3.3 – Slope
Dec 23, 2015
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.