Unit 5 notes

Unit 5 Notebook 2012.notebook

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December 18, 2012

Dec 51:49 PM

Unit 5:Foundations of Geometry

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Equations of Lines (and all the basics...)

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Let's start at the beginning...

Do you know where the x axis and y axis are on a coordinate axis system?

Where is the ORIGIN?

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Do you know where the 4 QUADRANTS are on a coordinate axis system?

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What is SLOPE? Slope is how much a line slants

What letter do we use to represent slope??? m

What is the SLOPE formula???

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Example:


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Do you recall what the 4 types of slope look like?

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What is the SLOPE INTERCEPT form of a line???

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What does m stand for?

What does b stand for?

m stands for SLOPE. m tells you how to MOVE from one

point to another.

b represents the Y INTERCEPT. b tells you where to BEGIN your graph of a line on the

coordinate axes.

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What is the slope (m) and y intercept (b) of this line?

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Examples:

1. Find the slope of the line formed by A(9, 3) and B(4, 5).

2. Write the equation of the line 5y 10x = 15 in slope intercept form. What is the slope and y intercept?

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3. If (k, 3) is on the line 2y = 9 x, what is the value of k?

4. What type of slope does the line y = 11 have?

5. Write the equation of the line 2y + 8 = x in slope intercept form. What is the slope and y intercept?


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December 18, 2012

Dec 52:21 PM

Homework:p. 1 2 oddsp. 3 #1 8

Dec 52:21 PM

Answers to Homework: p. 1 2 odds, p. 3 #1 8

1. B 3. C 5. B 7. C 9. D

11. C 13. C 15. 3/5

p. 31. y = 3/2x + 8 2. y = 13/11x + 12/11

3. y = 9/7x + 1 4. y = 1/3x 2

5. y = 6/5x 3 6. y = 4x 1

7. y = 11/4x 8 8. y = 11/8x + 6

Dec 53:43 PM

Graphing Lines

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Dec 53:43 PM Dec 53:43 PM

Homework: p. 5 & 6 evens


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December 18, 2012

Dec 53:46 PM

Writing Equations of Lines

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FORMULAS:

Slope:

Standard Form of a line: Ax + By = C

Slope Intercept Form of a line: y = mx + b

(m = slope, b = y intercept)

Point Slope Form of a Line: y – y1 = m(x – x1)

Recall...

Dec 53:46 PM

While y y1 = m(x x1) may be a new formula to some of you, it is actually just the slope formula

after you cross multiply!

And if you really do not want to memorize yet another formula, simply stick to your old

methods of writing equations of lines!

Dec 53:46 PM

How do you write the equation of the line if you are given

a slope and a y- intercept?

Plug directly into the point slope formula for m and b

Examples: Write the equation of the line give m and b.

1. m = 2, b = 5

2. m = 1/2, b = 7

3. m = 1, b = 9.2

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a slope and a point?

Find the y- intercept, & then

plug directly into the point slope formula for m and b

Examples: Write the equation of the line give the point and the slope.

1. slope = 2, point A(4, 6)

2. slope = 1/3, point B(3, 9)

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2 points?

Find the slope.

Use 1 point and your m to find the y- intercept, &

then plug directly into the point slope formula for m and b

Examples: Write the equation of the line give 2 points.

1. E(3, 2) and F(5, 0)

2. R(5, 1) and T(6, 2)


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December 18, 2012

Dec 53:46 PM

More Practice...

I: Write each equation in y = mx + b form. 1. y + 2x = 9 2. 3y 9 = 2x 3. 4y 8x = 12

II: Write the equation of the line in y = mx + b form with the given information.1. slope = 9, y intercept = 13

2. slope = 1/2, passes through the point E(4, 3)

3. passes through the points P(7, 6) and Q(8, 3)

Dec 53:46 PM

Homework: p. 12 #7, 8,

15 18

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p. 12 #7 18 Solutions:

7. y = 3/5x + 5 8. y = 9x + 4

15. y = x + 2 16. y = 1/7x + 26/7

17. y = x 1 18. y = 5x +4

Dec 53:50 PM

Writing Equations of Parallel and

Perpendicular Lines

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http://www.youtube.com/watch?v=Gbz7EEiWnv0

I'm sure you know something about parallel and

perpendicular lines...let's watch this video and check it

out...

