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11

FOUNDATIONS OF MATHEMATICS 11 COURSE OUTLINE

Ms Moon mskmoon.weebly.com

[email protected] In most cases, the results of a course are related to the amount of effort put forth on each and every day. Reviewing the day’s notes and examples is expected every evening in addition to the assigned homework. Daily Lessons and worksheets can be

downloaded online: mskmoon.weebly.com STUDENTS SHOULD DO A MINIMUM OF 30–60 MINUTES OF REVIEW & HOMEWORK IN MATHEMATICS five to six DAYS PER WEEK. If no new questions are assigned for homework students are to read old notes and review (or redo) previous assignments. I am happy to provide extra help if you want/need it. Please ask me for help. Course Overview: Textbook: Foundations of Mathematics 11

At the end of the course, you are expected to develop: • Spatial sense and proportional reasoning • Logical reasoning & Statistical reasoning • Algebraic and graphical reasoning through the study of relations • An appreciation of the role of mathematics in society

Evaluation: Marks are calculated based on your performance from the following:

Assignments/Participation /Quizzes/Tests/Exams Materials: The student is required to bring the following items to each class

Textbook & Notebook Completed Homework/Assignment Adequate supply of Lined Paper & Graph Paper in 1 1/2" - 2" Binder Blue & Red Pens, PENCILS, ERASER, Ruler, Highlighters Scientific Calculator (Casio or Sharp – no Smartphones/iPods) Graphing Calculator (TI 83 plus or TI 84) is strongly recommended

Chapter TOPIC 1 Inductive and Deductive Reasoning 2 Properties of Angles and Triangles 3 Acute Triangle Trigonometry 4 Oblique Triangle Trigonometry 5 Statistical Reasoning 6 Systems of Linear Equations 7 Quadratic Functions & Equations 8 Proportional Reasoning

DISCIPLINE PLAN FOR MS MOON’S CLASS

CLASSROOM RULES

1. Be Ready 2. Be Respectful 3. Be Responsible

EXPECTATIONS

You are expected to come to class prepared and on time. Having your textbook, completed homework, scientific calculators (no smartphones/iPods), red/blue pens, pencils and erasers is expected. You are expected to be seated in an assigned seating plan before class starts. Late entries will not be tolerated. If you continue to come to class late or unprepared, you will be sent to the office which will result missing important lessons, resulting unsuccessful outcomes.

You are expected to show respect the people, equipment, and furnishing of the classroom and school. You are to follow directions the first time they are given, and to provide immediate attention to me when called for. You are expected to remain quiet during instructional time and announcements, and to stay in your seats until dismissed by me at the end of class. You are to adjust your voice level to suit the activity. Work quietly during work time. Cell phones are not to be used for any reason in the classroom. You may listen to MP3 players when working independently provided that you do not disrupt the learning of yourself or others. No food or drinks (except water) are allowed in the classroom.

You are to take ownership for your learning. If you do not understand something, please set up a time to discuss this with me, but avoid first & last 5-10 minutes of the class time. I will extend my time to those who are using their class time wisely and are putting in time doing homework. PLEASE do not wait until just before exams to ask for help.

Binders/notebooks will be necessary for Math class. Notes are to be taken in class and examples are to be written down in your notebooks. A rule of thumb for note taking success: if I take the time to write an example on the board (whether you have asked the question or not), you should write the notes in your notebooks with extra notes explaining the examples.

You must submit all work to me neatly written and well labelled. Work that is not neat or not labelled will be rewritten. If you submit work that is copied from another source, it will be considered to be cheating. During quizzes, tests, and/or exams, copying answers from another student, allowing your work to be copied, or making use of unauthorized materials are all defined as cheating. You will receive a mark of zero and be sent to the office.

If you miss class, you are fully responsible for finding out specifically what you missed from another student in the class and be prepared to participate in the next class. For missed quizzes, tests, and/or exams, you will receive a score of zero unless a parent or guardian can substantiate the absence prior to the date (a hand-written note or an e-mail to Ms Moon). There are no re-tests. Missed tasks must be completed by the next day otherwise you will receive a mark of zero. It is your responsibility to find out what you have missed and see me to change your grade of zero.

======================================================================== Please sign below to indicate that you have read this syllabus and understand the expectations and classroom rules for Foundations of Mathematics 11.

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PLEASE KEEP THIS SHEET IN YOUR BINDER AT ALL TIMES My Best Wishes to You for a Very Rewarding Year. YOU CAN SUCCEED!!!

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Topic Page Assignment CHAPTER 1 – INDUCTIVE & DEDUCTIVE REASONING 1.1 – Making Conjectures: Inductive Reasoning ………… 1 p.12 #2, 3, 5-9, 11, 12

1.2 & 1.3 – Using Reasoning to Find a Counterexample to a Conjecture…… 5 p.22 #3-5, 10, 12, 14-16

1.4 – Proving Conjectures: Deductive Reasoning ………… 9 p.31 #1, 2, 4, 5, 8, 11, 13, 17

1.5 – Proofs That Are Not Valid ………… 11 p.42 #1-3, 5-7

1.6 – Reasoning to Solve Problems ………… 13 p.49 #2-4, 7, 9, 11, 14-16

1.7 – Analyzing Puzzles and Games ………… 16 p.55 #1, 2, 5, 7, 9, 10, 11, 13

CHAPTER 2 – PROPERTIES OF ANGLES AND TRIANGLES 2.1 – Exploring Parallel Lines ………… 20 p.72 #2, 5 & Worksheets

2.2 – Angles Formed by Parallel Lines ………… 24 p.78 #1-4, 15, 20 & WS

2.3 – Angles Properties in Triangles ………… 28 p.90 #3, 5-7, 10-15 & WS 2.4 – Angles Properties in Polygons ………… 31 P.99 #1-3, 6, 7, 10, 14, 16 & WS

CHAPTER 3 – ACUTE TRIANGLE TRIGONOMETRY 3.0 – Review on Right Triangle ………… 35 Worksheets

3.1 – Applying the Sine Law on Acute Triangles ………… 39 p.117 #1, 2, 4 & WS 3.2 – Proving the Sine Law & Problem Solving on Acute Triangles… 41 p.124 #1, 3-6, 8-13, 15, 16 3.3 – Applying the Cosine Law on Acute Triangles (Part 1/2)…… 44/47 p.136 #1, 4-6, 8, 11, 14 & WS

3.4 – Solving Problems Using Acute Triangles ………… 49 p.147 #1-7, 11, 13-15

CHAPTER 4 – OBLIQUE TRIANGLE TRIGONOMETRY 4.1 – Exploring the Primary Trigonometric Ratios of Obtuse Angles… 52 p.163 #1-4

4.2 – Proving the Sine & Cosine Laws for Obtuse Triangles (Part 1/2)……… 57/59 p.179 #1-5, 7-14 4.3 – The Ambiguous Case of the Sine Law (Part 1/2) ………… 63/69 p.183 #1-5, 10-13 4.4 – Solving Problems Using Obtuse Triangles ………… 72 p.193 #1-3, 5, 7-10

CHAPTER 5 – STATISTICAL REASONING 5.1 – Exploring Data ………… 75 p.211 #1-3 5.2 – Frequency Tables, Histograms, and Frequency Polygons… 77 p.222 #4, 6, 8, 9, 11 5.3 – Standard Deviation (Part 1/2) ………… 80/83 p.233 #5-7, 9, 11-13

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Topic Page Assignment

CHAPTER 5 – STATISTICAL REASONING – continued… 5.4 – The Normal Distribution ………… 85 P.251 #4, 6, 7, 10, 11, 13-15 5.5 – Z-Scores (Part 1/2) ………… 90/96 p.264 #5, 6, 8-11, 13, 16, 19, 21 5.6 – Confidence Intervals ………… 99 p.274 #4, 6-10 The Standard normal Distribution Table 103

CHAPTER 6 – SYSTEMS OF LINEAR EQUATIONS 6.1 – Graphing Linear Inequalities in Two Variables (Part 1/2)……… 105/110 p.303 #4-10, 12, 13 & WS 6.2 – Exploring Graphs of Systems of Linear Inequalities……… 113 p.307 #1, 2 6.3 – Graphing to Solve Systems of linear Inequalities ………… 115 p.317 #1-8, 10, 11 6.4 – Optimization Problems Part 1: Creating the Model……… 118 p.330 #1-4, 6-8 6.5 – Optimization Problems Part 2: Exploring Solutions……… 121 p.334 #1-3 6.6 – Optimization Problems Part 3: Linear Programming……… 124 p.341 #1-3, 5-8, 10, 11, 13, 16

CHAPTER 7 – QUADRATIC FUNCTIONS & EQUATIONS 7.1 – Exploring Quadratic Relations ………… 127 p.360 #1-3, 5 7.2 – Properties of Graphs of Quadratic Functions ………… 129 p.369 #4, 5, 9, 11, 13, 16-18 7.3 – Solving Quadratic Equations by Graphing with Technology…… 134 p.380 #5, 7, 8, 9, 10, 11, 13, 14 7.4 – Factored Form of a Quadratic Function (Part 1/2)………… 137/140 p.391 #2, 7, 9-12, 16, 18 & WS 7.5 – Solve Quadratic Equations by Factoring ………… 143 p.405 #3, 4, 6, 7, 9, 10, 12, 15, 16

7.6 – Vertex Form of a Quadratic Function ………… 146 p.417 #2, 5, 8, 11-13, 15, 16 7.7 – Solving Quadratic Equations Using the Quadratic Formula…… 151 p.428 #4, 6, 8-10, 12

7.8 – Solving Problems Using Quadratic Models ………… p.436 #2-4, 7, 8, 10

CHAPTER 8 – PROPORTIONAL REASONING 8.1 – Comparing and Interpreting Rates ………… 155 p.458 #1, 2, 4-8, 13, 14 8.2 – Solving Problems That Involve Rates ………… 158 p.466 #1-8, 13-15 8.3 – Scale Diagrams ………… 160 p.479 #1-3, 6-8, 13, 17 8.4 – Scale Factors and Areas of 2-D Shapes ………… 162 p.487 #1-4, 6, 9, 10, 13 8.5 – Similar Objects: Scale Models and Scale Diagrams……… 164 p.497 #1, 3-6, 11, 13, 15 8.6 – Scale Factors and 3-D Objects ………… 167 p.508 #1-6, 8, 12, 13

Foundations of Mathematics 11 Ms Moon

1.1 Making Conjectures: Inductive Reasoning Recognize Patterns Example 1. Which of the figures below the line of drawings best completes the series?

