chapter 6 concept -...

Chapter 6

Concept

Lines

Write Equations of Lines 2 different ways:

Slope-Intercept form Standard Form

Write Equations of Lines 2 different ways:

Slope-Intercept form Standard Form

y = 2x - 3 -2x + y = -3

Write Equations of Lines 2 different ways:

Slope-Intercept form Standard Form

y = 2x - 3 -2x + y = -3

Write Equations of Lines 2 different ways:

Slope-Intercept form Standard Form

y =

y =

2x - 3

- x + 4 3 2 2x + 3y = 12

-2x + y = -3

Write Equations of Lines 2 different ways:

Slope-Intercept form Standard Form

y =

y =

2x - 3

- x + 4 3 2 2x + 3y = 12

-2x + y = -3

Write Equations of Lines 2 different ways:

Slope-Intercept form Standard Form

y =

y =

y =

2x - 3

- x + 4

x 1

3 2

2 - x + y = 0

2x + 3y = 12

-2x + y = -3

1 2

Write Equations of Lines 2 different ways:

Slope-Intercept form Standard Form

y =

y =

y =

2x - 3

- x + 4

x 1

3 2

2 - x + y = 0

2x + 3y = 12

-2x + y = -3

1 2

Write Equations of Lines 2 different ways:

Slope-Intercept form Standard Form

y =

y =

y =

y =

2x - 3

- x + 4

x

4

1

3 2

2 - x + y = 0

2x + 3y = 12

-2x + y = -3

y = 4

1 2

Write Equations of Lines 2 different ways:

Slope-Intercept form Standard Form

y =

y =

y =

y =

2x - 3

- x + 4

x

4

1

3 2

2 - x + y = 0

2x + 3y = 12

-2x + y = -3

y = 4

1 2

Write Equations of Lines 2 different ways:

Slope-Intercept form Standard Form

y =

y =

y =

y =

2x - 3

- x + 4

x

4

1

3 2

2 - x + y = 0

2x + 3y = 12

-2x + y = -3

y = 4

N/A x = -4

1 2

Write Equations of Lines 2 different ways:

Slope-Intercept form Standard Form

y =

y =

y =

y =

2x - 3

- x + 4

x

4

1

3 2

2 - x + y = 0

2x + 3y = 12

-2x + y = -3

y = 4

x = -4

1 2

Slope-Intercept form Standard Form

y =

y =

y =

y =

2x - 3

- x + 4

x

4

1

3 2

2 - x + y = 0

2x + 3y = 12

-2x + y = -3

y = 4

x = -4

Ax + By = C y = mx + b

1 2

Slope-Intercept form Standard Form

y =

y =

y =

y =

2x - 3

- x + 4

x

4

1

3 2

2 - x + y = 0

2x + 3y = 12

2x - y = 3

y = 4

x = -4

Ax + By = C y = mx + b

1 2

Slope-Intercept form Standard Form

y =

y =

y =

y =

2x - 3

- x + 4

x

4

1

3 2

2 x - 2y = 0

2x + 3y = 12

2x - y = 3

y = 4

x = -4

Ax + By = C y = mx + b

Slope-Intercept form Standard Form

Ax + By = C y = mx + b

Slope

Gradient

how “steep” the line is

Slope-Intercept form Standard Form

Ax + By = C y = mx + b

Slope

Gradient

how “steep” the line is

m -A B

Slope of this

line is 1 = 1

1

The slope is becoming

less steep - slope is

getting smaller than 1.

The slope is becoming

less steep - slope is

getting smaller than 1.

The slope is becoming

less steep - slope is

getting smaller than 1.

The slope is becoming

less steep - slope is

getting smaller than 1.

The slope is becoming

so small - what is the

slope of this line?

The slope is becoming

so small - what is the

slope of this line?

Slope = 0

Slope of this

line is 1 = 1

1

The slope is becoming

more steep - slope is

getting bigger than 1.

The slope is becoming

more steep - slope is

getting bigger than 1.

The slope is becoming

more steep - slope is

getting bigger than 1.

The slope is becoming

more steep - slope is

getting bigger than 1.

The slope is becoming

more steep - slope is

getting bigger than 1.

The slope is

becoming so big-

what is the slope

of this line?

The slope is

becoming so big-

what is the slope

of this line?

Slope = doesn’t exist

The slope is

becoming so big-

what is the slope

of this line?