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Did you ever stop to really think about why it's important that

we have parallel lines?

How can you PROVE that 2 lines are parallel???

http://www.youtube.com/watch?v=gbz7eeiwnv0


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December 18, 2012

Dec 53:50 PM

Parallel lines have the SAME slopes. This means that the lines MOVE in the same way.

10 8 6 4 2 0 2 4 6 8 10

1098765432

12345678910

x

yIf you take the red point and MOVE it 3 units up and 4 units to the right, where will the new point be??? Now do the same thing with the green point.

Connect the red points and then connect the green points.

Check to see that the red line and the green line have the same slopes.

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Examples I:Find the slope of a line that is parallel to the given line.

1. y = 9x 3 2. y 2x = 3 3. 6x 2y = 8

Examples II:Write an equation of the line that is parallel to y = 7x + 2 with the given y intercept.

1. b = 1 2. b = 9 3. b = 0

Examples III:Write an equation that is parallel to the given line.

1. 2y = 6x 4 2. y 1 = x 3. 3x = y 7

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What types of lines do you see in these images?

Do these lines have the same slope???

What kinds of slope do these types of lines have???

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Perpendicular lines are lines that intersect and form a right angle. Since these types of lines do intersect, they clearly are not parallel, and therefore do not have the same slope. Because perpendicular lines meet and form a right angle, they have a very special relationship regarding slope. Perpendicular lines have negative reciprocal slope.

Example: m = 7 m =

m = m = 2

m = m =

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See if you can fill in the chart below using your knowledge of parallel and perpendicular lines.

Slope Parallel Slope Perpendicular Slope

3

4

1/2

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Examples: 1. Write the equation of the line that is perpendicular to y = 3x 5 and has the same yintercept as y + 2 = 5x.

2. Write an equation of the line that is perpendicular to y 4x = 8 and passes through (0, 2).

3. Write an equation of the line that is parallel to y 3 = 2x and passes through (6, 6).


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December 18, 2012

Dec 53:50 PM

Homework:

p. 7 8 #16 24

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Answers to p. 7 #16 24

16. C 17. D

18. C 19. B

20. B 21. B

22. 2/5 23. y 6 = 3(x 1)or y = 3x + 3

24. A

Dec 177:48 AM

Group Work!!!

Get a partner (or 2) and complete the following worksheet on writing equations of lines.

I will check this and go over this tomorrow!

Good luck.

Quiz #8 tomorrow!!!

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Solutions to Worksheet (review for Quiz #8)

1. C 2. A 3. B

4. 2 5. 3/7 6. 4

7. C 8. A 9. D

10. D 11. D 12. D

13. y = 2x 1 14. y = 1/2x 15.

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Quiz #8 today!!!

No CR due to the Holiday Break coming up.

Tomorrow we will take the Unit 5 Pre Test

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Solving Systems of Equations by Graphing


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December 18, 2012

Dec 53:50 PM

A system of equations is a set or collection of 2 or more equations with a set of unknown variables.

In solving a system of equations. we try to find values for each of the unknown variables that will

satisfy every equation in the system.

We can solve a system of equation using 4 methods:

1. Graphically2. Algebraically (Elimination)3. Algebraically (Substitution)4. Matrices

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Systems of equations can be given to you in a narrative as shown in the example below:

The price of a movie matinee ticket is $1.50 for kids and $4 for adults. If the theater took in $124 in ticket sales and 51 people attended the matinee, how many were adults and

how many were children?

Or a system of equation can be given algebraically, as shown below:

Today we are going to solve systems by graphing them!!!

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Graph these now please!!!

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December 18, 2012

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So basically, when you are asked to find the solutıon

to a system of equations, you simply fınd the poınt of ıntersectıon of the two

lines!!!

