Bell Ringer 10/8/14

Bell Ringer 10/9/14

Name the locations of the four quadrants on a graph.

Objective 1The student will be able to:

graph ordered pairs on a coordinate plane.

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

HW

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

Bell Work 10/10/14

1. Write the following equation in standard form and check if the ordered pair (4 , 4) is a solution the linear equation:

3(y - 5) + 2(x + 2) = 10

Bell Ringer 10/13/14

1. For the linear equation -2x + 3y = 8, determine whether the ordered pair is a solution.

• A. (-4 , 0)

• B. (2 , -4)

2. Solve for y

-4x + 5y = -1

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!

here

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

Bell Ringer 10/14/14Solve for y.

Evaluate the following expression

Domain: -4, -2, 0, 2, 4

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!

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)

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

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

ObjectiveThe student will be able to:

find the slope of a line given 2 points and a graph.

What is the meaning of this sign?

1. Icy Road Ahead

2. Steep Road Ahead

3. Curvy Road Ahead

4. Trucks Entering Highway Ahead

What does the 7% mean?

7% is the slope of the road. It means the road drops 7 feet vertically for every 100 feet

horizontally.

7%

So, what is slope???Slope is the steepness of a line.

7 feet

100 feet

Slope can be expressed different ways:

A line has a positive slope if it is going uphill from left to right.

A line has a negative slope if it isgoing downhill from left to right.

2 1

2 1

( ) vertical change

( ) horizontal change

y y risem

x x run

When given the graph, it is easier to apply “rise over run”.

1) Determine the slope of the line.

Start with the lower point and count how much you rise and run to get to the other

point!

Determine the slope of the line.

6

3

run

3

6= =

1

2

rise

Notice the slope is positive AND the line increases!

2) Find the slope of the line that passes through the points (-2, -2) and (4, 1).

(1 ( 2))

(4 ( 2))m

2 1

2 1

( )

( )

y ym

x x

(1 2)

(4 2)

When given points, it is easier to use the formula!

y2 is the y coordinate of the 2nd ordered pair (y2 = 1)

y1 is the y coordinate of the 1st ordered pair (y1 = -2)

13

6 2

Did you notice that Example #1 and Example #2 were the same problem

written differently?

(-2, -2) and (4, 1)6

31

2slope

You can do the problems either way!Which one do you think is easiest?

Find the slope of the line that passes through (3, 5) and (-1, 4).

1. 4

2. -4

3. ¼

4. - ¼

3) Find the slope of the line that goes through the points (-5, 3) and (2, 1).

m y2 y1

x2 x1

1 3

2 ( 5)m

1 3

2 5m

2

7m

Determine the slope of the line shown.

1. -2

2. -½

3. ½

4. 2

Determine the slope of the line.

The line is decreasing (slope is negative).

2

-1

rise

run

2

1 2

Find points on the graph. Use two of them and apply rise over run.

What is the slope of a horizontal line?

The line doesn’t rise!

m 0

number0

All horizontal lines have a slope of 0.

What is the slope of a vertical line?

The line doesn’t run!

All vertical lines have an undefined slope.

m number

0undefined

1

Draw a line through the point (2,0) that has a slope of 3.

1. Graph the ordered pair (2, 0).

2. From (2, 0), apply rise over run (write 3 as a fraction).

3. Plot a point at this location.

4. Draw a straight line through the points.

3