Experimenting with lines



Constructing Algebra

By Germán A. Moreno

1.1 Number Sets and the Structure of Algebra

Example 1

|x |C |

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Example 2

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Example 3

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Example 4

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Example 5

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Set Theory

Cantor described a set as a collection of definite, distinguishable objects of our perception to be conceived as a whole. The individual objects are called elements of the set. Sets are finite, infinite, or empty.

1. What do the words definite and distinguishable mean in every day language?

Example 6

The intersection of the set of all cars with the set of all trucks.

Example 7

Give an example of a finite, infinite and an empty set.

The set of vowels {a, e, i, o, u}

The set of years {…,-500 B.C.,…,0 A.D.,…500 A.D.}

Example 8

Example 9

Example 10

1.2 Fractions

Example 1

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Example 2

{15, 30, 45, 60,…}

Example 3

Multiples of 14: 14, 28, 42, 56,

Multiples of 21: 21, 42,63, 84,

Example 4

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Example 5

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Example 6

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Example 7

2 is a prime number.

Example 7

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Example 8

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1.3 Adding and Subtracting Real Numbers; Properties of Real Numbers

Example 1

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Example 2

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Example 3

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Example 4

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Example 5

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Using the Properties and Theorems

Example 6

Add or subtract the following by using the properties and theorems.

a. [pic]

b. [pic]

c. [pic]

d. [pic]

e. [pic]

1.4 Multiplying and Dividing Real Numbers; Properties of Real Numbers

Example 1

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Example 2

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Example 3

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Example 4

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Example 5

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Example 6

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Using the Properties and Theorems

Example 7

Evaluate.

a. [pic]

b. [pic]

c. [pic]

d. [pic]

e. [pic]

f. [pic]

1.5 Order of Operations

Warm Up Activity 1

Complete the statement so that it is true.

1. [pic]

2. [pic]

3. [pic]

4. [pic]

5.) [pic]

Example 1

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Example 2

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The Order of Operations is very important when simplifying expressions and equations. The Order of Operations is a standard that defines the order in which you should simplify different operations such as addition, subtraction, multiplication and division.

This standard is critical to simplifying and solving different algebra problems. Without it, two different people may interpret an equation or expression in different ways and come up with different answers. The Order of Operations is shown below.

Example 2

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Example 3

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Example 4

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Example 5

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Application Problems

Example 6

A teacher calculates his students’ grade as follows. Sixty percent of the final grade comes from their test average and 40 percent comes from their homework average. Roberto’s homework average is a 90 and his 4 test grades were 88, 89, 90, and 87. What will be his final grade?

1.7 Combining Like Terms

Example 1

Example 2

Evaluate Expressions

We have learned that, in an algebraic expression, letters can stand for numbers. When we substitute a specific value for each variable, and then perform the operations, it's called evaluating the expression.

Example 3

Evaluate [pic]when y = 5.

Example 4

Evaluate [pic] when u = 2 and v = 7.

Example 3

Combining Like Terms

When we talk about combining like terms we use the operations addition and subtraction.

Rule Number 1: With all operations you treat the coefficients as you would if they were just part of a problem that did not have variables.

 

 

Rule number 2: When you add or subtract like terms the power of the variable does not change. The only thing that changes is the coefficient like stated in rule 1.

Example 4

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Example 5

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Example 6

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Example 7

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2.1 Equations, Formulas, and the Problem Solving Process

How to Solve It

1. UNDERSTANDING THE PROBLEM

o First. You have to understand the problem.

o What is the unknown? What are the data? What is the condition?

o Draw a figure. Introduce suitable notation.

2. DEVISING A PLAN

o Second. Find the connection between the data and the unknown. You should obtain eventually a plan of the solution.

o Have you seen it before? Or have you seen the same problem in a slightly different form?

o Do you know a related problem? Do you know a theorem that could be useful?

o Could you restate the problem? Could you restate it still differently? Go back to definitions.

o If you cannot solve the proposed problem try to solve first some related problem. Could you solve a part of the problem?? Could you derive something useful from the data? Could you think of other data appropriate to determine the unknown?

o Did you use all the data? Did you use the whole condition? Have you taken into account all essential notions involved in the problem?

3. CARRYING OUT THE PLAN

o Third. Carry out your plan.

o Carrying out your plan of the solution, check each step. Can you see clearly that the step is correct? Can you prove that it is correct?

4. Looking Back

o Fourth. Examine the solution obtained.

o Can you check the result? Can you check the argument?

o Can you derive the solution differently? Can you see it at a glance?

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2.2 The Addition Principle

Example 1

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Example 2

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Example 3

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Example 4

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Example 5

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Application Problems

Example 7

Robert knows the distance from his home to work is 42 miles. Unfortunately he gets a flat tire 16 miles from work. How far did Robert drive before his flat tire?

Example 8

The perimeter of the trapezoid shown is 100 yards. Find the length of the missing side.

