Integrated Math 2 Section 6-2
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Transcript of Integrated Math 2 Section 6-2
![Page 1: Integrated Math 2 Section 6-2](https://reader030.fdocuments.in/reader030/viewer/2022013011/55895c11d8b42a543f8b4585/html5/thumbnails/1.jpg)
Section 6-2Slope of a Line
Thursday, November 12, 2009
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Essential QuestionsHow do you find the slope of a line?
How do you identify horizontal and vertical lines?
Where you’ll see it:
Business, science, transportation
Thursday, November 12, 2009
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Vocabulary1. Slope:
Thursday, November 12, 2009
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Vocabulary1. Slope: The ratio of vertical distance change to
horizontal distance change
Thursday, November 12, 2009
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Vocabulary1. Slope: The ratio of vertical distance change to
horizontal distance change
Let’s try again.
Thursday, November 12, 2009
![Page 6: Integrated Math 2 Section 6-2](https://reader030.fdocuments.in/reader030/viewer/2022013011/55895c11d8b42a543f8b4585/html5/thumbnails/6.jpg)
Vocabulary1. Slope: The ratio of vertical distance change to
horizontal distance change
Let’s try again.
1. Slope:
Thursday, November 12, 2009
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Vocabulary1. Slope: The ratio of vertical distance change to
horizontal distance change
Let’s try again.
1. Slope: How steep a line is, measured in “rise over run”
Thursday, November 12, 2009
![Page 8: Integrated Math 2 Section 6-2](https://reader030.fdocuments.in/reader030/viewer/2022013011/55895c11d8b42a543f8b4585/html5/thumbnails/8.jpg)
Vocabulary1. Slope: The ratio of vertical distance change to
horizontal distance change
Let’s try again.
1. Slope: How steep a line is, measured in “rise over run”
Formula:
Thursday, November 12, 2009
![Page 9: Integrated Math 2 Section 6-2](https://reader030.fdocuments.in/reader030/viewer/2022013011/55895c11d8b42a543f8b4585/html5/thumbnails/9.jpg)
Vocabulary1. Slope: The ratio of vertical distance change to
horizontal distance change
Let’s try again.
1. Slope: How steep a line is, measured in “rise over run”
Formula:
m =
y 2 −y1
x 2 − x1
, for points (x1 ,y1) and (x 2 ,y 2 )
Thursday, November 12, 2009
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MATH CALISTHENICS!
Thursday, November 12, 2009
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Example 1Graph the line the goes through the given points,
then find the slope of the line.
C = (−4,0)D = (4 ,4)
Thursday, November 12, 2009
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Example 1Graph the line the goes through the given points,
then find the slope of the line.
C = (−4,0)D = (4 ,4)
C
Thursday, November 12, 2009
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Example 1Graph the line the goes through the given points,
then find the slope of the line.
C = (−4,0)D = (4 ,4)
C
D
Thursday, November 12, 2009
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Example 1Graph the line the goes through the given points,
then find the slope of the line.
C = (−4,0)D = (4 ,4)
C
D
Thursday, November 12, 2009
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Example 1Graph the line the goes through the given points,
then find the slope of the line.
C = (−4,0)D = (4 ,4)
C
D
m =
y 2 −y1
x 2 − x1
Thursday, November 12, 2009
![Page 16: Integrated Math 2 Section 6-2](https://reader030.fdocuments.in/reader030/viewer/2022013011/55895c11d8b42a543f8b4585/html5/thumbnails/16.jpg)
Example 1Graph the line the goes through the given points,
then find the slope of the line.
C = (−4,0)D = (4 ,4)
C
D
m =
y 2 −y1
x 2 − x1
=
4 − 04 − (−4)
Thursday, November 12, 2009
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Example 1Graph the line the goes through the given points,
then find the slope of the line.
C = (−4,0)D = (4 ,4)
C
D
m =
y 2 −y1
x 2 − x1
=
4 − 04 − (−4)
=
48
Thursday, November 12, 2009
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Example 1Graph the line the goes through the given points,
then find the slope of the line.