Example 2. Which of the figures below the line of drawings best completes the series?

Try. Which is the right card to complete the series?

Try. Complete the series.

revised on May 11,

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We just used the inductive reasoning to figure out our answers. Inductive reasoning means to draw a general conclusion, known as a conjecture, by observing patterns and identifying properties in specific examples. A conjecture is a testable expression that is based on available evidence but is not yet proved. Use Inductive Reasoning to Make a Conjecture Example 3. Consider the following figures.

a) Complete the chart below.

Figure f 1 2 3

Number of Small Triangles n

b) Make a conjecture based on the pattern in the diagrams. c) One conjecture says that in Figure 10, there are 100 triangles. Determine

whether the conjecture is valid. Example 4. Using the September 2015 calendar, consider any two-by-two square

around four of the dates. Create a conjecture about what you notice. Test your conjecture with different two-by-two squares. Create as many conjectures as you can!

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Example 5. What conjecture can you make about the product of two odd integers?

3 × 9 = (1) (−17) =

(−3) × (5) = −11 · −7 =

Try. Suppose you multiply an odd number by an even number.

23 × 14 = 17 × 24 = 57 × 32 = What conjecture can you make? Example 6. Make a conjecture about the sum of two odd integers and one even

integer. Provide evidence to support the conjecture. Try. Make a conjecture about the sum of two even numbers and one odd number.

Provide evidence to support the conjecture.

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Example 7. What conjecture can you make about the difference between consecutive perfect squares?

Try. Suppose you add consecutive odd numbers starting at 1. What conjecture can

you make about the sum? 1 + 3 = 1 + 3 + 5 = 1 + 3 + 5 + 7 = 1 + 3 + 5 + 7 + 9 = Example 8. Suppose you add all the angle measures in a pentagon. What conjecture

can you make about the sum?

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1.2 & 1.3 Using Reasoning to Find a Counterexample to a Conjecture A palindrome is a word, phrase, number, or other sequence of units that can be read the same way in either direction. Some people have names that are palindromes. A few common palindrome names are: Maham, Ada, Anna, Bob, Eve, Hannah, and Otto. Explore the Validity of a Conjecture and Find a Counterexample Example 1: Examine and continue the number pattern.

12 = 1 112 = 121

1112 = 12321 11114 = 1234321

a) Make a conjecture based on the pattern. b) Is the conjecture valid? Provide an example if it is not valid. c) Revise the conjecture to make it valid.

Conjectures may seem valid based on initial evidence, but later can be shown to be invalid as new evidence is gathered. Conjectures can be revised based on new evidence or completely abandoned, depending on the situation. An example that proves a conjecture false is a counterexample. One counterexample is enough to disprove the conjecture.

Try. Is the square root of a number smaller than the number itself?

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Try. Decide whether each conjecture is true or false. If it is true, explain why. If it is false, give a counterexample.

a) A number that is not positive is negative. b) If 1 is added to an odd number, the result is always an even number. c) The square of a number is always greater than the number.

d) If two angles are acute, their sum is less than 180o.

e) Every rectangle is a square.

f) Every square is a rectangle.

Example 3. If two congruent regular heptagons (7-sided polygons) are positioned so that they

share a side, a dodecagon (12-sided polygon) is formed. If two congruent regular hexagons (6 sided polygons) are positioned so that they share a side, a decagon (10-sided polygon) is formed. If two congruent regular pentagons (5-sided polygons) are positioned so that they share a side, an octagon (8-sided polygon) is formed.

Starting Polygon has p sides Resulting Polygon has n sides

7

6

5

4

a) Make a conjecture about positioning two congruent regular quadrilaterals so that they share a side.

b) Is the conjecture valid? Provide a counterexample if

it is not valid. c) Revise the conjecture to make it valid.

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Example 3. Kerry created a series of circles. Each circle had points marked on its circumference and joined by chords.

As the number of points on the circumference increased, Kerry noticed a pattern for the number of regions created by the chords.

Number of Points 2 3 4 Number of Regions 2 4 8

She made the following conjecture:

As the number of connected points on the circumference of a circle increases by 1, the number of regions created within the circle increases by a factor of 2.

a) Is her conjecture valid? Is there a counterexample?

Even if you cannot find one, you cannot be certain that there is none. Any supporting evidence you develop while searching for a counterexample, does increase the likelihood that conjecture is true.

b) Revise the conjecture to make it valid.

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Try. Frank was working on the Pythagorean Theorem and came up with this conjecture: "In these problems I had to find the value of x. In the first triangle I got , using the Pythagorean Theorem. Then I noticed that 4 + 3 = 7. So, a faster way to find the value of x is just to add the two given lengths and take the square root. This method gives the correct answer for the other three triangles, too". Is his conjecture valid?

Try. Determine the validity of the conjecture - the difference between consecutive

perfect squares is odd. If you can only find more supporting evidence instead of a counterexample, the conjecture is most likely to be valid.

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1.4 Proving Conjectures: Deductive Reasoning Deductive reasoning involves starting with general assumptions that are known to be true and, through logical reasoning, arriving at a specific conclusion. Connect Conjectures with Reasoning Example 1. Consider the following sums of five consecutive integers.

Conjecture: The sum of any five consecutive integers is five times the median (middle

number when placed in order). Prove that the conjecture is valid. A demonstration using an example is called inductive reasoning. A demonstration using a proof is called deductive reasoning. Example 2. Prove that the difference between consecutive perfect squares is always

odd.

When you apply deductive reasoning correctly, you can be sure that the conclusion you draw is valid.

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Transitive property is often useful in deductive reasoning. It can be stated as follows: Things that are equal to the same thing are equal to each other.

If a = b and b = c, then a = c. Example 3. Use a two column proof to prove that when two straight lines intersect, the

vertically opposite angles are equal.

Statement Justification

Use Deductive Reasoning to Make a Valid Conclusion Example 4. All X-men are mutants. Wolverine is an X-man. What can be deduced

about Wolverine? Try. All students are teenagers. Lynn is a student. What can be deduced about

Lynn? Try. All musicians have long hair. Sharon is a musician. What can be deduced about

Sharon?

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1.5 Proofs That Are Not Valid Use Reasoning to Determine the Validity of an Argument Example 1. Ms Moon is Superman. Superman can fly. Therefore, Ms Moon can fly.

Examine whether her argument is correct by examining each statement individually.

Ms Moon is Superman. ______________ Superman can fly. ______________ Therefore, Ms Moon can fly. ________________________________ A single error in reasoning will break down the logical argument of a deductive reasoning. This will result in an invalid conclusion. Example 2. Here is a proof that 3 = 4. Where is the error in reasoning?

Division by zero always creates an error in a proof, leading to an invalid conclusion. Tupe of Errors * a false assumption or generalization * an error in reasoning * an error in calculation

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Try. Here is a proof that 2 = 1. Where is the error in reasoning?

Use Reasoning to Determine the Validity of a Proof Example 3. Choose any number. Add three. Double it. Add four. Divide the result by

two. Take away the number you started with. The result will always be five. Check the validity of the proof.

Example 4. Explain whether the conclusion follows logically from the given statement. Cars produce carbon dioxide gas. Carbon dioxide gas contributes to the

depletion of the ozone layer. Therefore, if there were no cars, the ozone layer would not be depleted.

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1.6 Reasoning to Solve Problems Inductive and deductive reasoning are useful in problem solving. Inductive reasoning involves solving a simpler problem, observing patterns, and drawing a logical conclusion from your observations to solve the original problem. Deductive reasoning involves using known facts or assumptions to develop an argument, which is then used to draw a logical conclusion and solve the problem. Use Inductive Reasoning to Solve a Problem Example 1. The members of a recently selected varsity basketball team met each other

at their first team meeting. Each person shook the hand of every other person. The team had 12 players and 2 coaches. How many handshakes were exchanged?

Try. Suppose there are 6 points on the circumference of a circle. How many line

segments can be made by joining dots?

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Use Deductive Reasoning to Solve a Problem Example 2. Sue signed up for games at her school’s fun night. Seven other people

were assigned to her group, making up four pairs of partners. The other members of her group were Dave, Angie, Josh, Tanya, Joy, Stu, and Linus. When the games started, Dave and his partner were to the left of Stu. Across from Dave was Sue, who was to the right of Josh. Dave’s brother’s partner, Tanya, was across from Stu. Joy was not on Stu’s right. Name the four pairs of partners.

