3.3 – Slope

Recall how to graph y = 2x – 3.

3.3 – Slope

Recall how to graph y = 2x – 3.

3.3 – Slope

Recall how to graph y = 2x – 3.

x

3.3 – Slope

Recall how to graph y = 2x – 3.

x

-1

0

1

3.3 – Slope

Recall how to graph y = 2x – 3.

x 2x – 3

-1

0

1

3.3 – Slope

Recall how to graph y = 2x – 3.

x 2x – 3 y

-1

0

1

3.3 – Slope

Recall how to graph y = 2x – 3.

x 2x – 3 y (x, y)

-1

0

1

3.3 – Slope

Recall how to graph y = 2x – 3.

x 2x – 3 y (x, y)

-1 2(-1) – 3 -5 (-1, -5)

0 2(0) – 3 -3 (0, -3)

1 2(1) – 3 -1 (1, -1)

3.3 – Slope

Recall how to graph y = 2x – 3.

x 2x – 3 y (x, y)

-1 2(-1) – 3 -5 (-1, -5)

0 2(0) – 3 -3 (0, -3)

1 2(1) – 3 -1 (1, -1)

3.3 – Slope

Recall how to graph y = 2x – 3.

x 2x – 3 y (x, y)

-1 2(-1) – 3 -5 (-1, -5)

0 2(0) – 3 -3 (0, -3)

1 2(1) – 3 -1 (1, -1)

3.3 – Slope

Recall how to graph y = 2x – 3.

x 2x – 3 y (x, y)

-1 2(-1) – 3 -5 (-1, -5)

0 2(0) – 3 -3 (0, -3)

1 2(1) – 3 -1 (1, -1)

3.3 – Slope

Slope

Slope = Change in y

Change in x

Slope = Change in y

Change in x

m = y2 – y1

x2 – x1

Slope = Change in y

Change in x

m = y2 – y1 ,

x2 – x1

for (x1,y1) and (x2,y2)

Slope = Change in y

Change in x

m = y2 – y1 ,

x2 – x1

for (x1,y1) and (x2,y2)

*See bottom p.256*

Slope = Change in y

Change in x

m = y2 – y1 , Slope =

x2 – x1

for (x1,y1) and (x2,y2)

*See bottom p.256*

Slope = Change in y

Change in x

m = y2 – y1 , Slope = rise

x2 – x1 run

for (x1,y1) and (x2,y2)

*See bottom p.256*

Slope = Change in y

Change in x

m = y2 – y1 , Slope = rise

x2 – x1 run

for (x1,y1) and (x2,y2)

*See bottom p.256*

Slope = Change in y

Change in x

m = y2 – y1 , Slope = rise

x2 – x1 run

for (x1,y1) and (x2,y2)

*See bottom p.256*

Slope = Change in y

Change in x

m = y2 – y1 , Slope = rise

x2 – x1 run

for (x1,y1) and (x2,y2)

*See bottom p.256*

Slope = Change in y

Change in x

m = y2 – y1 , Slope = rise

x2 – x1 run

for (x1,y1) and (x2,y2)

*See bottom p.256*

Slope = Change in y

Change in x

m = y2 – y1 , Slope = rise

x2 – x1 run

for (x1,y1) and (x2,y2)

*See bottom p.256*

Slope = Change in y

Change in x

m = y2 – y1 , Slope = rise

x2 – x1 run

for (x1,y1) and (x2,y2)

*See bottom p.256*

Slope = Change in y

Change in x

m = y2 – y1 , Slope = rise

x2 – x1 run

for (x1,y1) and (x2,y2)

*See bottom p.256*

Ex. 1 Find the slope of the lines through the following points:

a. (-1,2) and (3,4)

Slope = Change in y

Change in x

m = y2 – y1 , Slope = rise

x2 – x1 run

for (x1,y1) and (x2,y2)

*See bottom p.256*

Ex. 1 Find the slope of the lines through the following points:

a. (-1,2) and (3,4)

Slope = Change in y

Change in x

m = y2 – y1 , Slope = rise

x2 – x1 run

for (x1,y1) and (x2,y2)

*See bottom p.256*

Ex. 1 Find the slope of the lines through the following points:

a. (-1,2) and (3,4)

m = y2 – y1

x2 – x1

Slope = Change in y

Change in x

m = y2 – y1 , Slope = rise

x2 – x1 run

for (x1,y1) and (x2,y2)