C = (−4,0)D = (4 ,4)
C
D
m =
y 2 −y1
x 2 − x1
=
4 − 04 − (−4)
=
48 =
12
Thursday, November 12, 2009
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Example 1Graph the line the goes through the given points,
then find the slope of the line.
C = (−4,0)D = (4 ,4)
C
D
m =
y 2 −y1
x 2 − x1
=
4 − 04 − (−4)
=
48 =
12
Here, the slope tells us “Up 1, Right 2”Thursday, November 12, 2009
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Example 2Find the slope for the line containing the following:
a. (9, -2), (3, -2) b. (3, 12), (3, -4)
Thursday, November 12, 2009
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Example 2Find the slope for the line containing the following:
a. (9, -2), (3, -2) b. (3, 12), (3, -4)
m =
y 2 −y1
x 2 − x1
Thursday, November 12, 2009
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Example 2Find the slope for the line containing the following:
a. (9, -2), (3, -2) b. (3, 12), (3, -4)
m =
y 2 −y1
x 2 − x1
=−2 − (−2)
3 − 9
Thursday, November 12, 2009
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Example 2Find the slope for the line containing the following:
a. (9, -2), (3, -2) b. (3, 12), (3, -4)
m =
y 2 −y1
x 2 − x1
=−2 − (−2)
3 − 9
=
0−6
Thursday, November 12, 2009
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Example 2Find the slope for the line containing the following:
a. (9, -2), (3, -2) b. (3, 12), (3, -4)
m =
y 2 −y1
x 2 − x1
=−2 − (−2)
3 − 9
=
0−6 = 0
Thursday, November 12, 2009
![Page 25: Integrated Math 2 Section 6-2](https://reader030.fdocuments.in/reader030/viewer/2022013011/55895c11d8b42a543f8b4585/html5/thumbnails/25.jpg)
Example 2Find the slope for the line containing the following:
a. (9, -2), (3, -2) b. (3, 12), (3, -4)
m =
y 2 −y1
x 2 − x1
=−2 − (−2)
3 − 9
=
0−6 = 0
HorizontalThursday, November 12, 2009
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Example 2Find the slope for the line containing the following:
a. (9, -2), (3, -2) b. (3, 12), (3, -4)
m =
y 2 −y1
x 2 − x1
=−2 − (−2)
3 − 9
=
0−6 = 0
Horizontal
m =
y 2 −y1
x 2 − x1
Thursday, November 12, 2009
![Page 27: Integrated Math 2 Section 6-2](https://reader030.fdocuments.in/reader030/viewer/2022013011/55895c11d8b42a543f8b4585/html5/thumbnails/27.jpg)
Example 2Find the slope for the line containing the following:
a. (9, -2), (3, -2) b. (3, 12), (3, -4)
m =
y 2 −y1
x 2 − x1
=−2 − (−2)
3 − 9
=
0−6 = 0
Horizontal
m =
y 2 −y1
x 2 − x1
=−4 −12
3 − 3
Thursday, November 12, 2009
![Page 28: Integrated Math 2 Section 6-2](https://reader030.fdocuments.in/reader030/viewer/2022013011/55895c11d8b42a543f8b4585/html5/thumbnails/28.jpg)
Example 2Find the slope for the line containing the following:
a. (9, -2), (3, -2) b. (3, 12), (3, -4)
m =
y 2 −y1
x 2 − x1
=−2 − (−2)
3 − 9
=
0−6 = 0
Horizontal
m =
y 2 −y1
x 2 − x1
=−4 −12
3 − 3
=−16
0
Thursday, November 12, 2009
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Example 2Find the slope for the line containing the following:
a. (9, -2), (3, -2) b. (3, 12), (3, -4)
m =
y 2 −y1
x 2 − x1
=−2 − (−2)
3 − 9
=
0−6 = 0
Horizontal
m =
y 2 −y1
x 2 − x1
=−4 −12
3 − 3
=−16
0Undefined
Thursday, November 12, 2009
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Example 2Find the slope for the line containing the following:
a. (9, -2), (3, -2) b. (3, 12), (3, -4)
m =
y 2 −y1
x 2 − x1
=−2 − (−2)
3 − 9
=
0−6 = 0
Horizontal
m =
y 2 −y1
x 2 − x1
=−4 −12
3 − 3
=−16
0
Vertical
Undefined
Thursday, November 12, 2009
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Horizontal vs. Vertical
