10-7-14 Bellwork

Objective 1The student will be able to:

graph ordered pairs on a coordinate plane.

In the beginning of the year, I created a seating chart for my classes. I created 5 rows of desks with 4 desks in each row. Brian sits in the third row at the second desk (3,2) and Dwanda sits in the second row at the third

desk (2,3). Are these seats the same?

No!! The seats (3,2) and (2,3) are called ordered pairs because the order in which

the pair of numbers is written is important!!

Who is sitting in desk (4,2)?

A B C D E

F G H I J

K

P

L M N O

Q R S T

4

3

2

11 2 3 4 5

N

Ordered pairs are used to locate points in a coordinate plane.

x-axis (horizontal axis)

origin (0,0)

y-axis (vertical axis)5

5

-5

-5

In an ordered pair, the first number is the x-coordinate. The second number

is the y-coordinate. Graph. (-3, 2)

•

5

5

-5

-5

What is the ordered pair for A?

1. (3, 1)

2. (1, 3)

3. (-3, 1)

4. (3, -1)

5

5

-5

-5

• A

What is the ordered pair for B?

5

5

-5

-5

• B

1. (3, 2)

2. (-2, 3)

3. (-3, -2)

4. (3, -2)

What is the ordered pair for C?

1. (0, -4)

2. (-4, 0)

3. (0, 4)

4. (4, 0)

5

5

-5

-5

• C

What is the ordered pair for D?

5

5

-5

-5

• D

1. (-1, -6)

2. (-6, -1)

3. (-6, 1)

4. (6, -1)

Write the ordered pairs that name points A, B, C, and D.

A = (1, 3)

B = (3, -2)

C = (0, -4)

D = (-6, -1)

5

5

-5

-5

• A

• B• C

• D

The x-axis and y-axis separate the coordinate plane into four regions, called quadrants.

II

(-, +)

I

(+, +)

IV

(+, -)

III

(-, -)

Name the quadrant in which each point is located

(-5, 4)

1. I2. II3. III4. IV5. None – x-axis6. None – y-axis

Name the quadrant in which each point is located

(-2, -7)

1. I2. II3. III4. IV5. None – x-axis6. None – y-axis

Name the quadrant in which each point is located

(0, 3)

1. I2. II3. III4. IV5. None – x-axis6. None – y-axis

Linear Equations in Two Variables

Students will complete a table for a linear equation and graph ordered

pairs.

Objective 2 and 3

List some pairs of numbers that will satisfy the equation x + y = 4.

x = 1 and y = 3

x = 2 and y = 2

x = 4 and y = 0

What about negative numbers?

If x = -1 then y = ?

y = 5

x + y = 4What about decimals?

If x = 2.6 then y = ?

y = 1.4

Now, let’s graph the pairs of numbers we have listed.

(1, 3) (2, 2) (4, 0) (-1, 5) (2.6, 1.4)

Connect the points on your graph.

What does the graph look like?

• ••

•

•

It is a straight line! It is a linear relation.

All solutions for theequation x+y=4!

Is (3, -1) a solution to this equation?

NO! You can check by graphing it or plugging into the equation!

• ••

•

•What does the line

represent?

1) Which is a solution to2x – y = 5?

1. (2, 1)

2. (3, 2)

3. (4, 3)

4. (5, 4)

Answer NowAnswer Now

2) Which ordered pair is not a solution to the graph

shown?

1. (0, -1)

2. (3, 5)

3. (-2, -5)

4. (-3, -1)

Answer NowAnswer Now

Objectives 3 and 4The student will be able to:

1. graph linear functions.

2. write equations in standard form.

Graphing Steps

1) Isolate the variable (solve for y).

2) Make a t-table. If the domain is not given, pick your own values.

3) Plot the points on a graph.

4) Connect the points.

1) Review: Solve for y 2x + y = 4

• Draw “the river”

• Subtract 2x from both sides

- 2x - 2x

y = -2x + 4

2) Solve for y: 4x + 2y = -6• Subtract 4x• Simplify• Divide both sides by 2• Simplify

- 4x - 4x

2y = -4x - 6

2 2

y = -2x - 3

3) Solve for y: x - 3y = 6

• Subtract x• Simplify• Divide both sides by -3• Simplify

- x - x

-3y = -x + 6

-3 -3

6

3

xy

2

3

xy

or

4) Review: Make a t-tableIf f(x) = 2x + 4, complete a table using the domain {-2, -1, 0, 1, 2}.