Slope = undefined

Slope-Intercept form Standard Form

Ax + By = C y = mx + b

Slope

Gradient

how “steep” the line is

m -A B

Slope-Intercept form Standard Form

y = 2x - 3

y = mx + b Ax + By = C

slope = m slope = - A B

2x - y = 3

y =

y =

2x - 3

- x + 4 3 2 2x + 3y = 12

Slope-Intercept form Standard Form

slope = m slope = -

Ax + By = C y = mx + b

A B

2x - y = 3

y =

y =

y =

2x - 3

- x + 4

x 1

3 2

2

2x + 3y = 12

Slope-Intercept form Standard Form

slope = m slope = -

y = mx + b Ax + By = C

A B

2x - y = 3

x - 2y = 0

y =

y =

y =

y =

2x - 3

- x + 4

x

4

1

3 2

2

2x + 3y = 12

Slope-Intercept form Standard Form

slope = m slope = -

y = mx + b Ax + By = C

A B

2x - y = 3

x - 2y = 0

y = 4

y =

y =

y =

y =

2x - 3

- x + 4

x

4

1

3 2

2

2x + 3y = 12

x = -4

Slope-Intercept form Standard Form

slope = m slope = - A B

Ax + By = C y = mx + b

Undefined slope y = mx + b

2x - y = 3

x - 2y = 0

y = 4

(x,y)

point A (3,6) point B (1,2)

A

B

slope = change in y-values change in x-values

slope = difference in y-values difference in x-values

slope = x y

slope = = = 2 3-1 2 6-2 4

A

B

slope = change in y-values change in x-values

slope = difference in y-values difference in x-values

slope = x y

slope = = = 2 3-1 2

(x,y)

point A (3,6) point B (1,2)

6-2 4

A

B

slope = change in y-values change in x-values

slope = difference in y-values difference in x-values

slope = x y

slope = = = 2 3-1 2

(x,y)

point A (3,6) point B (1,2)

6-2 4

A

B

slope = change in y-values change in x-values

slope = difference in y-values difference in x-values

slope = x y

slope = = = 2 3-1 2

(x,y)

point A (3,6) point B (1,2)

6-2 4

A

B

slope = change in y-values change in x-values

slope = difference in y-values difference in x-values

slope = x y

slope = = = 2 3-1 2

(x,y)

point A (3,6) point B (1,2)

6-2 4

A

B

slope = change in y-values change in x-values

slope = difference in y-values difference in x-values

slope = x y

slope = = = 2 3-1 2

(x,y)

point A (3,6) point B (1,2)

6-2 4

A

B

slope = change in y-values change in x-values

slope = difference in y-values difference in x-values

slope = x y

slope = = = 2 3-1 2 6-2 4

(x,y)

point A (3,6) point B (1,2)

A

B

slope = change in y-values change in x-values

slope = difference in y-values difference in x-values

slope = x y

slope = = = 2 3-1 2

(x,y)

point A (3,6) point B (1,2)

6-2 4

A

B

slope = change in y-values change in x-values

slope = difference in y-values difference in x-values

slope = x y

slope = = = 2 3-1 2

(x,y)

point A (3,6) point B (1,2)

6-2 4

slope = = = 2 2-6 -4 1-3 -2

A

B

slope = change in y-values change in x-values

slope = difference in y-values difference in x-values

slope = x y

slope = = = - -4-4 -8 2 1--3 4 1

(x,y)

point A (-4,1) point B (4,-3)

A

B

slope = change in y-values change in x-values

slope = difference in y-values difference in x-values

slope = x y

slope = = = - -4-4 -8 2

(x,y)

point A (-4,1) point B (4,-3)

1--3 4 1

A

B

slope = change in y-values change in x-values

slope = difference in y-values difference in x-values

slope = x y

slope = = = - -4-4 -8 2

(x,y)

point A (-4,1) point B (4,-3)

1--3 4 1

A

B

slope = change in y-values change in x-values

slope = difference in y-values difference in x-values

slope = x y

slope = = = - -4-4 -8 2

(x,y)

point A (-4,1) point B (4,-3)

1--3 4 1

A

B

slope = change in y-values change in x-values

slope = difference in y-values difference in x-values

slope = x y

slope = = = - -4-4 -8 2

(x,y)

point A (-4,1) point B (4,-3)

1--3 4 1

A

B

slope = change in y-values change in x-values

slope = difference in y-values difference in x-values

slope = x y

slope = = = - -4-4 -8 2

(x,y)

point A (-4,1) point B (4,-3)

1--3 4 1

A

B

slope = change in y-values change in x-values

slope = difference in y-values difference in x-values

slope = x y

slope = = = - -4-4 -8 2

(x,y)

point A (-4,1) point B (4,-3)

1--3 4 1

-3-1 -4 1 slope = = = - 4--4 8 2

A

B

slope = = = = x -6-4 -10 5 y 0-4 -4 2 .

(x,y)

point A (-6,0) point B (4,4)

A

B

slope = = = = x -6-4 -10 5 y 0-4 -4 2 .

(x,y)

point A (-6,0) point B (4,4)

A

B

slope = = = = x -6-4 -10 5 y 0-4 -4 2 .

(x,y)

point A (-6,0) point B (4,4)

A

B

slope = = = = x -6-4 -10 5 y 0-4 -4 2 .

(x,y)

point A (-6,0) point B (4,4)

A

B

slope = = = = x -6-4 -10 5 y 0-4 -4 2 .

(x,y)

point A (-6,0) point B (4,4)

A

B

slope = = = = x -6-4 -10 5 y 0-4 -4 2 .