Example 3. Bahamas, Barbados, Jamaica and Trinidad and Tobago are four countries

in the Caribbean. All the following statements about their land areas are true. List the countries in order of increasing size. Barbados is smaller than Trinidad and Tobago. Bahamas is neither the largest nor the smallest. At least two countries are larger than Trinidad and Tobago.

Try. Alice, Brittany, Carol, Dani, and Elisa are on the school basketball team. All the

following statements are true. List the girls in order of increasing height. There are at least two girls shorter than Alice. Dani is shorter than Carol. Brittany is not the shortest girl. Dani is taller than Alice.

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Try. The number in each box of an arithmagon is the sum of the two numbers in the circles adjacent to the box.

Solve the three triangle arithmagons below.

Try. Once upon a time there were a Wolf, a Rabbit, and a basket of Cabbage. You are a farmer and you need to take all three to the county fair. As you are walking to the fair, you encounter a river. A boat man will take you across but he only has room for you and one other object (either the wolf, rabbit, or cabbage). How do you get everything across the river safely?

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1.7 Analyzing Puzzles and Games Use Reasoning Strategy to Solve a Puzzle Example 1. In a leapfrog puzzle, coloured counters are moved along a space on a

board. The goal is to move each set of coloured counters to the opposite side of the board. Counters can either move into the empty space or leapfrog over another counter into the empty space.

Based on the evidence as seen in the table above, the minimum number of moves to move each set of counters to the opposite side when there are six counters of each colour is: Try. Without lifting your pencil, connect all of the dots below with four straight lines.

Try. Can you move just two matchsticks and create seven squares?

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Use Reasoning Strategy to Determine Possible Winning Plays Example 2. Frank and Tara are playing darts, using the given rules. Their scores are

shown in the table below. To win, Frank must reduce his score to exactly zero and have his last counting dart be a double.

Rules: Each player's score starts at 501. The goal is to reduce your score to zero. Players take turns. Each player throws three darts per turn.

a) How can Frank win with a single dart on his next turn?

b) If Frank scores 18 on the first dart of the next turn, then how can he win on the next dart?

c) If Frank scores 18 and then 9 on the first two darts, is it possible for

him to win on the final dart of his next turn? Explain. Try. You are in a jail cell with two doors. One leads to freedom the other to the

dungeon. There are two guards; one always tells the truth while the other always lies. You can ask just one question to either guard. What is your question, and which guard will you ask?

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Example 3. Sudoku requires both inductive and deductive reasoning skills. The number that are used to complete a Sudoku puzzle relate to the size of the grid. For a 6-by-6 grid, the numbers 1 to 6 are used. For a 9-by-9 grid, the numbers 1 to 9 are used. The grid must be filled so that each column, row, or block contains all the numbers. No number can be repeated with any column, row, or block. Solve each of the Sudoku puzzle below.

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Try. Solve each of the Sudoku puzzle below.

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2.1 Exploring Parallel Lines

Vertically opposite angles are equal When two lines intersect, the opposite angles are equal.

Supplementary angles add to 180° Two (or more) adjacent angles on the same side of a line add to 180°.

Complementary angles add to 90° Two (or more) adjacent angles in a right angle add to 90°.

Example 1. Find the angle measure of angle b, using opposite angles.

revised on May 11, 2017 20


Example 2. Find the angle measure of angle b, using supplementary angles.

Example 3. Find the angle measure of angle b, using complementary angles.

Try. Find the angle measure of angle b.

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A transversal is a line that intersects two or more other lines at distinct points.

Explore Parallel Lines Example 4. Draw a transversal that crosses two parallel lines (below) at an angle other

than 90°. Label every angle formed between intersecting lines with a unique lower-case letter, and measure each angle in degrees.

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When a transversal intersects a pair of non-parallel lines, the corresponding angles are not equal.

Example 5. In each diagram, determine whether AB is parallel to CD. Explain how you

know.

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2.2 Angles Formed by Parallel Lines Any pair of parallel lines makes an F-shape with a transversal that crosses them.

The angles marked a° are equal and are called corresponding angles. Explore Corresponding Angles Example 1. On the diagrams below mark three other pairs of F-shaped corresponding

angles and label them b°, c°, and d°. (Note that the F-shape can be backwards or upside down).

Example 2. Determine the measures of the angles marked by letters.

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Any pair of parallel lines makes a Z-shape with a transversal that crosses them.

The angles marked a° are equal and are called alternate interior angles. The angles marked b° are equal and are called alternate exterior angles.

Explore Alternate Interior Angles Example 3. On the diagrams below mark two pairs of Z-shaped alternate interior

angles and label them b° and c°. (Note that the Z-shape can be backwards).

Example 4. Determine the measures of the angles marked by letters.

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Any pair of parallel lines makes a C-shape with a transversal that crosses them.

The angles marked o and x are called co-interior angles. They are NOT equal. They add to 180° Explore Co-Interior Angles Example 5. On the diagrams below mark two pairs of C-shaped co-interior angles and

label them o° and x°. (Note that the C-shape can be backwards).

Example 6. Calculate the measures of the three remaining angles 48° in the

parallelogram.

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Use Reasoning to Determine Unknown Angles Example 7. Determine the measures of a, b, c, and d.

Use Angle Properties to Prove that Lines Are Parallel Example 8. One side of a cell phone tower will be built as shown. Use the angle

measures to prove that braces CG, BF, and AE are parallel.

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2.3 Angles Properties in Triangles

In any triangle, the sum of the measures of the interior angles is proven to be 180°.

Use Angle Sums to Determine Angle Measures Example 1. Determine the measure of each indicated angle.

Use Reasoning to Determine the Relationship between the Exterior and Interior Angles of a Triangle Example 2. Determine the relationship between an exterior angle of a triangle and its

non-adjacent interior angles.

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Try. Prove

The measure of any exterior angle of a triangle is proven to be equal to the sum of the measures of the two non-adjacent interior angles.

Use Exterior Angle to Determine Angle Measure Example 3. Determine the measure of each indicated angle.

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Try. Determine the measure of each indicated angle.

Use Reasoning to Determine Solve Problems

Example 4. Determine the measures of , and .

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2.4 Angles Properties in Polygons Convex Polygon A convex polygon is any polygon in which each interior angle measures less than 180°

Concave Polygon A concave polygon is a polygon in which one or more interior angle measures more than 180°

Determine Convex and Concave Polygons Example 1. State if each polygon is concave or convex.

Determine Properties of Angles in Polygons Example 2. Use the following table to determine how to find the sum of the interior

angles of a polygon from the number of sides.

For a polygon with n sides, it takes a minimum of (n – 2) triangle(s) to cover it. Therefore, the sum of the interior angles is 180° (n – 2).

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Try. Calculate the sum of the measures of the interior angles of a polygon with 9 sides.

Try. Calculate the sum of the measures of the interior angles of a polygon with the

given sides. a) 12 b) 15

Example 3. Determine the number of sides of a polygon whose interior angle sum

equals 4140° Try. Determine the number of sides of a polygon whose interior angle sum equals:

a) 720° b) 1260° c) 2880°

For a regular polygon with n sides, the measure of each interior angle is °

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Example 4. A regular polygon is a polygon with all sides equal and all angles equal. How can you find the measure of each interior angles of a regular pentagon?

Try. What is the measure of each interior angle in a regular decagon (10-sided)? Example 5. Determine the number of sides of a regular polygon whose interior angle

measures 170°. Try. Determine the number of sides of a regular polygon whose interior angle

measures; a) 156° b) 175° c) 171°

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An exterior angle is formed between a ray formed by extending the sides of polygon in one direction and the next side of the polygon (adjacent to the ray).

Example 6. Determine the measure of each exterior angle of a regular pentagon?

The exterior angle of an n-sided regular polygon measures360°

The sum of the measures of the exterior angles of an n-sided regular polygon is 360° Try . Determine the number of sides of a regular polygon whose exterior angle

measures a) 15° b) 40° c) 1° Example 7. Visualizing Tessellations

A floor tiler designs custom floors using titles in the shape of regular polygons. Can the tiler use congruent regular octagons and congruent squares to tile a floor, if they have the same side length?

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Review on Right Triangles

Identify a Right Triangle Example 1. Is each triangle a right triangle? Explain.

a) a triangle has side lengths of 9 cm, 12 cm, and 15 cm

b) a triangle has side lengths of 5 cm,7 cm, and 9 cm

Try. A triangle has side lengths of 12 cm, 16 cm, and 20 cm. Is the triangle a right

triangle? Explain.

Apply the Pythagorean Theorem Example 2. Find the length of the unknown side. Use a calculator to

approximate to 1 decimal place.

Try. Find the length of the unknown side. Use a calculator to approximate each

answer to 1 decimal place.

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The side across from the right angle is called the hypotenuse (hyp). It is the

LONGEST side of a right-angled triangle.

If you stand at ∠A, the side BC is opposite to you and the side AB is next to you. We

call that BC is the opposite side (opp) and AB is the adjacent side (adj).

Name the Sides in a Right Triangle Example 3. Name the sides of each right triangle.

Try. Find the trigonometric ratio of each angle to the nearest thousandth.

a) cos (75°) b) sin (25°) c) tan (72°)

Try. Find ∠A to the nearest degree if trigonometric ratio is known.

a) tan ∠A = 1.327 b) cos ∠A =

1

3 c) sin ∠A =

3

4

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Recall the definitions of the three Primary Trigonometric Ratios:

The rules for determining the sine ratio, cosine ratio, and tangent ratio for an angle in

a right triangle can be memorized by using the acronym SOH CAH TOA.