[pic]

Example 9

In a survey respondents were given a statement and they could agree, disagree, or have no opinion regarding the statement. The results indicate [pic] of the respondents agree with the statement while [pic] disagree. What fraction had no opinion?

2.3 The Multiplication Principle

Example 1

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Example 2

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Example 3

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Example 4

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Example 5

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Application Problems

Example 5

Example 6

Example 7

2.5 Translating Word Sentences to Equations

Keywords that involve Addition

sum

total

more than

greater than

consecutive

increased by

plus

older than

farther than

Example 1:

|The sum of the length and width is 30. | |

|The length is 4 more than the width. | |

|The length is greater than the width by 5. | |

|The consecutive integer after the integer n is 20. | |

|Jane is two years older than Alice. | |

|Alice ran 5 kilometers farther than Jane. | |

Keywords that involve Subtraction

difference

diminished by

fewer than

less than

decreased by

minus

subtracted from

younger than

Example 2:

|The difference between Jane’s age and Alice’s age is 10. | |

|John has 10 fewer coins than Alice. | |

Keywords that involve Multiplication

product

twice

doubled

tripled

times

multiplied by

of

Example 3:

|The product of the length and the width is 50. | |

|The length is twice the width. | |

|John took half the number of marbles. | |

Keywords that involve Division

quotient

divided by

divided into

quotient of

in

per

Example 4:

|The length divided by the width is 5. | |

|John traveled 100 miles in 2 hours. | |

Example 5

Eight times the sum of eight and a number is equal to one hundred sixty.

Example 6

The product of three and the sum of a number and four added to twice the number yilds negative three.

Example 7

Five times the sum of a number and one-third is equal to three times the number increased by two-thirds.

Example 8

Translate the equation to a word sentence.

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Example 9

Translate the equation to a word sentence.

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2.6 Solving Linear Inequalities

Example 1

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Example 2

[pic] [pic] [pic] [pic]

Graphing Inequailties:

1. If the symbol is [pic] or [pic] draw a [ or a [pic] on the number line on the indicated number. If the symbol is < or > draw ( or an [pic] on the number line on the indicated number.

2. If the variable is > the indicated number, shade to the right. If the variable is < the indicated number, shade to the right.

Example 1

[pic] [pic]

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Example 2

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Example 3

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Example 4

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Example 5

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Example 6

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Example 7

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Example 8

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Application Problems

Example 9

The design of a storage box calls for a width of 27 inches and a length of 41 inches. If the surface area must be at least 4254 square inches, find the range of values for the height.

Example 10

In Aaron’s English course, the final grade is determined by the average of five papers. The department requires any student whose average falls below 75 to repeat the course. Aaron’s scores on the first four papers are 68, 78, 80 and 72. What range of scores on the fifth paper would cause him to have to repeat the course?

4.1 The Rectangular Coordinate System

Warm-Up Activity 2

Graph the following points.

1. (-5, -3.4)

2. (8.5,-3)

3. (-6.4,6)

4. (2/3,4/3)

Making Scatter Plots of Data

Make a different scatter plot for each of the following data sets.

1.

|x |y |

|3 |8 |

|5.5 |7.1 |

|8 |6.2 |

|12 |5 |

|16 |4.4 |

|18.5 |3.4 |

|23 |2 |

|25.5 |1 |

|30 |0.3 |

|31 |-1 |

2.

|x |y |

|-5 |10 |

|-15 |20 |

|-20 |30 |

|-25 |40 |

|-30 |50 |

|-40 |60 |

|-50 |70 |

3.

|x |y |

|-6 |5 |

|-2.5 |8 |

|-.5 |12 |

|0.9 |15 |

|1.8 |18 |

|3.5 |22 |

|5.9 |25 |

|8 |30 |

4.

|x |y |

|-2 |26 |

|-1 |17 |

|0 |10 |

|1 |5 |

|2 |2 |

|3 |1 |

|4 |2 |

|5 |5 |

|6 |10 |

|7 |17 |

|8 |26 |

5. Choose points that lie on a line and then make a scatter plot of the data.

|x |y |

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6. Choose points that make a pentagon then makea scatter plot of the data.

| x | y |

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4.7 Introduction to Relations and Functions

Example 1

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Example 2

Construct any relation with the specified number of elements.

1. 4 elements

2. 5 elements

3. 6 elements

Example 3

Identify the domain and range of each relation from Example 2

Example 4

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The Vertical Line Test

The vertical line test is a way to determine whether or not a relation is a function. The vertical line test simply states that if a vertical line intersects the relation's graph in more than one place, then the relation is a not a function.

Example 5

Activity I

Relations and Functions

Determine whether or not the following relations are functions. Explain how you know that it is or is not a function.