2(-2) + 4 = 0 (-2, 0)

2(-1) + 4 = 2 (-1, 2)

2(0) + 4 = 4 (0, 4)

2(1) + 4 = 6 (1, 6)

2(2) + 4 = 8 (2, 8)

x f(x)-2

-1

0

1

2

ordered pair

5) Given the domain {-2, -1, 0, 1, 2}, graph 3x + y = 6

-3(-2) + 6 = 12 (-2, 12)

-3(-1) + 6 = 9 (-1, 9)

-3(0) + 6 = 6 (0, 6)

-3(1) + 6 = 3 (1, 3)

-3(2) + 6 = 0 (2, 0)

x -3x + 6 ordered pair

1. Solve for y: 3x + y = 6

Subtract 3x - 3x - 3x

y = -3x + 62. Make a table

-2

-1

0

1

2

Bonus questions!What is the x-intercept?

(2, 0)What is the y-intercept?

(0, 6)Does the line increase or decrease?

Decrease

5) Given the domain {-2, -1, 0, 1, 2}, graph 3x + y = 6

3. Plot the points(-2,12), (-1,9), (0,6), (1,3), (2,0)

4. Connect the points.

1. .

2. .

3. .

4. .

Which is the graph of y = x – 4?

Standard FormAx + By = C

A, B, and C have to be integers

An equation is LINEAR (the graph is a straight line) if it can be written in standard form.

This form is useful for graphing (later on…).

Determine whether each equation is a linear equation.

1) 4x = 7 + 2y

Can you write this in the form

Ax + By = C?

4x - 2y = 7

A = 4, B = -2, C = 7

This is linear!

2) 2x2 - y = 7Can you write it in standard form?

NO - it has an exponent!Not linear

3) x = 12x + 0y = 12

A = 1, B = 0, C = 12Linear

Determine whether each equation is a linear equation.

Here’s the cheat sheet! An equation that is linear does NOT contain the following:

1. Variables in the denominator

2. Variables with exponents

3. Variables multiplied with other variables.

xy = 12

32y

x

2 3y x

Is this equation linear?

1. Yes

2. No

4 3x y

Standard Formx – 4y = 3

Is this equation linear?

1. Yes

2. No

29 4y x

Exponents are not allowed!

Is this equation linear?y = -3

1. Yes

2. No

Standard Form0x + y = -3

Objective 4 and 5The student will be able to:

find the x- and y-intercepts of linear equations.

What does it mean to INTERCEPT a pass in football?

The path of the defender crosses the path of the thrown football.

In algebra, what are x- and y-intercepts?

What are the x- and y-intercepts?The x-intercept is where

the graph crosses the x-axis. The y-coordinate is always 0.

The y-intercept is where the graph crosses the y-axis. The x-coordinate is always 0.

(2, 0)

(0, 6)

Find the x- and y-intercepts.1. x - 2y = 12

x-intercept: Plug in 0 for y.

x - 2(0) = 12

x = 12; (12, 0)

y-intercept: Plug in 0 for x.

0 - 2y = 12

y = -6; (0, -6)

x-intercept: Plug in 0 for y.

-3x - 5(0) = 9

-3x = 9

x = -3; (-3, 0)

y-intercept: Plug in 0 for x.

-3(0) + 5y = 9

5y = 9

y = ; (0, )9

5

9

5

Find the x- and y-intercepts.2. -3x + 5y = 9

x-intercept: Plug in 0 for y.

Does 0 = 7?

No! There is no x-intercept. None

What type of lines have no x-intercept?

Horizontal!

Remember VUXHOY?

Horizontal lines…y = 7…y-int = (0, 7)

Find the x- and y-intercepts.3. y = 7 ***Special case***

What is the x-intercept of3x – 4y = 24?

1. (3, 0)

2. (8, 0)

3. (0, -4)

4. (0, -6)

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What is the y-intercept of-x + 2y = 8?

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30 31 32

1. (-1, 0)

2. (-8, 0)

3. (0, 2)

4. (0, 4)

What is the y-intercept ofx = 3?

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30 31 32

1. (3, 0)

2. (-3, 0)

3. (0, 3)

4. None