Determine the Unknown Angle in a Right Triangle

Example 4. In right ∆ABC , find tan ∠C , then ∠C .

Try. In right ∆PQR , find the measures

of ∠P .

Try. In right ∆ABC , find ∠C .

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Determine the Unknown Side Length in a Right Triangle

Example 5. Use trigonometric ratios to determine the unknown side in the given triangle. Answer correct to the nearest tenth of an unit.

Example 6. Use trigonometric ratios to determine the unknown side in the given

triangle. Answer correct to the nearest tenth of an unit.

Try. Find the unknown side to the nearest tenth.

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3.1 Applying the Sine Law on Acute Triangles

We use primary trigonometric ratios, SOH CAH TOA, to solve right triangles. What about the triangles that are not right-angled, say acute triangles?

Note: lower-case letters a, b, c represent the opposite sides from ∠A, ∠B and ∠C respectively.

Apply Sine Law to Find Unknown Side Length Example 1. Find each measurement indicated. Round your answers to the nearest

tenth.

Try. Find each measurement indicated. Round your answers to the nearest tenth.

a) Find x b) Find AC

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Apply Sine Law to Find Unknown Angle Measure

Example 2. Find each measurement indicated. Round your answers to the nearest whole degree.

Try. Find each measurement indicated. Round your answers to the nearest whole

degree. a) Find m R b) Find m x

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3.2 Proving the Sine Law and Problem Solving on Acute Triangles

Use Reasoning to Prove Sine Law Example 1. Prove sine law using deductive reasoning.

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Use Sine Law to Solve Problem

Example 2. Toby uses chains attached to hooks on the ceiling and a winch to lift engines at his father’s garage. The chains, the winch, and the ceiling are arranged as shown. Determine the angle that each chain makes with the

ceiling to the nearest degree.

Example 3. Allison is flying a kite. She has released the entire 150 m ball of kite

string. She notices that the string forms a 70° angle with the ground. Marc

is on the other side of the kite and sees the kite at an angle of elevation of 30°. How far is Marc from Allison, to the nearest tenth of a metre?

Directions are often stated in terms of north and south on a compass. For example,

N30°E means travelling in a direction 30° east of north. S45°W means travelling in a direction 45° west of south.

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Use Sine Law to Solve Direction Problem

Example 4. Janice is sailing from Gimli on Lake Winnipeg to Grand Beach. She had planned to sail 26.0 km in the direction S71°E; however, the wind and current pushed her off course. After several hours, she discovered that

she had actually been sailing S79°E. She checked her map and saw that she must sail S18°W to reach Grand Beach. Determine, to the nearest

tenth of a kilometre, the distance remaining to Grand Beach.

Example 5. The captain of a small boat is delivering supplies to two lighthouses, as

shown. His compass indicates that the lighthouse to his left is located at N30°W and the lighthouse to his right is located at N50°E. Determine the

compass direction he must follow when he leaves lighthouse B for lighthouse A.

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3.3 Applying the Cosine Law on Acute Triangles (Part I)

Example 1. Find each measurement indicated. Round your answers to the nearest tenth.

The sine law cannot always help you determine unknown angle measures or side

lengths. You can use the sine law to solve a problem modelled by an acute triangle when you know:

- two sides and the angle opposite a known side

- two angles and any side

You can use the cosine law to solve a problem that can be modelled by an acute

triangle when you know: – two sides and the contained angle – all three sides

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Apply Cosine Law to Find Unknown Side Length Example 2. Find each measurement indicated. Round your answers to the nearest

tenth.

Try. Find each measurement indicated. Round your answers to the nearest tenth.

(Obtuse angle)

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Apply Cosine Law to Find Unknown Angle Measure

Example 3. Find each measurement indicated. Round your answers to the nearest whole degree.

Try. Find each measurement indicated. Round your answers to the nearest whole

degree.

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3.3 Proving the Cosine Law and Problem Solving on Acute Triangles (Part II)

Use Reasoning to Prove Cosine Law

Example 1. Prove cosine law using deductive reasoning.

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Solve a Problem Using the Cosine Law

Example 2. The radar screen of a Coast Guard rescue ship shows that two boats are in the area, N45°E and S50°E. How far apart are the two boats, to the nearest tenth of a kilometre?

Try. Two aircraft, A and B, leave an airport at the same time. A flies on a course of N90°E at 700 km/h, and B flies on a course of N70°W at 600 km/h. Calculate

their distance apart in 12 minutes to the nearest kilometre.

Example 3. Two airplanes leave the Hay River airport in the Northwest Territories at

the same time. One airplane travels at 355 km/h. The other airplane travels at 450 km/h. About 2 h later, they are 800 km apart. Determine

the angle between their paths, to the nearest degree.

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3.4 Solving Problems Using Acute Triangles

To decide whether you need to use the sine law or the cosine law, consider the information given about the triangle and the measurement to be determined.

Solve a 2-D Problem Example 1. Ryan is in a police helicopter, 400 m directly above the Sea to Sky

highway near Whistler, British Columbia. When he looks north, the angle of depression to a car accident is 65°. When he looks south, the angle of

depression to the approaching ambulance is 30°. How far away is the ambulance from the scene of the accident, to the nearest tenth of a metre?

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Example 2. The first hole at a golf course is 210 yards long in a direct line from the tee

to the hole. Andrew Duffer hit his first shot at an angle of 15° off the direct line to the hole. The angle between his first shot and his second shot was 105°. His second shot landed in the hole. What was the length of his

second shot, to the nearest yard?

Example 3. An oil company drilling off shore has pipelines from platform Beta to the same shore station Delta. Platform Alpha is 180 km on a bearing of 50°

from Delta and platform Beta is 250 km on a bearing of 125° from Delta. Calculate the distance between platform Alpha and platform Beta to the nearest km.

Example 4. From a point A, level with the foot of a hill, the angle of elevation of the top of the hill is 16°. From a point B, 950 metres nearer the foot of the hill, the

angle of elevation of the top is 35°. What is the height of the hill, DC, to the nearest metre?

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Example 5. P and Q are two bases for a mountain climb. PQ is 600 m and QR is a

vertical stretch of a rock face. The angle of elevation of Q from P, (i.e.

∠QPS) is 31°. The angle of elevation of R from P, (i.e. ∠RPS) is 41°. Mark

these measurements on the diagram and state the measures of ∠RPQ and

∠PRQ. Use the sine law in ∆PQR to calculate the height of the vertical climb, QR, to the nearest metre.

Solve a 3-D Problem Example 6. On June 30, 1956, the world’s largest free standing totem pole was

erected in Beacon Hill Park in Victoria. Recently, a surveyor took measurements to verify the height, h, of the totem pole. In the diagram, triangle ABC lies in a vertical plane and triangle BCD lies in a horizontal

plane.

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4.1 Exploring the Primary Trigonometric Ratios of Obtuse Angles Use Primary Trigonometric Ratios on Acute Angles Example 1. Use primary trigonometric ratios to determine the measure of the

indicated angle to the nearest degree.

Example 2. Calculate, to the nearest tenth of a cm, the length of the third side of ∆ .

Until now, we have used the primary trigonometric ratios only acute angles. However, there are primary trigonometric ratios for obtuse angles as well.

Investigate Sine Ratios for Acute Angles and Obtuse Angles Example 3. Round to three decimal places. How does the value of sin 35° compare to

the value of sin 145 °? How are the two angles 35° and 145° related?

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Try. How does the value of cos 25° compare to the value of cos 155°? Round to three decimal places.

Try. How does the value of tan 45° compare to the value of tan 135°? Round to three

decimal places. Example 4. Completed the following table. Round to three decimal places.

sin cos tan 180° sin 180° cos 180° tan 180°

110°

120°

130°

140°

150°

160°

170°

180°

Summarize any patterns you observed as the measure of the obtuse angle increased.

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Sine Graph

sin 30° sin 150°

Cosine Graph

cos 60° cos 120°

Tangent Graph

tan 45° tan 135°

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Be careful when using the sine ratio to determine the measure of an angle. For any angle , sin sin 180° . The inverse sine of a ratio always gives an acute angle, but the supplementary angel has the same ratio. Determine the Ambiguous Case of the Sine Law Example 5. Determine two solutions, one on acute angle and one on obtuse angle, for

each of the following equations. Answer to the nearest degree.

a) sin

b) sin 0.788

Try.

c) sin

d) sin

Example 6. Determine two angles between 0° and 180° that have each sine ratio:

a) 0.46

b)

Try. Determine two angles between 0° and 180° that have each sine ratio:

c) 0.59

d)

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Relationships between the value of a primary trigonometric ratio for an acute angle and the value of the same primary trigonometric ratio for the supplement of the acute angle.

For any angle sin sin 180° cos cos 180° tan tan 180°

Example 7. Calculate each ratio to four decimal places. Predict another angle that will

have an equal or opposite trigonometric ratio a) sin 25°

b) tan 45°

c) cos 72°

d) sin 160°

Try. Determine each trigonometric ratio. Predict another angle that has an equal or

opposite ratio. Check your prediction. e) sin 122°

f) cos 100°

g) sin 58°

h) tan 15°

Example 8. Determine the angle measure between 0° and 180° that have each

trigonometric ratio. a) cos 0.46

b) tan

Because the cosine and tangent ratios for an angle and its supplement are not equal (they are opposites), the inverse cosine or tangent of a ratio always gives only one possible answer.

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4.2 Proving the Sine and Cosine Laws for Obtuse Triangles (Part 1) Prove Sine Law in an Obtuse Triangle Example 1. Prove sine law using deductive reasoning.