1.

|X |Y |

|1 |1 |

|3 |3 |

|5 |4 |

|7 |7 |

|9 |9 |

|11 |0 |

|13 |8 |

|15 |9 |

2.

|X |Y |

|-10 |-8 |

|-6 |-3 |

|-4 |0 |

|-1 |1 |

|0 |3 |

|1 |-2 |

|2 |-4 |

3.

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4.

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5.

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6.

|X |Y |

|1 |1 |

|3 |3 |

|1 |4 |

|7 |7 |

|9 |9 |

|3 |0 |

|13 |8 |

|15 |9 |

7.

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8.

[pic]

9. [pic]

|X |-3 |

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A. Choose any 5 positive integers as inputs.

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B. Choose any 5 integers as inputs. Choose positives and negatives.

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Example 4

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Example 5

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Horizontal Lines

Example 6

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Vertical Lines

Example 7

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Application Problems

Example 8

Example 9

4.3 Graphing Using Intercepts

Example 1

Finding the x-intercepts

To find an x intercept.

1. Replace y with 0 in the given equation.

2. Solve for x

Finding the y-intercepts.

To find a y-intercept

1. Replace x with 0 in the given equation.

2. Solve for y.

Example 2

Equations in standard form.

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Example 3

Equations in slope-intercept form.

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Example 4

Equations in y = mx form.

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Example 5

Equations in y = c form.

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Example 6

Equations in x = c form.

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Application Problem

Example 7

Slope-Intercept Form.

Example 1

1. Sally and her mom are riding bikes to the park. The graphs below show the distance traveled over time. Who is traveling faster?

[pic][pic]

Example 2

Experimenting with lines

Slope

1. In your calculator or on a graph, graph any line with a positive slope. If you are using a calculator sketch the graph on you paper.

2. Choose a different number for the slope (choose a positive number) and graph the new line on the calculator or on the same graph. If you are using a calculator sketch the graph on you paper.

3. Choose a different number for the slope (choose a positive number) and graph the new line on the calculator or on the same graph. If you are using a calculator sketch the graph on you paper.

4. Follow steps 1-3, but now use negative numbers for all the slopes. When you are done with this you should have 6 different lines.

6. In a few sentences explain the effect of changing the value for slope on the line.

Y-intercept

1. In your calculator or on a graph, graph any line. If you are using a calculator sketch the graph on you paper.

2. Choose a different number for the the y-intercept and graph the new line on the calculator or on the same graph. If you are using a calculator sketch the graph on you paper.

3. Choose a different number for the y-intercept and graph the new line on the calculator or on the same graph. If you are using a calculator sketch the graph on you paper.

4. In a few sentences explain the effect of changing the value for y intercept on the line.

[pic] [pic]

Example 3

Compute the slope of the line that goes through the given points. Then plot the points and the line that goes through them on a graph.

1. (1, 2), (6, 5) 2. (1, 3), (2,-1)

3. (-5,-6), (10, 11) 4. (-1, 6), (3, 15)

Example 4

The graph shows how much gasoline will be used on the family vacation this year. Determine the slope of the line.

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Example 5

After spending a day at a water park, Sue returned her inner tube with mud on it and now she must pay a cleaning fee. The solid line represents the cost of renting an inner tube. The dashed line represents the rental cost plus the cleaning fee. What is the slope of each line? How much is the cleaning fee?

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Graphing Equations in Slope-Intercept Form

Example 4

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Determine the equation of a line given two points on the line.

Example 5

[pic], [pic]

[pic], [pic]

[pic], [pic]

[pic], [pic]

Pattern Block Activity

Algebra can be used to describe patterns. In this activity you will write linear equations that model a pattern. For each of the pictures below you must suppose that the object is lying on the floor and that you are being asked to determine how many faces of the object are exposed. For each of the pages:

1. Fill in the table shown.

2. Determine the function rule for the pattern that expresses how many faces are exposed.

3. Make a scatter plot and graph the function rule in the calculator. Sketch the graph of the scatter plot and the function rule.

|N |Picture |Explanation |Process |F(n) |

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|N |Picture |Explanation |Process |F(n) |

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|N |Picture |Explanation |Process |F(n) |

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|n |Picture |Explanation |Process |F(n) |

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Point Slope Form

Point-Slope form

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Example 1

Standard Form

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Example 2

Write the equation of the line represented by the table and graph in standard form.

|x |y |

|-3 |1.1 |

|-2 |1.4 |

|-1 |1.7 |

|0 |2 |

|1 |2.3 |

|2 |2.6 |

|3 |2.9 |

[pic]

Example 3

Write the equation of the line that passes through (2,3) and is parallel to the line below.

[pic]

|x |y |

|-3 |-5 |

|-2 |-3 |

|-1 |-1 |

|0 |1 |

|1 |3 |

|2 |5 |

|3 |7 |

Example 4

Write the equation of the line that passes through (2,3) and is perpendicular to the line below.