*See bottom p.256*

Ex. 1 Find the slope of the lines through the following points:

a. (-1,2) and (3,4)

x1

m = y2 – y1

x2 – x1

Slope = Change in y

Change in x

m = y2 – y1 , Slope = rise

x2 – x1 run

for (x1,y1) and (x2,y2)

*See bottom p.256*

Ex. 1 Find the slope of the lines through the following points:

a. (-1,2) and (3,4)

x1 y1

m = y2 – y1

x2 – x1

Slope = Change in y

Change in x

m = y2 – y1 , Slope = rise

x2 – x1 run

for (x1,y1) and (x2,y2)

*See bottom p.256*

Ex. 1 Find the slope of the lines through the following points:

a. (-1,2) and (3,4)

x1 y1 x2

m = y2 – y1

x2 – x1

Slope = Change in y

Change in x

m = y2 – y1 , Slope = rise

x2 – x1 run

for (x1,y1) and (x2,y2)

*See bottom p.256*

Ex. 1 Find the slope of the lines through the following points:

a. (-1,2) and (3,4)

x1 y1 x2 y2

m = y2 – y1

x2 – x1

Slope = Change in y Change in x

m = y2 – y1 , Slope = rise

x2 – x1 run

for (x1,y1) and (x2,y2)*See bottom p.256*

Ex. 1 Find the slope of the lines through the following points:a. (-1,2) and (3,4)

x1 y1 x2 y2

m = y2 – y1

x2 – x1

= (4 – 2) (3 – (-1))

Slope = Change in y Change in x

m = y2 – y1 , Slope = rise

x2 – x1 run

for (x1,y1) and (x2,y2)*See bottom p.256*

Ex. 1 Find the slope of the lines through the following points:a. (-1,2) and (3,4)

x1 y1 x2 y2

m = y2 – y1

x2 – x1

= (4 – 2) = 1 (3 – (-1)) 2

Slope = Change in y

Change in x

m = y2 – y1 , Slope = rise

x2 – x1 run

for (x1,y1) and (x2,y2)

*See bottom p.256*

Ex. 1 Find the slope of the lines through the following points:

a. (-1,2) and (3,4)

x1 y1 x2 y2

m = y2 – y1

x2 – x1

= (4 – 2) = 1

(3 – (-1)) 2

Slope = Change in y

Change in x

m = y2 – y1 , Slope = rise

x2 – x1 run

for (x1,y1) and (x2,y2)

*See bottom p.256*

Ex. 1 Find the slope of the lines through the following points:

a. (-1,2) and (3,4)

x1 y1 x2 y2

m = y2 – y1

x2 – x1

= (4 – 2) = 1

(3 – (-1)) 2

Slope = Change in y

Change in x

m = y2 – y1 , Slope = rise

x2 – x1 run

for (x1,y1) and (x2,y2)

*See bottom p.256*

Ex. 1 Find the slope of the lines through the following points:

a. (-1,2) and (3,4)

x1 y1 x2 y2

m = y2 – y1

x2 – x1

= (4 – 2) = 1

(3 – (-1)) 2

Slope = Change in y

Change in x

m = y2 – y1 , Slope = rise

x2 – x1 run

for (x1,y1) and (x2,y2)

*See bottom p.256*

Ex. 1 Find the slope of the lines through the following points:

a. (-1,2) and (3,4)

x1 y1 x2 y2

m = y2 – y1

x2 – x1

= (4 – 2) = 1

(3 – (-1)) 2

Slope = Change in y

Change in x

m = y2 – y1 , Slope = rise

x2 – x1 run

for (x1,y1) and (x2,y2)