Thursday, November 12, 2009
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Horizontal vs. VerticalHorizontal lines have slopes of
Thursday, November 12, 2009
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Horizontal vs. VerticalHorizontal lines have slopes of zero
Thursday, November 12, 2009
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Horizontal vs. VerticalHorizontal lines have slopes of zero
(Think “horizon”)
Thursday, November 12, 2009
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Horizontal vs. VerticalHorizontal lines have slopes of zero
(Think “horizon”)
Vertical lines have a slope that is
Thursday, November 12, 2009
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Horizontal vs. VerticalHorizontal lines have slopes of zero
(Think “horizon”)
Vertical lines have a slope that is undefined
Thursday, November 12, 2009
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Horizontal vs. VerticalHorizontal lines have slopes of zero
(Think “horizon”)
Vertical lines have a slope that is undefined(It’s neither uphill, downhill, or level)
Thursday, November 12, 2009
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Example 3Graph the line that passes through P = (-1, 1) and
has a slope of -2.
Thursday, November 12, 2009
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Example 3Graph the line that passes through P = (-1, 1) and
has a slope of -2.
−2 =
−21
Thursday, November 12, 2009
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Example 3Graph the line that passes through P = (-1, 1) and
has a slope of -2.
−2 =
−21
Down 2, right 1
Thursday, November 12, 2009
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Example 3Graph the line that passes through P = (-1, 1) and
has a slope of -2.
−2 =
−21
Down 2, right 1P
Thursday, November 12, 2009
![Page 42: Integrated Math 2 Section 6-2](https://reader030.fdocuments.in/reader030/viewer/2022013011/55895c11d8b42a543f8b4585/html5/thumbnails/42.jpg)
Example 3Graph the line that passes through P = (-1, 1) and
has a slope of -2.
−2 =
−21
Down 2, right 1P
Thursday, November 12, 2009
![Page 43: Integrated Math 2 Section 6-2](https://reader030.fdocuments.in/reader030/viewer/2022013011/55895c11d8b42a543f8b4585/html5/thumbnails/43.jpg)
Example 3Graph the line that passes through P = (-1, 1) and
has a slope of -2.
−2 =
−21
Down 2, right 1P
Thursday, November 12, 2009
![Page 44: Integrated Math 2 Section 6-2](https://reader030.fdocuments.in/reader030/viewer/2022013011/55895c11d8b42a543f8b4585/html5/thumbnails/44.jpg)
Example 3Graph the line that passes through P = (-1, 1) and
has a slope of -2.
−2 =
−21
Down 2, right 1P
Thursday, November 12, 2009
![Page 45: Integrated Math 2 Section 6-2](https://reader030.fdocuments.in/reader030/viewer/2022013011/55895c11d8b42a543f8b4585/html5/thumbnails/45.jpg)
Example 3Graph the line that passes through P = (-1, 1) and
has a slope of -2.
−2 =
−21
Down 2, right 1P
Thursday, November 12, 2009
![Page 46: Integrated Math 2 Section 6-2](https://reader030.fdocuments.in/reader030/viewer/2022013011/55895c11d8b42a543f8b4585/html5/thumbnails/46.jpg)
Example 3Graph the line that passes through P = (-1, 1) and
has a slope of -2.
−2 =
−21
Down 2, right 1P
Thursday, November 12, 2009
![Page 47: Integrated Math 2 Section 6-2](https://reader030.fdocuments.in/reader030/viewer/2022013011/55895c11d8b42a543f8b4585/html5/thumbnails/47.jpg)
Example 3Graph the line that passes through P = (-1, 1) and
has a slope of -2.