[pic]

|x |y |

|-3 |-5 |

|-2 |-3 |

|-1 |-1 |

|0 |1 |

|1 |3 |

|2 |5 |

|3 |7 |

4.6 Graphing Linear Inequalities

Graphing Linear Inequalities in Two Variables

To graph a linear inequality in two variables (say, x and y), first get y alone on one side. Then consider the related equation obtained by changing the inequality sign to an equals sign. The graph of this equation is a line.

If the inequality is strict ([pic] or [pic]), graph a dashed line. If the inequality is not strict ([pic] or [pic]), graph a solid line.

Finally, pick one point not on the line ((0, 0) is usually the easiest) and decide whether these coordinates satisfy the inequality or not. If they do, shade the half-plane containing that point. If they don't, shade the other half-plane.

Example 1

[pic]

Example 2

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Application Problems

Example 3

Pg. 334 #41

Example 4

Pg. #42

8.1 Solving Systems of Linear Equations

Example 1

Problem 1

The school’s rocket club wants to travel to Washington D.C. for a competition. They have built a 4 ft. rocket for the competition and they think they can send it up 2000 ft in the air. The club must hold a fund raiser to raise the money. They decide to sell chocolates door to door to raise money, but they have found two companies that sell the chocolates. Chocolates-Are-Us charges a $50 base fee and then charges $0.90 per chocolate. Chocolates-Chocolates charges an $80 base fee and then charges $0.81 per chocolate. Answer the following questions.

1. Fill out the table below to determine which is the better offer.

|Number of chocolates |Process column |Cost at |Process column |Cost at |

| |(Chocolates-Are-Us) |Chocolates-Are-Us |(Chocolates-Chocolates) |Chocolates-Chocolates |

|10 |  |  |  |  |

|20 |  |  |  |  |

|30 |  |  |  |  |

|60 |  |  |  |  |

|90 |  |  |  |  |

|120 |  |  |  |  |

| 240 |  |  |  |  |

| 480 |  |  |  |  |

|N |  |  |  |  |

2. Write a function rule for the cost of chocolates from Chocolates-Are-Us.

3. Write a function rule for the cost of chocolates from Chocolates-Chocolates.

4. How many chocolates (approximately) can you buy from Chocolates-Are-Us for

$190? $310?

5. How many chocolates (approximately) can you buy from Chocolates-Chocolates for $190? $310?

6. Graph both deals on the same graph below.

7. Determine at which point the cost is the same for the same amount of chocolates (approximately) and write down the number of chocolates and the cost below.

Example 2

On Your Own

Problem 2

The school choir would like to travel to a competition in from El Paso to Washington D.C. The choir must hold a fund raiser to raise the money. They decide to sell candies at a school fair to raise money, but they have found two companies that sell the candies. Company X charges a $40 base fee and then charges $0.50 per candy. Company Y charges a $100 base fee and then charges $0.35 per candy. They will be selling the candies for $2.00.

Answer the following questions.

2. Fill out the table below to determine which is the better offer.

|Number of roses |Process column (Company X) |Cost at company X |Process column (Company |Cost at Company Y |

| | | |Y) | |

|10 |  |  |  |  |

|20 |  |  |  |  |

|30 |  |  |  |  |

|60 |  |  |  |  |

|90 |  |  |  |  |

|120 |  |  |  |  |

| 240 |  |  |  |  |

| 480 |  |  |  |  |

|N |  |  |  |  |

2. Write a function rule for the cost of candies for Company X.

3. Write a function rule for the cost of candies from company Y.

4. How many candies can you buy from Company X for $200? $300? $350?

5. How many candies can you buy from Company Y for $200? $300? $350?

6. Graph both deals on the same graph below.

7. Determine at which point the cost is the same for the same amount of candies approximately and write down the number of candies and the cost below.

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Walking Experiment

The objective of this experiment is to obtain data about you and your partners walking speed. Once you have collected raw data from your experiment, you are to determine the equation that models your walking.

Materials:

4 partners

timer

calculator

notepad

Procedure:

1. When you get to the track, practice walking at a constant rate. That means that you should not speed up or slow down as you walk.

2. Choose one of the partners to be the first walker.

3. Someone should be chosen to write down the data using the table provided below.

4. Begin timing him/her from the 0 yard line.

5. Another partner should walk along with the walker and notice the distance the walker has traveled.

6. Every thirty seconds the timer should request the distance of the walker from the partner mentioned in 4.

7. Data should be collected every thirty seconds so that there are at least 10 data points.

8. Repeat steps 1-7 for a different walker at a different speed.

x f(x) x f(x)

Example 1

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or

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Example 2

[pic] is a solution to the following system:

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Example 3

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Example 4

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Application Problems

Example 6

Example 7

8.2 Solving Systems of Linear Equations by Substitutuion

Substitution Method for Solving Systems of Equations with Two Equations

1. Choose one equation and isolate one variable; this equation will be considered the first equation. (Use the method detailed in earlier units to do this.)