(x,y)

point A (-6,0) point B (4,4)

slope = = = = y 4-0 4 2 . x 4--6 10 5

A

B

slope = = = = 7 x -3--1 -2 y -9-5 -14

(x,y)

point A (-3,-9) point B (-1,5)

A

B

slope = = = = 7 x -3--1 -2 y -9-5 -14

(x,y)

point A (-3,-9) point B (-1,5)

A

B

slope = = = = 7 x -3--1 -2 y -9-5 -14

(x,y)

point A (-3,-9) point B (-1,5)

A

B

slope = = = = 7 x -3--1 -2 y -9-5 -14

(x,y)

point A (-3,-9) point B (-1,5)

A

B

slope = = = = 7 x -3--1 -2 y -9-5 -14

(x,y)

point A (-3,-9) point B (-1,5)

slope = = = = 7 x -1--3 2 y 5--9 14

B A

slope = = = = 0 x 3--3 6 y 3-3 0

(x,y)

point A (3,3) point B (-5,3)

B A

slope = = = = 0 x 3--5 6 y 3-3 0

(x,y)

point A (3,3) point B (-5,3)

B A

slope = = = = 0 x 3--5 8 y 3-3 0

(x,y)

point A (3,3) point B (-5,3)

B A

slope = = = = 0 x 3--5 8 y 3-3 0

(x,y)

slope = = = = 0 y 3-3 0 x -5-3 -8

point A (3,3) point B (-5,3)

B

A

slope = = = = undefined x 3-3 0 y 3--3 6

(x,y)

point A (3,3) point B (3,-3)

Can’t divide by 0

Zero cannot be on bottom of fraction

B

A

slope = = = = undefined x 3-3 0 y 3--3 6

(x,y)

point A (3,3) point B (3,-3)

Can’t divide by 0

Zero cannot be on bottom of fraction

B

A

slope = = = = undefined x 3-3 0 y 3--3 6

(x,y)

point A (3,3) point B (3,-3)

Can’t divide by 0

Zero cannot be on bottom of fraction

B

A

x 3-3 0 y 3--3 6

(x,y)

point A (3,3) point B (3,-3)

Can’t divide by 0

Zero cannot be on bottom of fraction

slope = = = = undefined

B

A

slope = = = = undefined x 3-3 0 y 3--3 6

(x,y)

point A (3,3) point B (3,-3)

Can’t divide by 0

Zero cannot be on bottom of fraction

B

A

slope = = = = undefined x 3-3 0 y 3--3 6

(x,y)

point A (3,3) point B (3,-3)

Can’t divide by 0

Zero cannot be on bottom of fraction

B A

slope = = = = 0 x 3--5 8 y 3-3 0

(x,y)

point A (3,3) point B (-5,3)

Yes,

Zero can be on top of fraction

B

A

slope = = = = undefined x 3-3 0 y 3--3 6

(x,y)

point A (3,3) point B (3,-3)

NO

Zero cannot be on bottom of fraction

Write Equations of Lines 2 different ways:

Slope-Intercept form Standard Form

y =

y =

y =

y =

2x - 3

- x + 4

x

4

1

3 2

2

2x + 3y = 12

x = -4

2x - y = 3

x - 2y = 0

y = 4

y = 2x + 2

y = 2x + 2 y = 2x - 3

y = 2x + 2 y = 2x - 3

y = 2x - 6

y = 2x + 2 y = 2x - 3

y = 2x - 6

y = 2x + 2 y = 2x - 3

y = 2x - 6

y = 2x + 7

y = 2x + 2 y = 2x - 3

y = 2x - 6

y = 2x + 7

y = 2x + 2 y = 2x - 3

y = 2x - 6

y = 2x + 7

y = 2x - 9

y = 2x + 2 y = 2x - 3

y = 2x - 6

y = 2x + 7

y = 2x - 9

y = 2x + 2 y = 2x - 3

y = 2x - 6

y = 2x + 7

y = 2x - 9

y = 2x + 0

y = 2x + 2 y = 2x - 3

y = 2x - 6

y = 2x + 7

y = 2x - 9

y = 2x + 0

y = 2x + 2 y = 2x - 3

y = 2x - 6

y = 2x + 7

y = 2x - 9

y = 2x + 0

Parallel lines have the same slope.

y = 2x + 2

y = 2x + 2

y = 2x + 2

y =- x + 4 1 2

y = 2x + 2

y =- x + 4 1

Slopes of perpendicular lines are

1) Reciprocals 2) Opposite signs

2

y = x + 4 3 2

Slopes of perpendicular lines are

1) Reciprocals 2) Opposite signs

y = x + 4 3 2

Slopes of perpendicular lines are

1) Reciprocals 2) Opposite signs

y = x + 4

y =- x + 5 2 3

3 2

Slopes of perpendicular lines are

1) Reciprocals 2) Opposite signs

y = x + 4

y =- x + 5

y = x - 2

2 3

5 2

3 2

Slopes of perpendicular lines are

1) Reciprocals 2) Opposite signs

y = x + 4

y =- x + 5

y = x - 2

2 3

5 2

3 2

Slopes of perpendicular lines are

1) Reciprocals 2) Opposite signs

y = x + 4

y =- x + 5

y =- x - 4

y = x - 2

2 3

5 2

3 2

2 5

Slopes of perpendicular lines are

1) Reciprocals 2) Opposite signs