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Prove Cosine Law in an Obtuse Triangle Example 2. Prove cosine law using deductive reasoning.

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4.2 Applying the Sine and Cosine Laws for Obtuse Triangles (Part II) The sine law and cosine law can be used to determine unknown side lengths and angle measures in obtuse triangles. They are used with the obtuse triangles in the same way that they are used with acute triangles. Use the sine law when you know… Use the cosine law when you know…

The lengths of two sides and the measure of the angle that is opposite a known side

The lengths of two sides and the measure of the contained angle

The measures of two angles and the length of any side

The lengths of all three sides

Be careful when using the sine law to determine the measure of an angle. For any angle , sin sin 180° . The inverse sine of a ratio always gives an acute angle, but the supplementary angle has the same ratio.

Example 1. Determine two angles between 0° and 180° that have each sine ratio.

a) sin 0.46

b) sin

Try. Determine two angles between 0° and 180° that have each sine ratio.

c) sin

d) sin 0.59

Because the cosine ratios for an angle and its upplement are not equal (they are opposites), the inverse cosine of a ratio always gives the answer.

Example 2. Determine two angles between 0° and 180° that have each cosine ratio

a) cos 0.64

b) cos

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Use Reasoning and the Sine Law to Determine the Meaure of An Obtuse Angle Example 3. Find two possible measures of C, in ∆ , where 50°, 7.5 cm,

and 9.5 cm.

Try. Find two possible measures of C, in ∆ , where 45°, 9.5 cm, and

7.5 cm.

You must decide whether the acute angle, , or the obtuse angle, 180° , is the correct angle for your triangle.

Try. Determine the measure of A of an obtuse triangle to the nearest tenth of a

degree.

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Because the cosine ratios for an angle and its supplement are not equal (they are opposites), the inverse cosine of a ratio always gives the answer.

Use Reasoning and the Cosine Law to Determine the Measure of An Obtuse Angle Example 4. The roof of a house consists of two slanted sections, as shown. A roofing cap is being made to fit the crown of the roof, where the two slanted sections meet. Determine the measure of the angle needed for the roofing cap, to the nearest tenth of a degree.

Try. Two ships set sail from port, P, heading in different directions. The first ship

sails 7 km to R and the second ship sails 8 km to Q. If the distance between R and Q is 13 km, determine the angle between the directions of the two ships.

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Use Reasoning and the Sine and Cosine Law to Determine a Side Length Example 5. Determine the length of x to the nearest tenth of a centimetre.

Try. Find the length of BC, to the nearest thenth of a centimetre.

Example 6. Colleen and Juan observed a tethered balloon advertising the opening of a new

finess centre. They were 250 m apart, joined by a line that passed directly below the balloon, and were on the same side of the balloon. Juan observed the balloon at an angle of elevation of 7° while Colleen observed the balloon at an angle of elevation of 82°. Determine the height of the balloon to the nearest metre.

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4.3 The Ambiguous Case of the Sine Law – SSA Situations (Part I) Determine the Height Using Given Information Example 1. To the nearest tenth, determine the height of each right triangle with the

given information. a) 53° and 15cm in ∆

b) 81° and 9cm in ∆

c) 28° and 8.4cm in ∆ Try. To the nearest tenth, determine the height of each triangle with the given

information. d) 40° and 5cm in ∆

e) 15° and 71cm in ∆

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SSA Situation (Two sides and one non-contained angle) Two side lengths and the measure of an angle that is opposite one of these sides.

SSA Situation Example 2. Decide whether each description of a triangle involves the SSA situation.

a) In ∆ , 81°, 9cm and f = 8 cm

b) In ∆ , 100°, 8cm and b = 10 cm Try. Decide whether each description of a triangle involves the SSA situation.

c) In ∆ , 40°, 5cm and g = 4 cm

d) In ∆ , 15°, 71cm and k = 36 cm

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Connect the SSA Situation to the Number of Possible Triangles Example 3. Given each SSA situation for ∆ , determine how many triangles are

possible (zero, one, or two), if 30° and 12m.

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In ∆ below, where h is the height of the triangle, A and the lengths of sides a and b are given, and A is acute, there are four possibilities to consider:

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4.3 The Ambiguous Case of the Sine Law – Problem Solving (Part II) Solving a Problem with the Ambiguous Case Example 1. The Raven’s Song, a traditional Tsimshian cedar canoe, is paddled away

from a dock, directly toward a navigational buoy that is 5 km away. After reaching the buoy, the direction of the canoe is altered and it is paddled another 3 km. From the dock, the angle between the buoy and the canoe’s current position measures 12°. How far is the Raven’s Song from the dock?

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Example 2. In an extreme adventure triathlon, participants swim 1.7 km from a dock to one end of an island, run 1.5 km due north along the length of the island, and then kayak back to the dock. From the dock, the angle between the lines of sight to the ends of the island measures 15°. How long is the kayak leg of the race?

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Try. Martina and Carl are part of a team that is studying weather patterns. The team is about to launch a weather balloon to collect data. Martina’s rope is 7.8 m long and makes an angle of 36.0° with the ground. Carl’s rope is 5.9 m long. Assuming that Martina and Carl form a triangle in a vertical plane with the weather balloon, what is the distance between Martina and Carl, to the nearest tenth of a metre?

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4.4 Solving Problems Using Obtuse Triangles

In each problem involving obtuse triangles, be sure to make a diagram and label all the given sides and angles. Remember that if the given information is ASA, AAS, or SSA, The Sine Law can be used as a first step to solve the triangle in the problem, but if you are given SSA, there may be zero, one or two solutions (as it may be the ambiguous case). If the given information is SSS or SAS, then the Cosine Law can be used as a first step to solve the triangle in the problem. If the triangle has a right angle, then simple trigonometric ratios, and the Pythagorean Theorem can be used to solve the triangle in the problem.

Visualize A Triangle to Solve a Problem Example 1. A surveyor in a helicopter would like to know the width of Garibaldi Lake

in British Columbia. When the helicopter is hovering at 1610 m above the forest, the surveyor observes that the angles of depression to two points on opposite shores of the lake measure 45° and 82°. The helicopter and the two points are in the same vertical plane. Determine the width of the lake, to the nearest metre.

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Example 2. Shannon wants to build a regular pentagonal sun deck. She is going to use five 2-by-6s, each 12 ft long, to frame the perimeter. She plans to finish the deck with 4 in. cedar planks, laid side by side and parallel to one of the sides. Determine the length of the longest cedar plank.

Try. ∆ is an equilateral triangle with a perimeter of 36 cm. Three triangles are

created when A is divided into three equal angles. Two of these triangles are obtuse. Determine the side lengths of the obtuse triangles, to the nearest centimetre.

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Try. The following describes the location of a buried treasure. From the pine tree, walk 30 paces N20E°, then turn and walk 15 paces until the tree is due south. How many paces would you need to walk due north of the tree to reach the buried treasure?

Try. From the window of a building, 45 m up, the angles of depression to two

different intersections measure 76° and 65°. The measure of the angle between the lines of sight to the two intersections is 135°. Calculate, to the nearest metre, the distance between the two intersections.

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5.1 Exploring Data

Statistics

Statistics is a field of Mathematics dealing with the collecting and summarizing of data.

Once the data has been gathered, the information is evaluated and then analyzed so that

a decision based on these measurable events can be made.

Mean, median and mode are all measures of central tendency or average.

The mean, or arithmetic average, is calculated by finding the sum of the data and dividing by the number of pieces of data.

The mode represents popularity. It is the number that occurs most frequently. (NOTE: A set of data may have no mode or more than one mode)

The median is the middle value when the data are arranged in numerical order. (NOTE: If the number of pieces of data is even, there are 2 middle values. The median is the arithmetic average of these 2 values)

The range, a measure of dispersion of data, tells how spread out the data is. There range is the difference between the highest and lowest values.

Calculate Mean, Median, Mode and Range

Example 1. Determine the mean, median, mode and range. Round to 1 decimal place, if

necessary.

a) 15, 18, 16, 21, 18, 14, 12, 19,

11, 16

b) 80 min, 1 hr, 65 min, 50 min, ¾

hr

Try. Determine the mean, median, mode and range. Round to 1 decimal place, if

necessary.

c) 2°C, -6°C , -6°C, 0°C, 1°C, 3°C, -1°C

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Compare Sets of Data

Example 2. Suppose you need a new battery for your car. Two different brands of battery

have the same price. The lifespan, in years, of 30 batteries of each brand, is

shown below.

a) Determine the mean, median, mode and range. Round to 2 decimal places, if

necessary. Describe any similarities and differences between the two sets of data.

b) Explain why the mean and median do not fully describe the difference between

these two brands of batteries. Consider the range, which is one measure of

dispersion for data. Explain what additional information can be learned from the

range of the data.

c) Is the mode useful to compare in this situation? Explain.

d) Suppose that one battery included in the set of data for brand Y is defective, and

its lifespan is 0.5 years instead of 5.9 years. How would this affect your decision?

When analyzing two sets of data, it is important to look at both similarities and

differences in the data.

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5.2 Frequency Tables, Histograms, and Frequency Polygons

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5.3 Standard Deviation Part I

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When data is concentrated close to the mean, the standard deviation, σ is low. When

data is spread far from the mean, the standard deviation is high. As a result, standard

deviation is a useful statistic to compare the dispersion of two or more sets of data.