2. Substitute the solution from step 1 into the second equation and solve for the variable in the equation.

3. Using the value found in step 2, substitute it into the first equation and solve for the second variable.

4. Substitute the values for both variables into both equations to show they are correct.

Example 1

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Example 2

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Example 3

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Example 4

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Application Problem

Example 5

Example 6

8.3 Solving Systems of Linear Equations by Elimination

Solving a System of Linear Equations by Elimination by Addition:

Elimination by Addition is the second method that is used in this unit to get an exact solution to a system of equations.  The steps for solving by elimination are:

1. Decide which variable you are going to eliminate. The signs of that variable have to be opposite.

2. If the coefficients of that variable have different values, then make them the same but opposite in sign. This can be done easily by multiplying the top equation by the coefficient of that variable in the second equation. Multiply the second equation by the coefficient of that variable in the first equation.

3. Add the two equations. One of the variables should disappear.

4. Solve for the remaining variable.

5. Solve for the other variable using either equation.

Example 1

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Example 2

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Example 3

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Example 4

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Application Problem

Example 5

Example 6

5.1 Exponents

Example 1

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Example 2

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Example 3

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Example 4

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5.2 Introduction to Polynomials

Example 1

Example 2

Example 3

Example 4

 

Example 5

Example 6

Evaluating Polynomials

Example 7

x = 4

[pic]=

u = -2

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Writing polynomials in descending order of degree

Example 8

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Combining Like Terms

Example 9

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5.3 Adding and Subtracting Polynomials

Adding Polynomials

1. Change all subtraction to adding the opposite.

2. Drop the parentheses.

3. Combine like terms.

4. State the answer in standard form.

Example 1

Example 2

Subtracting Polynomials

1. Change all subtraction to adding the opposite when necessary.

2. The negative in front of the second set of parentheses is understood to be a –1.

3. To eliminate the parentheses, distribute the –1. Remember that distributing by -1 will change each term to its opposite.

4. Combine like terms.

5. State the answer in standard form.

Example 3

Example 4

Activity

Pg. 409 Collaborative Exercise Building Furniture and Profits

5.4 Exponent Rules and Multiplying Monomials

Example 1

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Monomials

Example 2

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Example 3

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Example 4

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Example 5

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5.5 Multiplying Polynomials

Multiplying a Monomial by a Polynomial

To multiply a Polynomial by a Monomial, use the distributive property to multiply each term in the polynomial by the monomial.

Example 1

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2.

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4.

|r |w |l |A |

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Activity

Quadratic Equations

Suppose you have 200 feet of fencing available to enclose a playground. Suppose the play area will be against a large wall. What dimensions will give the playground the greatest area.

1. Understanding the Problem

a. Draw a sketch of the rectangular playground and its location against the wall. Use variables to label the sides.

b. Write any equations you can think of with the given information.

c. Make a table of data.

d. Graph the data.

2. Devise a plan.

a. Explain your plan for answering the question.

3. Carry out the plan.

a. Show your work to find the answer to the question.

4. Looking back.

a. Check you work or solve the problem a different way.

Activity 2- Quadratic Functions:

Pipe Cleaners and Area

Objectives:

1. To explore functions in Geometry

2. To investigate the relationship between area and height

3. To develop an intuitive feel for the behavior of related variable quantities

4. To understand maximum area, and upper and lower bounds

on the value of height

5. To make connections between the mathematical world and the real-world

6. To make connections between the geometric representation,(the graph), of the real-world phenomenon and the algebraic representation, (the function)

Initial Investigation:

I.

1. Using graph paper of 1x1 unit size, measure the length of the pipe cleaner.

2. Bend each end of the pipe cleaner upward so that each piece bent upward is one unit long. Each upward bent piece is called the height of the pipe cleaner.

3. The bottom part of the pipe cleaner is called the base . Using the graph paper, what is the length of the base when the height is 1 unit long?

Length of Base:______

4. The area (of the figure made from the pipe cleaner), is the number of squares above the pipe cleaner, but no higher then the tops fo the pieces that are bent upward. What is the area of the figure when the height is 1 unit long?

Area:_____________

II.

Now, straighten the pipe cleaner out. This time, turn th eend pieces upward so that the height of the figure is 2 units long.

a. Find the length of the base.

b. Find the area of the figure.

III.

Continue to straighten and bend the pipe cleaner upward one unit longer each time. When there is no longer enough pipwe cleaneer to continue, you may stop. Record your results below.

Height Base Area

1. What are the variables in this activity?

Even though there are several variables in this activity, it is necessary to focus our attention on only one at a time. For this reason, we will look at height.

2. Are there any constants in this activity?

3. Looking at the height, base, and area, do you see a relationship between the three measurements?

Application with Spreadsheet:

1. Using the relationship that you have found between the base, height, and area, make a spreadsheet from what you have found.

