 2 together eliminating one of the variables. 4. Solve for the remaining variable. 5. Substitute this solution into either of the original equations. 6. Solve for the second variable. 7. Write the solution set and check.

Solve each system by elimination.

A. 3 5 6

2 7 4

x y

x y

B.

6 5

4 3 7

x y

x y

C.

9 12 3

3 4 2

x y

x y


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V. Solve by Graphing. Review: Graph by using x and y intercepts. To find the x-intercept: To find the y-intercept: Substitute a 0 for y and solve for x. Substitute a 0 for the x and solve for y. (x, 0) (0, y) 1. Graph using x and y intercepts.

A. 2 4 8x y B. 1

32

x y

D. Sketch the graph. Find the coordinates of C. Solve by graphing. the figure formed or find the coordinates of the

feasible region

3 6

3 2 18

3

x y

x y

y

3

2

1

x

y

y x


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Common Core Math 3

Notes - Unit 1 Day 2

“Linear Programming” Problem 1: A calculator company produces a scientific calculator and a graphing calculator. Long-term projections indicate an expected demand of at least 100 scientific and 80 graphing calculators each day. Because of limitations on production capacity, no more than 200 scientific and 170 graphing calculators can be made daily. To satisfy a shipping contract, a total of at least 200 calculators must be shipped each day. If each scientific calculator sold result in a $2 loss, but each graphing calculator produces a $5 profit, how many of each type should be made daily to maximize profits?

X: number of scientific calculators Y: number of graphing calculators produced

1. What do the following constraints mean?

. 100

. y 80

. 200

. y 170

. 200

a x

b

c x

d

e x y

2. The above constraints are graphed below. One of the vertices of the “feasible region” is (120,80). Name the rest of the vertices and indicate what each one means. (There are 5 total) a. b. c. d. e.


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3. Each scientific calculator sold results in a $2 loss, but each graphing calculator produces a $5 profit. The equation 2 5P x y represents this situation. Explain each part of the equation.

a. P represents? b. -2x represents? c. 5y represents? d. If 50 scientific calculators were sold and 80 graphing calculators were sold, what would be the profit? 4. Using the profit equation and the vertices you found in step 2, find the profit for each vertex and explain what that means. a. b. c. d. e. 5. How many of each calculator must be sold to generate the largest profit? What is the largest profit? Problem 2: A backpack manufacturer produces an internal frame pack and an external frame pack. Let x represent the number of internal frame packs produced in one hour and let y represent the number of external frame packs produced in one hour. The inequalities describe the constraints for manufacturing both packs.

3 18

2 16

0

0

x y

x y

x

y

Graph the constraints (on the next page)


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1. Graph the constraints. (on the next page)

2. Name the vertices of the feasible region (there are 4). 3. Find the equation that represents the profit if each internal frame pack profits $50 and each external frame pack produces a profit of $80.

4. Using the vertices and the profit function, determine the maximum profit and the number of each type of backpack needed to produce. Summarize Steps in Linear Programming: 1. 2. 3. 4.


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Problem 3: Suppose you make and sell skin lotion. A quart of regular skin lotion used 2 cups of oil and 1 cup of cocoa butter. A quart of extra-rich skin lotion contains 1 cup of oil and 2 cups of cocoa butter. You make a profit of $10 per quart on regular lotion and a profit of $8 per quart on extra-rich lotion. You have 24 cups of oil and 18 cups of cocoa butter. How many quarts of each lotion should you make to maximize your profit? What is the maximum profit?

5 10 15 20


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Problem 4: Two raw materials are needed to make one of the products produced by Dartmouth Inc. The product must contain no more than 9 units of material A and at least 18 units of material B. The company can spend no more than $300 on materials for each piece produced. Material A costs $4 per unit and weighs 10 pounds per unit. Material B costs $12 per unit and weighs 20 pounds per unit. How much of each material should be used to maximize the weight of the product?

10 20 30 40 50 60 70


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Problem 5: Jerry spends no more than 20 hours a week working at two jobs during the school year. He is paid $10 an hour for tutoring Algebra 2 students and $7 an hour for delivering pizzas for Pizza Hut. He wants to spend at least 3 hours, but no more than 8 hours a week tutoring. Find Jerry’s maximum weekly earning.

5 10 15 20


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Common Core Math 3 Notes - Unit 1 Day 3 “Arithmetic Sequences and Series” I. Introduction

Towering Numbers

Complete the table to the right of the tower.

1. There are six rows in the tower pictured above. How many bricks would be in the seventh row?

2. Suppose you wanted to build a tower with 25 rows using the same design. Describe how you could figure out how many bricks you would need for the twenty-fifth row.

3. If somebody told you how many rows of bricks were in a tower, how could you figure out the

number of bricks in the longest row?

4. A very large tower was build using the same design. The longest row had 299 bricks in it. How

many rows of bricks did the tower have?

5. If somebody told you how many bricks were in the longest row of a tower, how could you figure out how many rows there were?