*See bottom p.256*

Ex. 1 Find the slope of the lines through the following points:

a. (-1,2) and (3,4)

x1 y1 x2 y2

m = y2 – y1

x2 – x1

= (4 – 2) = 1

(3 – (-1)) 2

1

Slope = Change in y Change in x

m = y2 – y1 , Slope = rise

x2 – x1 run

for (x1,y1) and (x2,y2)*See bottom p.256*Ex. 1 Find the slope of the lines through the following points:

a. (-1,2) and (3,4)

x1 y1 x2 y2

m = y2 – y1

x2 – x1

= (4 – 2) = 1 (3 – (-1)) 2

1

Slope = Change in y Change in x

m = y2 – y1 , Slope = rise

x2 – x1 run

for (x1,y1) and (x2,y2)*See bottom p.256*Ex. 1 Find the slope of the lines through the following points:

a. (-1,2) and (3,4)

x1 y1 x2 y2

m = y2 – y1

x2 – x1

= (4 – 2) = 1 (3 – (-1)) 2

2

1

Slope = Change in y Change in x

m = y2 – y1 , Slope = rise

x2 – x1 run

for (x1,y1) and (x2,y2)*See bottom p.256*Ex. 1 Find the slope of the lines through the following points:

a. (-1,2) and (3,4)

x1 y1 x2 y2

m = y2 – y1

x2 – x1

= (4 – 2) = 1 (3 – (-1)) 2

2

1

b.

b. Slope =

b. Slope =

b. Slope = -1

b. Slope = -1

b. Slope = -1

3

b. Slope = -1

3

c. (1,2) and (-1,2)

b. Slope = -1

3

c. (1,2) and (-1,2) x1 y1 x2 y2

b. Slope = -1

3

c. (1,2) and (-1,2) x1 y1 x2 y2

m = y2 – y1

x2 – x1

b. Slope = -1

3

c. (1,2) and (-1,2) x1 y1 x2 y2

m = y2 – y1 = (2 – 2)

x2 – x1 -1 – 1

b. Slope = -1

3

c. (1,2) and (-1,2) x1 y1 x2 y2

m = y2 – y1 = (2 – 2) = 0

x2 – x1 -1 – 1 -2

b. Slope = -1

3

c. (1,2) and (-1,2) x1 y1 x2 y2

m = y2 – y1 = (2 – 2) = 0 = 0

x2 – x1 -1 – 1 -2

Ex. 2 In 1990, the average annual household income was $55,280. It then rose to $62,424 in 1998. Based on this information, at what rate did the average annual household income change per year.

Ex. 2 In 1990, the average annual household income was $55,280. It then rose to $62,424 in 1998. Based on this information, at what rate did the average annual household income change per year.

Ex. 2 In 1990, the average annual household income was $55,280. It then rose to $62,424 in 1998. Based on this information, at what rate did the average annual household income change per year.

Ex. 2 In 1990, the average annual household income was $55,280. It then rose to $62,424 in 1998. Based on this information, at what rate did the average annual household income change per year.

***Time is always the independent (x) variable!

Ex. 2 In 1990, the average annual household income was $55,280. It then rose to $62,424 in 1998. Based on this information, at what rate did the average annual household income change per year.

***Time is always the independent (x) variable!

m = y2 – y1

x2 – x1

Ex. 2 In 1990, the average annual household income was $55,280. It then rose to $62,424 in 1998. Based on this information, at what rate did the average annual household income change per year.

***Time is always the independent (x) variable!

m = y2 – y1

x2 – x1

Ex. 2 In 1990, the average annual household income was $55,280. It then rose to $62,424 in 1998. Based on this information, at what rate did the average annual household income change per year.

***Time is always the independent (x) variable!

m = y2 – y1 = (62,424 – 55,280)

x2 – x1 (1998 – 1990)

Ex. 2 In 1990, the average annual household income was $55,280. It then rose to $62,424 in 1998. Based on this information, at what rate did the average annual household income change per year.

***Time is always the independent (x) variable!

m = y2 – y1 = (62,424 – 55,280)

x2 – x1 (1998 – 1990)

= 7144

8

Ex. 2 In 1990, the average annual household income was $55,280. It then rose to $62,424 in 1998. Based on this information, at what rate did the average annual household income change per year.

***Time is always the independent (x) variable!

m = y2 – y1 = (62,424 – 55,280)

x2 – x1 (1998 – 1990)

= 7144

8

= 893

Ex. 2 In 1990, the average annual household income was $55,280. It then rose to $62,424 in 1998. Based on this information, at what rate did the average annual household income change per year.

***Time is always the independent (x) variable!

m = y2 – y1 = (62,424 – 55,280)

x2 – x1 (1998 – 1990)

= 7144

8

= 893

= $893/yr.