HEIGHT AREA BASE

2. a. The largest possible value of a variable is called its maximum. What is the maximum value for the area in this activity?

b. What is the value of the height when this maximum area is reached?

c. Does this value of the height also produce a maximum value for the base? Explain why or why not.

d. Using the spreadsheet, make a graph of the height and the area. Which one is the dependent variable? Which one is the independent variable? Explain the difference between the two concepts.

e. Describe what the graph looks like. What shape is the graph?

f. Describe where the maximum value for area is on the graph.

12. A rectangular buildings floor has a width that is three times as long as the length.

What is the area of the buildings floor?

10.3 Solving Quadratic Equations Using the Quadratic Formula

The Quadratic Formula

[pic]

Example 1

Example 2

Example 3

Example 4

Application Problem

Example 5

A cylinder is to be made so that its volume is equal to that of a sphere with a radius of 3 inches. If a cylinder is to have a height of 4 inches find its radius.

Example 6

The equation [pic]models the trajectory of a ball thrown upwards and outwards from a height of 6 feet. Determine the horizontal distance from where the ball was thrown when the ball hits the ground.

10. 4 Graphing Quadratic Equations

A quadratic equation in two variables is an equation of the form

y = ax2 + bx + c. The graph of a quadratic equation is called a parabola.

Look at the graph of y = x2 This is the starting point for every parabola.

The direction of a parabola

If a > 0, then the parabola opens up

If a < 0, then the parabola opens down

The x-intercepts of a parabola

Every parabola passes through the x-axis either 0, 1, or 2 times,

determined by how many real zeros the equation has.

To find the x-intercepts of a parabola, solve the equation ax2 + bx + c = 0

The vertex of a parabola

The vertex of a parabola is the highest point (if it opens down) or lowest

point (if it opens up) of the parabola

Find the vertex by one of the following methods. 1. Take the average of the x coordinates of the x intercepts. This is the x coordinate of the vertex. Use the equation to find y coordinate. 2. The x coordinate of the vertex is given by x=-b/2a. Then find the y coordinate by using the given equation.

Axis of symmetry

Every parabola is symmetric across the vertical line passing through the

vertex, i.e. the line x = − b/2a

The y-intercept of a parabola

Every parabola has exactly 1 y-intercept

To find the y-intercept, plug 0 in to the equation for x

Example 1

Example 2

Example 3

Application Problem

Example 4

Example 5

Cylinders-1

Step 1. Create a cylinder with a circumference of 11 in using an 8.5x11 plain white paper. Cut the cylinder to any height you choose.

Step 2. Measure the height and radius of several cylinders and fill out the table below.

|Height |Radius |Volume |

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Step 3. Input the data into the table in your calculator.

Step 4. Create a scatter plot of the data in your calculator using height as the input and volume as the output. Sketch the scatter plot below.

Step 5. Determine whether the data is linear or quadratic by finding first and second differences from the data (you must show this above in the table). Then use the stat-calc buttons to find the equation for the data. Make sure that [pic]. If not, you chose the wrong type of curve. Write the equation and r value below.

Step 6. Using the second 8.5x11 sheet of paper to create the cylinder with the greatest height possible and using the third sheet create a cylinder with the smallest value for height that you can make. Record these values below.

Step 7. Determine the domain and range using the values from Step 6. Write the domain and range below.

Step 8. Use the domain to graph the function below.

Answer the following questions using the above information.

1. Tell what kind of function best represented the data.

2. Using the equation, predict what the volume will be of a cylinder with a height of .0833… feet . (Make sure to convert to inches).

3. Discuss whether it is possible to make cylinders with volumes of 2 in sq., 20 in sq., .1 ft. sq., and 50 ft. sq.

4. Suppose that you change the circumference to 10 in. Show how the equation will be different from the one where the circumference was 11 in.

5. Using the circumference of 10 inches, show how the graph will be different to the one that uses a circumference of 11 inches.

6. Explain how the domain and range will differ from that which you found using the circumference of 11 inches if you use a circumference of 10 inches. (Assume that you can only use an 8.5x11 paper)

Cylinders-2

Step 1. Create a cylinder with a height of 9 in. using an 8.5x11 plain white paper.

The surface area of the side of the cylinder should be either 9, 18, 27, 36, 45, 54, or 63 in. sq. as specified by the teacher.

Step 2. Calculate the radius and volume and fill out the table below.

|Radius |Area |Height |Volume |

| | | | |

| | | | |

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Step 3. Input the data into the table in your calculator.

Step 4. Create a scatter plot of the data in your calculator using radius as the input and volume as the output. Sketch the scatter plot below.

Step 5. Determine whether the data is linear or quadratic by finding first and second differences from the data (you must show this above in the table). Then use the stat-calc buttons to find the equation for the data. Make sure that [pic]. If not, you chose the wrong type of curve. Write the equation and [pic] value below.

Step 6. Using the second 8.5x11 sheet of paper to create the cylinder with the greatest volume possible and using the third sheet create a cylinder with the smallest value for volume that you can make (The height should remain 9 in.). Record the volume and radii of each of the cylinders.