1

1 2 1

3 2 2 1 1

4 3 3 2 2 1 1

5 4 4 3 3 2 2 1 1

6 5 4 3 2 4 1 3 2 1 5

Row # of Bricks

1

2

3

4

5

6


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II. Arithmetic Sequences Definition – A sequence in which each term after the first is found by ADDING a constant to the previous term called the Common Difference. Formula: Examples:

1. Find the first five terms if 1 5 and 4.a d

2. What term is -3, 1, 5, 9 …29? 3. -10 is the _______th term of 14, 12.5, 11, 9.5 …-10

4. Find the indicated term: 15 for 3,3,9...a

5. Write an equation for the nth term of the sequence: 8, 17, 26, 35 … 6. Find the missing terms: 3, _____, _____, 20 7. Find the missing terms: _____, 4 , _____, _____, _____, 29


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III. Arithmetic Series Definition: A sequence written as a sum. Formula: Examples: 1. Find the sum of 5, 7, 9, … , 27

2. Find the sum of the first 50 terms where 1 5 and 25.a d

3. Name the first three terms of the series where 1 14 , = -85 and 1207n na a S .

4. Name the first three terms of the series where 1 5 , = 100 and 1050n na a S

IV. Sigma Notation – Summation Notation

Types and their formulas

Constant Linear Quadratic

Examples:


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Expand the series and find the sum.

1. 10

2

2k

k

2. 120

4

3 6x

x

3. 240

5

3 4a

a

Expand the series:

Method 1:

Method 2:

Find n Find a1 Find an Find the sum


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Common Core Math 3 Notes - Unit 1 Day 4 “Geometric Sequences and Series” I. Introduction 1. Nigel saves $2 in the first week of the New Year. If he doubles the amount he saves every week after that, how much will he save in the 2nd, 3rd, 4th, and 5th week of the year?

Week # 1st Week 2nd Week 3rd Week 4th Week 5th Week

Amount Saved

2. The price of an item triples on the first day of every month. It costs $2.00 on the 1st of January. Calculate its cost on the first day of the following four months.

Date 1st February 1st March 1st April 1st May

Cost

3. A town had a population of 100. In January 2010, a new factory was being built in the town and consequently, the local authority expected that the town’s population would increase by 20% year-on-year for the net 5 years. a. Explain clearly how the town’s population in January 2011 is calculated.

Date Jan 2011 Jan 2012 Jan 2013 Jan 2014 Jan 2015

Population

4. Examine the completed tables from the previous three questions. a. Do the numbers you calculated follow any particular pattern in each table? Table 1? Table 2? Table 3?


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II. Geometric Sequences Definition: A sequence in which each term after the first is found by multiplying a constant called a Common Ratio. Formula:

To find r:

1. Find the seventh term of a geometric sequence in which 3 96 and 4.a r

2. Find the nth term where 4

110, 10 and .

2a n r

3. Find the nth term where 6 75, =15 and 21a a n

4. Write an equation for the nth term of the geometric sequence: 3, 12, 48, 192, … 5. Find the missing geometric means: _____ , _____ , 64 , _____ , _____ , _____ , _____ , 2 ,_____ 6. Find the missing terms: _____ , 3 , _____ , _____ , _____ , 48


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III. Finite Geometric Series Formula: A sequence that is written as a sum. Examples:

1. Find the sum where 1 512, 972, and 3a a r

2. Find the sum where 1 5, r 3, and 7a n

3. Find the sum of 160 + 80 + 40 + … n = 6.

4. Find the sum: 5

1

1

12

4

n

n

5. Find the sum: 16

1

3

2 3x

x

6. Find 1 if 39,360 , 4 and 3na S n r


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IV. Infinite Geometric Series How can something infinite have a sum?

Formula:

where 1 1r . Examples: 1. Find the sum of 8 + 4 + 2 + … 2. Find the sum of 4/3 - 2/3 + 1/3 – 1/6 …

3. 1

1

3(2)a

a

4. 1

1

1(4)

2

a

a


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Common Core Math 3 Notes - Unit 1 Day 5 “Arithmetic and Geometric Sequences and Series Application” I. Introduction

To Babysit or Not To Babysit?

This summer, Shara is looking to babysit to make some extra money. She has the choice between

babysitting for two families.

The Pi Family offered Shara a flat $100 stipend for gas money plus $75 a day to babysit.

The Radical Family had a different approach. Since they wanted to ensure Shara would stay with

them for the entire month, they offered to only pay her whenever she stopped working for them in

one lump sum. She would be paid an initial amount of a penny for choosing their family and then

her pay would double each day she babysat until she decided to stop babysitting for them or when

the month ended. (For example, she would earn $0.01 initially, $0.02 for the one day, $0.04 for

two days, and so on. This means that the starting pay is after one day. ) If at any time, she wanted

to stop babysitting, they would give her the money she earned up until that point. Which family

should Shara work for, assuming she would babysit on weekdays only?