−2 =
−21
Down 2, right 1P
Thursday, November 12, 2009
![Page 48: Integrated Math 2 Section 6-2](https://reader030.fdocuments.in/reader030/viewer/2022013011/55895c11d8b42a543f8b4585/html5/thumbnails/48.jpg)
Example 3Graph the line that passes through P = (-1, 1) and
has a slope of -2.
−2 =
−21
Down 2, right 1P
Thursday, November 12, 2009
![Page 49: Integrated Math 2 Section 6-2](https://reader030.fdocuments.in/reader030/viewer/2022013011/55895c11d8b42a543f8b4585/html5/thumbnails/49.jpg)
Example 3Graph the line that passes through P = (-1, 1) and
has a slope of -2.
−2 =
−21
Down 2, right 1P
Thursday, November 12, 2009
![Page 50: Integrated Math 2 Section 6-2](https://reader030.fdocuments.in/reader030/viewer/2022013011/55895c11d8b42a543f8b4585/html5/thumbnails/50.jpg)
Example 3Graph the line that passes through P = (-1, 1) and
has a slope of -2.
−2 =
−21
Down 2, right 1P
Thursday, November 12, 2009
![Page 51: Integrated Math 2 Section 6-2](https://reader030.fdocuments.in/reader030/viewer/2022013011/55895c11d8b42a543f8b4585/html5/thumbnails/51.jpg)
Example 3Graph the line that passes through P = (-1, 1) and
has a slope of -2.
−2 =
−21
Down 2, right 1P
Thursday, November 12, 2009
![Page 52: Integrated Math 2 Section 6-2](https://reader030.fdocuments.in/reader030/viewer/2022013011/55895c11d8b42a543f8b4585/html5/thumbnails/52.jpg)
Example 3Graph the line that passes through P = (-1, 1) and
has a slope of -2.
−2 =
−21
Down 2, right 1P
Thursday, November 12, 2009
![Page 53: Integrated Math 2 Section 6-2](https://reader030.fdocuments.in/reader030/viewer/2022013011/55895c11d8b42a543f8b4585/html5/thumbnails/53.jpg)
Example 3Graph the line that passes through P = (-1, 1) and
has a slope of -2.
−2 =
−21
Down 2, right 1P
Thursday, November 12, 2009
![Page 54: Integrated Math 2 Section 6-2](https://reader030.fdocuments.in/reader030/viewer/2022013011/55895c11d8b42a543f8b4585/html5/thumbnails/54.jpg)
Example 3Graph the line that passes through P = (-1, 1) and
has a slope of -2.
−2 =
−21
Down 2, right 1P
Thursday, November 12, 2009
![Page 55: Integrated Math 2 Section 6-2](https://reader030.fdocuments.in/reader030/viewer/2022013011/55895c11d8b42a543f8b4585/html5/thumbnails/55.jpg)
Example 3Graph the line that passes through P = (-1, 1) and
has a slope of -2.
−2 =
−21
Down 2, right 1P
Thursday, November 12, 2009
![Page 56: Integrated Math 2 Section 6-2](https://reader030.fdocuments.in/reader030/viewer/2022013011/55895c11d8b42a543f8b4585/html5/thumbnails/56.jpg)
Example 3Graph the line that passes through P = (-1, 1) and
has a slope of -2.
−2 =
−21
Down 2, right 1P
Thursday, November 12, 2009
![Page 57: Integrated Math 2 Section 6-2](https://reader030.fdocuments.in/reader030/viewer/2022013011/55895c11d8b42a543f8b4585/html5/thumbnails/57.jpg)
Example 3Graph the line that passes through P = (-1, 1) and
has a slope of -2.
−2 =
−21
Down 2, right 1P
Thursday, November 12, 2009
![Page 58: Integrated Math 2 Section 6-2](https://reader030.fdocuments.in/reader030/viewer/2022013011/55895c11d8b42a543f8b4585/html5/thumbnails/58.jpg)
Example 4a. Find the slope of AB and CD for the given points.