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5.3 Standard Deviation Part II

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5.4 The Normal Distribution

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5.5 Z-Scores Part I

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5.5 Z-Scores Part II

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5.6 Confidence Intervals

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Foundations of Mathematics 11

Chapter 5 Sample Note Sheet

- 68% of the data within one standard deviation

- 95% of the data within two standard deviations

- 99.7% of the data within three standard deviations

- The area under the curve = 1 unit (100% of the data)

If the Mean is less than the Median, sku

If the Mean is greater than the Median

Positive Z-Score = Above the Mean

Negative Z Score = Below the Mean

Mean

Median

Mode

Range

Determine Interval Width

12)

Frequency Distribution

Histogram (bar graph)

Marginal error: results are accurate within

- How far off the results of the survey could be from the true value in the population

- As you increase the sample size, you decrease the margin of error

- As you decrease the sample size, yo

Confidence Interval: plus or minus x points/percent

- Chances that the true value in the population are in this range, could be a %

Confidence Level: “X” times out of “Y” (X divided by Y = %)

- Interval in which the true value

a +/- notation, such as 54% +/-

- When you decrease the sample size, you decrease the confidence level

- When you increase the sample size, you incr

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one standard deviation of the Mean

two standard deviations of the Mean

three standard deviations of the Mean

area under the curve = 1 unit (100% of the data)

, sku RIGHT

Median, sku LEFT

Z-Score Formula

Mean Formula

Mean: arithmetic average

Median: middle value (or arithmetic average of the 2 middle values)

Mode: popular or repetitive number

Range: dispersion of data (biggest number minus smallest number)

Determine Interval Width: Range divided by interval

12)

Frequency Distribution: Set of intervals displayed as a

Histogram (bar graph), or Frequency Polygon (line graph)

- Frequency Data plotted on the vertical axis

- Intervals are plotted on the horizontal axis

- Title: Vertical Axis VS. Horizontal Axis

- Frequency Polygon – joining midpoints of the intervals using straight

: results are accurate within Confidence Interval + Confidence Level.

How far off the results of the survey could be from the true value in the population

As you increase the sample size, you decrease the margin of error

As you decrease the sample size, you increase the margin of error

x points/percent

Chances that the true value in the population are in this range, could be a %

” (X divided by Y = %)

Interval in which the true value is estimated to be found within a certain amount of certainty, expressed as

- 3.5%, or ranging from 50.5% - 57.5%

When you decrease the sample size, you decrease the confidence level

When you increase the sample size, you increase the confidence level

Ms Moon

Formula

Formula

: middle value (or arithmetic average of the 2 middle values)

dispersion of data (biggest number minus smallest number)

(interval is between 5 –

: Set of intervals displayed as a Table (tally chart),

lygon (line graph):

joining midpoints of the intervals using straight

How far off the results of the survey could be from the true value in the population

is estimated to be found within a certain amount of certainty, expressed as

Line Plot

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6.1 Graphing Linear Inequalities in Two Variables (Part 1) Review of Graphing Linear Equations:

Most linear equations will be written in one of two different forms: 0 General form

Slope y-intercept form (slope = m & y-intercept = b) The method used to graph a linear relation without technology depends on the form in which the linear equation is written.

Example 1: Use slope and y-intercept method to sketch: a) 2 3 b) 2

Example 2: Use x- and y-intercepts slope method to sketch:

a) 5 3 15 0 b) 2 4 12 0

Equations with only one variable produce either a horizontal or vertical line.



Example 2. For which inequalities is (3, 1) a possible solution? a) 13 3 4 b) 2 5

c) 10 d) 9 Try. Which point(s) is/are in the solution region of the inequality 3 5 8?

a) (0, 0) b) (5, 1)

c) (6, 2) d) 2, 4

The following linear inequalities can be graphed on a Cartesian Plane: 2 3 6 4 3 10 2 0

The solution region to a linear inequality in one or two variables can be represented on a coordinate plane using a boundary line and shading one side of the line. The boundary line will be solid or broken according to the following rules: A solid boundary line is used to represent or . A broken or dotted boundary line is used to represent > or <.

The following procedure can be used to graph the solution region of a two variable linear inequality. Step 1: On a coordinate plane, graph the corresponding linear equation using a

table of values, intercepts, or point and slope. Draw the line solid or broken according to the rule above.

Step 2: The line divides the coordinate plane into two regions, called half planes. The solution region will be on one side of the line. To determine which side, choose the coordinates of a point not on the line, called a test point, and determine if the coordinates of the point satisfy the inequality. If the inequality is satisfied, then the solution is the region from which the point was chosen. If not, the solution region is the other region.

Step 3: Shade the appropriate region.

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Graph Linear Inequalities with Continuous Variables Example 3: Graph the solution set for the following linear inequalities

a) , |2 6, ∈ , ∈ Slope and y-intercept Method

b) , | 2 5 10, ∈ , ∈ x- and y-intercepts Method

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To graph a linear inequality with 0 or 0, the inequatlity must be re-written in the form or . Example 4. Graph the solution set for each linear inequality on a Cartesian plane.

a) , |3 6 0, ∈ , ∈ b) , |4 2 0, ∈ , ∈

c) , |4 8 0, ∈ , ∈ d) , |6 3 0, ∈ , ∈

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Equations with only one variable produce either a horizontal or vertical line.

Example 5. Graph the solution set for each linear inequality on a Cartesian plane.

a) , | 3 0, ∈ , ∈ b) , | 2 0, ∈ , ∈

c) , | 4 0, ∈ , ∈ d) , | 5 0, ∈ , ∈

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6.1 Graphing Linear Inequalities in Two Variables (Part 2)

Graph Linear Inequalities with Discrete Number Variables Example 1. Graph the solution set for the following inequalities.

a) , |2 6, ∈ , ∈ b) , | 2 5 10, ∈ , ∈

c) , |2 7, ∈ , ∈ d) , |2 5 10, ∈ , ∈

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When interpreting the solution region for a linear inequality, consider the restrictions on the domain and range of the variables.

1. If the solution set is continuous, all the points in the solution region are in the solution set.

2. If the solution set is discrete, only specific points in the solution region are in the solution set. This is represented graphically by stippling.

Graph Linear Inequalities Vertical or Horizontal Boundaries Example 2. Graph the solution set for each linear inequality on a Cartesian plane.

a) , | 2 0, ∈ , ∈ b) , | 3 6 6 , ∈ , ∈

c) , | 7 14, ∈ , ∈ d) , |5 10, ∈ , ∈

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Try. Match each linear inequality to its graph.

Most linear inequalities representing real-world problem situations have graphs that are restricted to the first quadrant because the values of the variables in the system must be positive.

Solve By Graphing a Linear Inequality with Discrete Whole-Number Solutions Example 3. A sports store has net revenue of $100 on every pair of downhill skis sold

and $120 on every snowboard sold. The manager’s goal is to have net revenue of more than $600 a day from the sales of these two items. What combinations of ski and snowboard sales will meet or exceed this daily sales goal? Choose two combinations that make sense, and explain your choices.

Define Variables Constraints

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6.2 Exploring Graphs of Systems of Linear Inequalities

System of Linear Inequalities A set of two or more linear inequalities that are graphed on the same coordinate plane; the intersection of their solution regions represents the solution set for the system.

Solve System of Linear Inequalities Example 1. Graph the system of linear inequalities. Justify your representation of the

solution set. a) , | 2 5, ∈ , ∈ and , | 5, ∈ , ∈

b) , | , ∈ , ∈ and , | 5, ∈ , ∈

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c) , | 10, ∈ , ∈ and , | 2, ∈ , ∈

d) , | 3 6, ∈ , ∈ and , | 1, ∈ , ∈

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6.3 Graphing to Solve Systems of Linear Inequalities Solve Graphically a System of Linear Inequalities with Continuous Variables Example 1. Graph the solution set for the following system of inequalities, and choose

two possible solutions from the set. a) 3 2 6 and 3

Use an open dot to show that an intersection point of a system’s boundaries is excluded from the solution set. An intersection point is excluded when a dashed line intersects either a dashed or solid line.

b) 2 1 3 and 2 5 10

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Solve Graphically a System of Linear Inequalities with Discrete Whole-Number Variables Example 2. Graph the system and determine a possible solution.

a) , |3 2 6, ∈ , ∈ and , | 3, ∈ , ∈

b) , | 10, ∈ , ∈ and , | 6, ∈ , ∈

Most linear inequalities representing real-world problem situations have graphs that are restricted to the first quadrant because the values of the variables in the system must be positive.

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Example 3. A restaurant owner, Moon, has two part-time employees: K and J. K is skilled at cooking but has limited experience with customers. Moon pays her $18 an hour. J has experience with customers but not much with cooking. Moon pays him $10 an hour. Moon has a budget of $470 for their wages. Moon can hire both of these employees for no more than 30 hrs. a week, in total. Both employees are scheduled in whole numbers of hours. Determine two possible combinations of numbers of hours scheduled for K and J.


Try. A company makes two types of very fancy cakes: cheesecakes and tiramisus.

The company can make a maximum of 10 cakes in a day. Cheesecakes are more profitable than tiramisus; the company makes at least 3 more cheesecakes than tiramisus each day. Determine two possible combinations of numbers of cheesecakes and tiramisus can be made in one day.


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6.4 Optimization Problems Part 1: Creating the Model Create a Model for an Optimization with Whole-Number Variables Example 1. On a flight between Winnipeg and Vancouver, there are business class

and economy seats. - At capacity, the airplane can hold no more than 145 passengers. - No fewer than 130 economy seats (e) are sold, and no more than 8 business

class seats (b) are sold. - The airline charges $615 for business class seats and $245 for economy

seats. The flight company wants to know the combination of economy and business class seats that will generate the minimum and maximum revenues. Create a model to represent this situation and graph the feasible region.