1. What function describes the Pi Family’s offer?

2. What function describes the Radical Family’s offer?

3. If Shara could only babysit for the first three weeks, would this change your choice? Why or why

not? Justify your answer using good mathematics.

4. Is there a point at which both families would pay her the same amount?

5. The Radical Family asked Shara to babysit four additional days that same summer and continue

on her pay system. If she worked the three weeks and four additional days, would that change

your mind?


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II. Identify whether the sequence is arithmetic, geometric, or neither. If the sequence is arithmetic or geometric identify the common difference or common ratio.

1. 40, 43, 46, 49,52...

2. 4,12, 36,108, 324...

3. 4,16,36,64,100...

4. 163 200na n

5. 2(2 )na n

6. 14( 3)n

na

III. Application Geometric Formulas: Use if given the initial amount. Use if given the first term. (Amount at n = 1)

0

n

na a r 1

1

n

na a r

1. Edgar is getting better at math. On his first quiz he scored 57 points, then he scores 61 and 65 on his next two quizzes. If his scores continued to increase at the same rate, what will be his score on his 9th quiz? Show all work. 2. A recovering heart attack patient is told to get on a regular walking program. The patient is told to walk a distance of 5 km the first week, 8 km the second week, 11 km the third week and so on for a period of 10 weeks. At that point the patient is to maintain the distance walked during the 10th week. How far will the patient walk during the 10th week?

3. Suppose you drop a tennis ball from a height of 15 feet. After the ball hits the floor, it rebounds to 85% of its previous height. How high will the ball rebound after its third bounce? Round to the nearest tenth.


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4. A virus reproduces by dividing into two, and after a certain growth period, it divides into two again. As the virus continues to reproduce, it will continue to divide in two. How many viruses will be in a system starting with a single virus AFTER 10 divisions? 5. A house worth $350,000 when purchased was worth $335,000 after the first year and $320,000 after the second year. If the economy does not pick up and this trend continues, what will be the value of the house after 6 years? 6. Viola makes gift baskets for Valentine’s Day. She has 13 baskets left over from last year, and she plans to make 12 more each day. If there are 15 work days until the day she begins to sell the baskets, how many baskets will she have to sell? 7. Sam has purchased a $30,000 car for his business. The car depreciates 30% every year. Depreciation means the value of the car goes down by that percent each year. What will be the value of the car after the 5th year? Note: The car is 0 years old when purchased so the first year is the second entry in the sequence.

8. Allen is on the football team this year but he has poor time management skills. His mother told him that he is off the team if he fails anything in school. On his first math quiz he earned a 90, then he earned an 86 and an 82 on his next two quizzes. If his grades continue at this rate, what will his quiz grade be after the 8th quiz? Will he still be on the team?

9. In a certain region, the number of highway accidents increased by 20% over a four year period. How many accidents were there in 2006 if there were 5120 in 2002? Hint: When the percent increases, you want the original 100% plus the additional 20%.


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Common Core Math 3 Notes – Unit 1 Day 6 “Recursive and Explicit Formulas” I. Introduction


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II. Special Sequences 1, 1, 2, 3, 5, 8… Can you find the next term?____________ Can you find the 30th term without having to fill in the blanks?_____________ This sequence is a famous sequence called ___________________________________.

Term Represented by

First Term

Second Term

Third Term

Nth Term

The pattern is an example of a _______________________________ which means that ___________ _____________________________________________________________________________________ Example 1:

Find the first five terms of the sequence in which 1 15 and 2 +7 and 1.n na a a n

n = 1 n = 2 n = 3 n = 4

First five terms: ____________________________________________ Example 2:

Find the first five terms of the sequence in which 1 13 ; and 1n na a a n n


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III. Arithmetic Explicit and Recursive Formulas

Explicit Formula: Recursive Formula:

Example 1: 7, 10, 13, 16, …


Example 2: 386, 365, 344, 323, …


IV. Geometric Explicit and Recursive Formulas


Example 1: 297, 99, 33, 11, …


Example 2: 1, -6, 36, -216, …


1

1

start

n n

a

a a d

Hopefully you are reminded of:

Now = start

Next = now + d

1 ( 1)na a n d

1

1

start

n n

a

a a r

Hopefully you are reminded of:

Now = start

Next = (now)(r)

1

1

n

na a r


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Determine whether the sequence is arithmetic or geometric. Then write the explicit and recursive formula. 1. 40, 43, 46, 49… Explicit Recursive 2. 3, 12, 48, 192… Explicit Recursive 3. -34, -26, -18, -10… Explicit Recursive 4. 236, 118, 59, 29.5… Explicit Recursive Determine whether the formula is arithmetic or geometric and explicit or recursive. Then write the opposite formula for each sequence.

1. 1( 3)n

na 2. 1

1

8

2

n na a

a

3. 1

1

10

6

n na a

a

4. 43na n

common core math 3 notes unit 1 day 1 i. systems of linear...

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