Step 7. Determine the domain and range using the values from Step 6. Write the domain and range below.

Step 8. Use the domain to graph the function below.

Answer the following questions using the above information.

7. Tell what kind of function best represented the data.

8. Using the equation, predict what the volume will be of a cylinder with an area of 22.5 in. sq.

9. Discuss whether it is possible to make cylinders with volumes of 2 in cu. and 200 in. cu. with a cylinder that has a 9 in. height and using an 8.5x11 in paper.

10. Suppose that you change the height to 2 inches. Show how the equation will be different from the one where the height was 9 in.

11. Suppose that you change the height to 2 inches. Show how the graph will be different to the one that uses a height of 9 inches.

Straw Activity

Step 1. Cut 4 of the straws at any height. Keep one of the pieces and discard the other.

Step 2. Measure the height and radius of each piece and fill out the table below.

|Height |Radius |Volume |

| | | |

| | | |

| | | |

| | | |

| | | |

Step 3. Input the data into the table in your calculator.

Step 4. Create a scatter plot of the data in your calculator using height as the input and volume as the output.

Step 5. Determine whether the data is linear or quadratic by finding first and second differences from the data. Then use the stat-calc buttons to find the equation for the data. Make sure that [pic]. If not you chose the wrong type of curve.

Step 6

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Theorem: If a and b are elements of the real numbers, a < b and c > 0, then [pic]. If a < b and c < 0, then [pic]

Theorem: If a and b are elements of the real numbers and a < b, then a + c < b + c.

Definition: The square root of x is the number a that, when multiplied by itself, gives the number, x.

Definition: The greatest common factor, or GCF, is the greatest factor that divides two numbers.

Series1

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Definitions: A line of symmetry is a line that divides a graph into two symmetrical halves. The vertex is the lowest pint on a parabola that opens up or highest point on a parabola that opens down.

Definition: A quadratic equation in two variables is an equation that can be written in the form [pic], where a, b, and c are real numbers and [pic].

Factoring a Difference of Two Squares

[pic]

Note that the sum of two squares DOES NOT factor.

Just like the perfect square trinomial, the difference of two squares  has to be exactly in this form to use this rule.   When you have the difference of two bases being squared, it factors as the product of the sum and difference of the bases that are being squared.

Factoring a Perfect Square Trinomial

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OR

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It has to be exactly in this form to use this rule.  When you have a base being squared plus or minus twice the product of the two bases plus another base squared, it factors as the sum (or difference) of the bases being squared. 

Rule: Perpendicular Lines-The slope of a perpendicular line with a slope of a/b will be –b/a.

Rule: Parallel lines-The slopes of parallel lines are equal.

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The slope formula. [pic]

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Definition: The empty set is the set which has no elements belonging to it.

1. Parentheses and Brackets -- Simplify the inside of parentheses and brackets before you deal with the exponent (if any) of the set of parentheses or remove the parentheses.

2. Exponents, Square Roots and Absolute Values -- Simplify the exponent of a number or of a set of parentheses, find square roots and find absolute values, before you multiply, divide, add, or subtract it.

3. Multiplication and Division -- Simplify multiplication and division in the order that they appear from left to right.

4. Addition and Subtraction -- Simplify addition and subtraction in the order that they appear from left to right.

Definition 1. The set of integers is the set consisting of the natural numbers, their negatives, and the number 0.

Definition 1. The set of rational numbers is the set of numbers that can be expressed in the form [pic] , where a and b are integers, and [pic].

Definition The set of real numbers is the set which consist of all numbers representable by all the points of a number line.

Definition: A set is said to be finite if it contains n elements or if it is the empty set. A set which is not finite is said to be infinite.

Definition: The domain is the set of all input values for a relation. The range is the set of all output values for a relation.

Definition: A relation is a set of ordered pairs.

Definition. An ordered pair [pic] of elements is a grouping together of two elements in a definite order, so that equality is defined as [pic]means that [pic]and [pic].

Definition: A variable is a symbol for which any element of a specified set, called the domain may be substituted.

Definition: Two binomials are conjugates of each other if the terms are same but the operation between the terms differs.

Definition: Let a and b be elements of the real numbers. To subtract a from b means to find the unique x in the real numbers such that [pic].

Determining if an ordered pair is a solution.

To determine whether a given ordered pair is a solution for an equations with two variables:

1. Replace the variables in the equation with the corresponding coordinates.

2. Verify that the equation is true.

Theorem 1. If a and b are elements of the real numbers and[pic], then for any c in the set of real numbers [pic]

Theorem 1. If a and b are elements of the real numbers and [pic] , then for any c in the set of real numbers [pic]

As noted before, when we work with real numbers we use two operations: addition and multiplication. Division can be defined in terms of multiplication. Also, when we work with real numbers we know that when we multiply two real numbers we get a real number. This is called closure.