Objective Function

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Example 2. A school is organizing a track and field meet. - There will be no more than 250 events and no fewer than 100 events to be

scheduled. - The organizers allow 15 min. for each track event (t) and 45 min. for each

field event (f). - They are considering different combinations of track and field events. The school wants to determine the least and greatest amounts of time they should allow. Create a model to represent this situation and graph the feasible region.


Objective Function

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Try. Sophie has two summer jobs. - She works no more than a total of 32 hrs. a week. Both jobs allow her to have

flexible hours but in whole hours only. - At one job, Sophie works no less than 12 hrs. and earns $8.75/hr. At the

other job, Sophie works no more than 24 hrs. and earns $9.00/hr. She would like to know the combination of numbers of hours that will allow her to maximize her earnings. Create a model to represent this situation and graph the feasible region.


Objective Function

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6.5 Optimization Problems Part 2: Exploring Solutions

The solution to an optimization problem (maximum or minimum) is usually found at one of the vertices of the feasible region.

Explore the Feasible Region of a System of Linear Inequalities Example 1. Ribbon flowers and crepe-paper rosettes are being made as decorations.

- At least 50 ribbon flowers and no more than 75 rosettes are needed. - Altogether, no more than 140 decorations are needed - Each ribbon flower takes 6 min. to make, and each rosette takes 9 min. to

make. What combinations of ribbon flowers and rosettes will take the least and the most amount of time to make? What are the minimum and maximum time needed to make these decorations?


Objective Function

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Example 2. A transportation company leases vehicles. - It has 10-passenger vans and 16-passenger minibuses to lease. - At most, 5 minibuses are available to lease. - There are 120 or fewer people to be transported. - Each minibus plus a driver costs $730 to lease, and each van plus a driver

costs $550. What combinations of vans and minibuses will allow the transportation company to minimize and maximize the value of leases? What will the minimum and maximum values be? How many people can be transported?


Objective Function

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Try. The stylists in a hair salon cut hair for women and men. - The salon books at least 4 women’s appointments more than men’s

appointment. - Usually there are 90 or fewer appointments, in total, during a week. - The salon is trying to reduce the number of hours the stylists work. - A women’s cut takes about 75 min., and a man’s cut takes about 30 min. What combination of women’s and men’s appointments would minimize and maximize the number of hours the stylists work? How many hours would this be?


Objective Function

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6.6 Optimization Problems Part 3: Linear Programming Solve Optimization Problems Example 1. A toy company manufactures two types of toy vehicles: racing cars and

SUVs. - Because the supply of materials is limited, no more than 60 SUVs (s) and 40

racing cars (r) and can be made each day. - However, the company can make 70 or more vehicles, in total, each day. - It costs $12 to make a SUV (s) and $8 to make a racing car (r). There are many possible combinations of SUVs and racing cars that could be made. The company wants to know what combination will result in the minimum cost, and what cost that will be?


Objective Function

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Linear Programming A mathematical technique used to determine which solutions in the feasible region result in the optimal solutions of the objective function.

Try. Chubby Cubbies Education Technologies (CCET) manufactures packages of

pattern blocks and linking cubes. - CCET can produce at least 60 packages of pattern block and linking cubes

per day. - Due to the amount of material at hand, CCET can produce at most 30

packages of pattern block and 50 packages of linking cubes per day. - The sale price of the pattern blocks is $7 per pack; the sale price of the

linking cubes is $5 per pack. The company wants to know what combinations will result in the maximum revenue, and what revenue that would be.


Objective Function

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Example 2. L&G Construction is competing for a contract to build a fence. - The fence will be no longer than 50 yd. and will consist of narrow boards that

are 6.in. wide and wide boards that are 8 in. wide. - There must be no fewer than 100 wide boards and no more than 80 narrow

boards. - The narrow boards cost $3.56 each, and the wide boards costs $4.36 each. Determine the minimum cost for the lumber to build the fence.


Objective Function

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7.1 Exploring Quadratic Relations Sketch a Parabola Example 1. Sketch using table of values.

3

2

1

0

1

2

3

Direction of Opening Lowest (Min.) or Highest (Max.) Vertex (Max. or Min.) Vertical Line of Symmetry

A quadratic relations is a relation that can be written in the standard form , where 0. The degree of all quadratic functions is 2, e.g.

4 2 1. The graph of any quadratic function is a parabola with a single vertical line of symmetry. The highest or lowest point lies on the symmetry.

Example 2. Which of the following relation is/are quadratic?

a)

b)

c) 2 1 4 d) √3 0.7


Foundations of Mathematics 11 Ms Moon Determine the Effect of “a” Value Example 3. Roughly sketch 4 3 and 4 3. Describe their

directions of opening.

Determine the Effect of “b” Value Example 4. Roughly sketch 4 6 and 8 6. Describe their locations

of the vertical line of symmetry.

Determine the Effect of “c” Value Example 5. Roughly sketch 2 3 5 and 2 3 7. Describe their y-

intercepts.

To find y-int. set x = 0, and solve for y.

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7.2 Properties of Graphs of Quadratic Functions

Properties of Graph of Quadratic Function Example 1. Determine the vertex, axis of symmetry, x-intercept(s) and y-intercept(s).

Vertex – the lowest (min.) or highest

(max.) point of the parabola. Axis of symmetry – the parabola is

symmetrical about a vertical line, through its vertex.

x-intercept(s) – where the parabola passes

through the x-axis. y-intercept(s) – where the parabola passes

through the y-axis.

Try. Determine the vertex, axis of symmetry, x-intercept(s) and y-intercept(s).

Vertex Axis of symmetry x-intercept(s) y-intercept(s)

** Two points on the same y value of the parabola will be equal distance from Axis

of symmetry.

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Foundations of Mathematics 11 Ms Moon Use Two Points on the Parabola to Determine the x-Coordinate of the Vertex Example 2. Determine the equation of the axis of symmetry for the parabola, if both

points, 2.5, 3 and 5.5, 3 , are located on the same parabola. Try. Determine the equation of the axis of symmetry for the parabola, if both points,

7.25, 6 and 2.75, 6 , are located on the same parabola. Use Symmetry to Estimate the Coordinates of the Vertex Example 3. Determine the axis of symmetry and estimate the coordinates of the

vertex.

Try. Determine the axis of symmetry and estimate the coordinates of the vertex.

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Foundations of Mathematics 11 Ms Moon For all quadratic functions, the domain is the set of real numbers, and the range is a subset of real numbers.

Graph a Quadratic Function Using a Table of Values Example 4. Sketch the graph of the function: 2. Determine the coordinates

of the vertex, the domain and range of the function.

3

2

1

0.5

0

1

2

3

Domain: Range:

Try. Sketch the graph of the function: 3 7. Determine the coordinates of

the vertex, the domain and range of the function.

3

2

1.5

1

0

1

2

3

Domain: Range:

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Foundations of Mathematics 11 Ms Moon When a problem can be modelled by a quadratic function, the domain and range of the function may need to be restricted to values that have meaning in the context of the problem.

Reason about the Maximum Value of a Quadratic Function Example 5. The path of water from a water jet forms an arch that can be defined by

the function: 9 , where x represents the horizontal distance from the opening in the ground in feet and is the height of the sprayed water, also measured in feet. Graph the function, and state its domain and range for this context. What is the maximum height of the arch of water, and how far from the opening in the ground can the water reach?

0 1 2 3 4 4.5 5 6 7 8 9

The width of the opening in problem solving can be determined from , if 0.

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Foundations of Mathematics 11 Ms Moon Try. The path of water from a water jet forms an arch that can be defined by the

function: 0.12 3 , where x represents the horizontal distance from the opening in the ground in feet and is the height of the sprayed water, also measured in feet. Graph the function, and state its domain and range for this context. What is the maximum height of the arch of water, and how far from the opening in the ground can the water reach?

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7.3 Solving Quadratic Equations by Graphing with Technology

2 is a quadratic function where as 2 0 is a quadratic equation. Solving an equation means to solve for x. A quadratic equation can be solved by graphing the corresponding quadratic function. The solution of x is called a root. The root(s) of a quadratic equation is/are the x-intercept(s) of the graph. They are also known as the zeroes. Therefore, zeroes = x – intercepts = roots.

Solve a Quadratic Equation in Standard Form Example 1. Solve for x in each equation by graphing.

a) 2 8 1 0

b) 4 4 0

c) 2 5 6 0

For any quadratic equation, there can be zero, one, or two real roots. This is because a parabola can intersect the x-axis in zero, one, or two places.

Determine the zeroes using TI 83 Step 1. Graph the quadratic function

Press Y=

Enter Quadratic Function Press Graph

Step 2. Finding one zero

Press 2nd key

Select CALC

Select #2 ZERO Select Left bound Press Enter

Select Right Bound Press Enter

Guess Press Enter

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Try. Solve each equation by graphing a) 2 3 2 0

b) 5 8 3 0

If the quadratic function is not in standard form – you can graph the expression on the left side and the expression on the right side as functions on the same axes – the x-coordinates of the points of intersection of the two graphs are the roots of the equation.