Now what....

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Remember our friends from WSHS??? They did the "Quad Solve" rap? They have

another good one for Systems of Equations...Let's watch!!!

http://www.youtube.com/watch?v=1qHTmxlaZWQ

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Let's try this example...Can someone please do it on the board for me??? PLEASE..........

Solve the following system of equations GRAPHICALLY:

5 4 3 2 1 0 1 2 3 4 5

5

4

3

2

1

1

2

3

4

5

x

y

#1:

Dec 53:50 PM

Let's try this example...Can someone please do it on the board for me??? PLEASE..........

Solve the following system of equations GRAPHICALLY:

5 4 3 2 1 0 1 2 3 4 5

5

4

3

2

1

1

2

3

4

5

x

y

#2:

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Can you check your solution set to question #2?

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Homework: p. 13 #1 4

http://www.youtube.com/watch?v=1qhtmxlazwq


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December 18, 2012

Dec 53:50 PM

Solutions to p. 13 #1 4

1. x = 3, y = 2

2. x = 1, y = 1

3. x = 4, y = 3

4. x = 2, y = 1

Dec 53:55 PM

Solving Systems of Equations by

Elimination

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The Elimination Method used to be called the "Addition/ Elimination Method." When you solve a system of equations using the Elimination Method, you will try to get rid of one variable by adding the system together.

Steps: 1. Make sure the system is set up for elimination (all variables will align, as will the = sign)2. Use simple multiplication to ensure that you can eliminate one of the variables by addition. (Sometimes you can skip this step)3. Solve the remaining linear equation.4. Substitute your answer back into one of the equations and solve for the other variable.5. Check your solution.

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Solve the following system of equations and check:

4x + 8y = 204x + 2y = 30

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Solve the following system of equations and check:

4x + 9y = 9x 3y = 6

Try the "Criss Cross" method

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Solve the following system of equations:

3x + 7y = 169x + 5y = 16

Try the "Criss Cross" method


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December 18, 2012

Dec 53:55 PM

Homework: p. 15 #3, 5, 7, 9

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Solutions to p. 15

3. x = 10, y = 1

5. x = 1, y = 3

7. x = 5, y = 6

9. x = 1, y = 4

Dec 53:55 PM

Solving Systems of Equations by Substitution

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The method of solving "by substitution" works by solving one of the equations (you choose which one) for one of the variables (you choose which one), and then plugging

this back into the other equation.

***Most of the time, your system will already have one

equation solved for (one of the variables).



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December 18, 2012

Dec 53:55 PM

Homework:p. 17 #5 8

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Solutions to p. 17 #5 8

5. x = 3, y = 2

6. x = 2, y = 3

7. x = 3. y = 6

8. x = 3, y = 4

Dec 183:19 PM

Review for Unit 5 Post Assessment

Get into groups of 4 and work on the review sheet.

Answers will be on the smart board. Check the answers as

you go along.

Good luck :)

Dec 183:19 PM

Answers to Review Sheet to Unit 5 Post Assessment:

1. k = 4 2. y = 7 3. m = 1

4. D 5. B 6. A

7. y = x + 1 8. y = 4x 1 9. y = 2x 1

10. b = 0 11. m = 2/5 12. k = 8

13. y = 3x + 3 14. D 15. A

16. Graph 17. x = 2, y = 3 (Quad IV) 18. x = 4, y = 2

19. x = 1, y = 2 20. x = 2, y = 2

Dec 53:57 PM

Trig Ratios???

Dec 53:56 PM

Trigonometry Finding Missing Sides


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December 18, 2012


Trigonometry Finding the Missing Angle


Trigonometry Word Problems

Dec 52:20 PM

y = mx + bSlope formulaIs a poitn on a line?What is the slope and y intercept?get y by itself

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December 18, 2012



Unit 5 notes

Documents

line y

y intercept

y axis

pm dec53

givena slope

pm dec52

point slope formula

kinds of slope