Multiplication Properties of Real Numbers

Multiplication Properties of Real Numbers

Commutativity of Multiplication [pic]

Associativity of Multiplication [pic]

Existence of an Multiplicative Identity [pic]

Existence of an Multiplicative Inverse For each a element of the real numbers, with [pic], there is an element [pic] in the real numbers such that[pic].

Distributive Property [pic]

Definition Let a and b be elements of the real numbers. To divide b by a means to find the unique x such that [pic]

Theorem 1. For each element a, of the real numbers, [pic]

Theorem 1 For any two elements a and b of the real numbers, there is a unique element in the real numbers such that [pic].

When we work with real numbers we use two operations: addition and multiplication. Subtraction can be defined in terms of addition. Also, when we work with real numbers we know that when we add two real numbers we get real number. This is called closure.

Addition Properties of Real Numbers

Commutativity of Addtition [pic]

Associativity of Addition [pic]

Existence of an Additive Identity [pic]

Existence of an Additive Inverse For each a element of the real numbers, with [pic], there is an element [pic] in the real numbers such that[pic].

Theorem [pic]

Theorem [pic]

Theorem [pic]

Theorem [pic]

Definition: If a and b are elements of the real numbers, then

1. [pic]if and only if [pic] is an element of the positive real numbers

2. [pic]if and only if [pic] is an element of the positive real numbers

Definition: A constant is a symbol which does not vary in value.

Definition 1. An expression is a constant, a variable or any combination of constants, variables and arithmetic operations that describes a calculation.

Definition: An equation is a mathematical relationship that contains an equal sign.

Definition: An inequality is a mathematical relationship that contains an inequality symbol.

Defintion: A fraction is a quotient of two numbers or expressions a and b having the form [pic], where [pic]

Definition: A multiple of a given integer is the product of n and an integer.

Definition: The least common multiple is the smallest number that is a multiple of each number in a given set of numbers.

Definition: The least common denominator is the least common multiple of the denominators of a given set of fractions.

Definition: If [pic], then a and b are factors of c.

Definition: A prime number is a natural number that has exactly two different factors, 1 and the number itself.

Definition: A factorization that contains only prime factors is called the prime factorization.

Defnition: Given a fraction [pic], if the only factor common to both a and b is 1, then the fraction is in lowest terms.

Definition: A term is a product of a number and some variables

Definition: If the variable part is the same in two terms, they're called like terms.

Definition: The coefficients are the numbers in front of the variables.

Definition: An equation is a mathematical relationship that contains an equal sign.

Definition: A solution is a number that makes an equation true when it replaces the variable in the equation.

Definition: A linear equation is an equation in which each variable term contains a single variable raided to an exponent of 1.

Definition: An x-intercept is a point where a graph intersects the x-axis

Definition: A y-intercept is a point where a graph intersects the y-axis.

Definition: The slope is the ratio of the vertical change between any two points on a line to the horizontal change between these points.

Definition: An exponent is a symbol written to the upper right of a base number that indicates how many times to use the base as a factor. The base is the number that is repeatedly multiplied.

Definition: A linear inequality is an inequality containing expressions in which each variable term contains a single variable with an exponent of 1.

Theorem: For a and b in the real nubers:

1. [pic] if and only if [pic]and [pic] or [pic]and [pic].

2. [pic] if and only if [pic]and [pic] or [pic]and [pic].

Definition: A relation f is a function if and only if it is a relation in which no element of the domain appears as a first element of more than one ordered pair.

Definition: Lines in the same plane that do not intersect are said to be parallel.

Definition: A system of equations is a group of two or more equations.

Definition: A solution for a system of equations is an ordered set of numbers that makes all equations in the system true.

Definition: When a system of equations has no solution we say that the system of equations is inconsistent. When there is at least one solution, we say that the system is consistent.

Theorem: Let a and b be elements of the real numbers with [pic]and n an element of the natural numbers. Then [pic].

Theorem: If a is an element of the real numbers and [pic], then [pic].

Theorem: If a is an element of the real numbers with [pic] and n an element of the natural numbers, then [pic].

Theorem: If a is an element of the real numbers with [pic] and n an element of the natural numbers, then [pic].

Definition: A polynomial is the sum or difference of terms which have variables raised to positive integer exponents and which have coefficients that may be real or complex.

Definition: A monomial is a polynomial with one term.

Definition: A coefficient is the number multiplied times a product of variables or powers of variables in a term.

Definition: The degree of a polynomial is the highest degree of any term in the polynomial.

Definition: A binomial is a polynomial with two terms which are not like terms.

Definition: A trinomial is a polynomial with three terms which are not like terms.

Theorem: If a is an element of the real numbers and m and n are integers then [pic]

Theorem: If a is a real number and m and n are integers, then[pic].

Theorem: If a and b are real numbers and n is an integers, then [pic].

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