Solve a Quadratic Equation in Non-Standard Form Example 2. Determine the root(s) of this quadratic equation. Verify your answers. 3 6 5 2 4

Try. Solve by graphing 4 1 3

Determine the zeroes using TI 83 Step 1. Graph the quadratic function

Press Y=

Enter 1st Quadratic Function under Y1 Enter 2nd Quadratic Function under Y2 Press Graph

Step 2. Finding one zero

Press 2nd key

Select CALC

Select #5 INTERSECT Select First Curve Move closer Press Enter

Select Second Curve Press Enter

Guess Press Enter

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You may have to change window in order to see a better graph. Press Window to change its setting. Xmin= left, Xmax = right, Ymin = low, Ymax = high

Graph to Determine the Number of Roots Example 3. Lamont runs a boarding kennels for dogs. He wants to construct a

rectangular play space for the dogs, using 40 m of fencing and an existing fence as one side of the play space.

length

a) Write a function that describes the area, A, in square metres, of the play space for any width, w in metres.

b) Write equations you could use to determine the widths for areas of 250 m2, 200 m2, and 150 m2.

c) Determine the number of possible widths for each equation using a graph.

width

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7.4 Factored Form of a Quadratic Function (Part 1) Factor Binomials and Trinomials Example 1. Factor each trinomial.

a) 3 40 b) 6 16

c) 6 d) 6 9

e) 6 7 f) 5 24

g) 3 7 6 h) 2 5 7 Example 2. Factor each binomial.

a) 1 b) 25

c) 64 d) 121 Example 3. Factor each binomial.

a) 2 b) 25

c) 64 d) 121

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When a quadratic function is written in factored form , each factor can be used to determine a zero of the function by setting each factor equal to zero

and then solving. The x-intercepts are and . The symmtry is . The y-

intercept is ∙ ∙ .

Graph a Quadratic Function Given in Standard Form Example 1. Rewrite the following functions in factored form .

Determine (1)the x-intercept(s), (2)axis of symmetry, (3)vertex and (4)y-intercept. Sketch and state the (5)domain and range.

a) 2 8 6

b) 3 6 3

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If a quadratic function has only one x-intercept, the factored form can be written as follows: Try. Rewrite the following functions in factored form . Determine

(1)the x-intercept(s), (2)axis of symmetry, (3)vertex and (4)y-intercept. Sketch and state the (5)domain and range.

a) 2 8 6

b) 3 6 3

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7.4 Factored Form of a Quadratic Function (Part 2) Determine the Equation of a Quadratic Function Given its Graph Example 1. Determine the functions that define these parabolas. Write the function in

standard form. a)

b)

Try. Determine the functions that define these parabolas. Write the function in

standard form.

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Use a Partial Factoring Strategy to Sketch the Graph of a Quadratic Function Example 2. Sketch the graph of the following quadratic functions. State the domain

and range of the function.

a) 6 10

b) 3 12

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Solve a Problem Modelled by a Quadratic Function in Factored Form Example 3. That members of a church hold a fundraiser every Friday night in the

summer. They usually charge $6 for a burger meal. They know, from previous Fridays, that 120 burger meals can be sold at the $6 price but, for each $1 price increase, 10 burger meals will be sold. What should the members charge if they want to raise as much money as they can for the church?

Try. A student who wishes to use a computer lab at a local library must buy a

membership. They library charges $20 for membership. Four hundred students purchase the membership. The library estimates that for every $4 increase in the membership fee, 40 fewer students will become members. What membership fee will provide the maximum revenue to the library?

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8.1 Comparing and Interpreting Rates Rate: Unit rate: Ex. 1: Determine each unit rate:

a) $4.20 for 12 cans of coke. b) 608 km in 8 hours

Ex. 2: Natasha wants to buy a 12 kg turkey from her local butcher for $42.89. The

local supermarket has turkeys advertised for $1.49/lb. There are about 2.2lb in 1 kg. Which store has the lower price?

Comparing prices Comparing unit prices

Your Turn: Determine which store has the better price. Store A: 30 cans of spam for $20.70 Store B: 25 cans of spam for $16.00

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Ex. 3: On one day Sue drove 240 km in 4 h. She stopped for lunch for 2 h and then drove back in 3 h. Draw a graph to compare her driving rates.

Ex. 4: For each of the following, compare the two rates and determine the higher rate.

a) Running speeds: 8 mph or 4 km in 12 min

b) fuel efficiency: 12.4 L/ 100 km or 41.3 L of fuel needed to travel 350 km

c) driving speeds: 15 km in 10 min or 26 m/s

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Ex. 5: It takes one person 2 min and 15 s to run 600 m and another person takes 5 min and 25 s to run 1.5 km. Who has the faster running speed?

8.2 Solving Problems That Involve Rates Ex. 1: Jeff lives in White Rock. The gas tank of his truck holds about 90 L. He can

either buy in town at $1.34CAD/L or travel across the border and pay $4.17USD/gal. Which option makes the most sense economical? Note: 1 gal = 3.79 L and the current exchange rate is $1 USD = $1.03 CAD.

Ex. 2: Describe a situation in which each unit rate might be used.

a) 0.05 mg/kg

b) 98.5 ¢/L

c) 7.2 MBps

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Ex. 3: Paula is asked to order snacks for an office meeting of 180 people. She decides to order cookies, which come in a box of 24. She estimates that she will need 2.5 cookies/person. How many boxes should she buy?

Ex. 4: Amelia walks briskly, at 6 km/h. When she walks at this rate for 2 h, she

burns 454 Cal. Bruce walks at a slower rate, 4 km/h, burning 62 Cal. in 30 min. If Amelia walks for 3 h, how much longer will Bruce have to walk in order to burn the same amount of Calories?

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8.3 Scale Diagrams Scale Diagram: Scale: Scale Factor: Ex. 1: The following two polygons are similar. Determine the lengths of sides g, h, x,

and y to the nearest tenth of a unit.

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Ex. 2: Understanding the idea of a scale of 1:5 or scale factor of 2.

Original

Key Ideas:

You can multiply any linear dimension of a shape by the scale factor to calculate the corresponding dimension of a similar shape.

When determining the scale factor, k, used for a scale diagram, the measurement from the original shape is place in the denominator.

When a scale factor is between 0 and 1, the new shape is a of the original shape.

When a scale factor is greater than 1, the new shape is an of the original shape.

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8.4 Scale Factors and Areas of 2-D Shapes When two 2-D shapes are similar and their dimensions are related by a scale factor k:

Area of similar shape 2-D shape = k2 (Area of original shape) Ex. 1: a) Jasmine is making a kite from a 2:25 scale

diagram. The area of the scale diagram is 20 cm2. How much fabric will she need for her kite?

b) If the scale diagram for the kite had been drawn using

a scale ratio of 1:20, and the area of the scale diagram had been 30 cm2, how much fabric would Jasmine have needed for her kite?

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Ex. 2: Jim’s laptop has a monitor with the dimensions 9 in. by 12 in. The image on his laptop is projected onto the screen of a whiteboard. According to the documentation for the whiteboard, its screen area is 2836.6875 in.2.

a) The image on the whiteboard is similar to the image on the laptop. Determine

the scale factor used to project the images on the laptop to the whiteboard.

b) Determine the dimensions of the whiteboard. Note:

When the area of a similar 2-D shape and the area of the original shape are known:

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8.5 Similar Objects: Scale Models and Scale Diagrams Similar objects: Ex. 1: A chef bought a pair of new frying pans that are

different sizes. The larger pan has a bottom diameter of 30 cm, a depth of 6 cm, and a handle length of 24 cm. the smaller one has a bottom diameter of 20 cm a depth of 4 cm and a handle length of 16 cm. Are the pans similar?

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Ex. 2: Esmerelda bought this toy tractor to give to her younger brother for his birthday. The dimensions of the toy are given in the diagram to the right. The scale ratio on the package is 1:16. She knows that her brother will want to know the size of the real tractor. How can she determine the dimensions of the real tractor?

Ex. 3: Nadia has found plan for a bookend in the woodworking magazine. The plans include a scale diagram, with a scale ratio of 1:5. Determine the dimensions (length, width, height, and base thickness) of the actual bookend.

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Ex. 4: Determine if the two objects are similar. Ex. 5: A juice company plans to enlarge this can by a scale factor of

1.5. Determine the measurements of the enlarged can. To determine the scale factor:

When a scale factor is between 0 and 1, the new object is a reduction of the original object. When a scale factor is greater than 1, the new object is an enlargement of the original object.

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8.6 Scale Factors and 3-D Objects Ex. 1: I enlarged a rectangular prism with dimensions 6

cm by 10 cm by 8 cm to a similar one with dimensions 12 cm by 20 cm by 16 cm.

a) Determine the scale factor used.

b) Determine the surface area of each prism.

c) By what factor is the surface area of the second one bigger than the first?

d) Determine the volume of each prism.

e) By what factor is the volume of the second one bigger than the first?

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Notes: For Surface Area: For Volume: Ex. 2: The Great Pyramid of Giza in Egypt was

built on a square base, with the dimensions shown. An artist who works with plate glass wants to build a replica of the pyramid for an installation at an art gallery. The artist is restricted by the floor dimensions, which are 6.0 m by 6.0 m, and the ceiling height of 3.5 m. as well, the glass sculpture must have room for a 1.0 m walkway around its base.

a) What scale factor might the artist use the build the sculpture?

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b) How much glass will the artist need to build the sculpture? Ex. 3: The smaller tank in the photograph has a

capacity of 1400m3, and the larger tank has a capacity of 4725m3.

a) During the refining process, both tanks are

filled with oil from a pumping station at the same rate. How many times longer will it take to fill the larger tank than it will take to fill the smaller tank?

b) How many times greater is the radius of the larger tank than the radius of the smaller tank?

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