A = (0,−1), B = (2,2), C = (−3,1), D = (−1,4)
Thursday, November 12, 2009
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Example 4a. Find the slope of AB and CD for the given points.
A = (0,−1), B = (2,2), C = (−3,1), D = (−1,4)
m(AB ) =
y 2 −y1
x 2 − x1
Thursday, November 12, 2009
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Example 4a. Find the slope of AB and CD for the given points.
A = (0,−1), B = (2,2), C = (−3,1), D = (−1,4)
m(AB ) =
y 2 −y1
x 2 − x1 =
2 − (−1)2 − 0
Thursday, November 12, 2009
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Example 4a. Find the slope of AB and CD for the given points.
A = (0,−1), B = (2,2), C = (−3,1), D = (−1,4)
m(AB ) =
y 2 −y1
x 2 − x1 =
32
=2 − (−1)
2 − 0
Thursday, November 12, 2009
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Example 4a. Find the slope of AB and CD for the given points.
A = (0,−1), B = (2,2), C = (−3,1), D = (−1,4)
m(AB ) =
y 2 −y1
x 2 − x1 =
32
m(CD ) =
y 2 −y1
x 2 − x1
=
2 − (−1)2 − 0
Thursday, November 12, 2009
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Example 4a. Find the slope of AB and CD for the given points.
A = (0,−1), B = (2,2), C = (−3,1), D = (−1,4)
m(AB ) =
y 2 −y1
x 2 − x1 =
32
m(CD ) =
y 2 −y1
x 2 − x1
=
2 − (−1)2 − 0
=
4 −1−1− (−3)
Thursday, November 12, 2009
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Example 4a. Find the slope of AB and CD for the given points.
A = (0,−1), B = (2,2), C = (−3,1), D = (−1,4)
m(AB ) =
y 2 −y1
x 2 − x1 =
32
m(CD ) =
y 2 −y1
x 2 − x1 =
32
=
2 − (−1)2 − 0
=
4 −1−1− (−3)
Thursday, November 12, 2009
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Example 4
A = (0,−1), B = (2,2), C = (−3,1), D = (−1,4)b. Graph the two lines. What do you notice?
Thursday, November 12, 2009
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Example 4
A = (0,−1), B = (2,2), C = (−3,1), D = (−1,4)b. Graph the two lines. What do you notice?
A
Thursday, November 12, 2009
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Example 4
A = (0,−1), B = (2,2), C = (−3,1), D = (−1,4)b. Graph the two lines. What do you notice?
A
B
Thursday, November 12, 2009
![Page 68: Integrated Math 2 Section 6-2](https://reader030.fdocuments.in/reader030/viewer/2022013011/55895c11d8b42a543f8b4585/html5/thumbnails/68.jpg)
Example 4
A = (0,−1), B = (2,2), C = (−3,1), D = (−1,4)b. Graph the two lines. What do you notice?
A
BC
Thursday, November 12, 2009
![Page 69: Integrated Math 2 Section 6-2](https://reader030.fdocuments.in/reader030/viewer/2022013011/55895c11d8b42a543f8b4585/html5/thumbnails/69.jpg)
Example 4
A = (0,−1), B = (2,2), C = (−3,1), D = (−1,4)b. Graph the two lines. What do you notice?
A
BC
D
Thursday, November 12, 2009
![Page 70: Integrated Math 2 Section 6-2](https://reader030.fdocuments.in/reader030/viewer/2022013011/55895c11d8b42a543f8b4585/html5/thumbnails/70.jpg)
Example 4
A = (0,−1), B = (2,2), C = (−3,1), D = (−1,4)b. Graph the two lines. What do you notice?
A
BC
D
Thursday, November 12, 2009
![Page 71: Integrated Math 2 Section 6-2](https://reader030.fdocuments.in/reader030/viewer/2022013011/55895c11d8b42a543f8b4585/html5/thumbnails/71.jpg)
Example 4
A = (0,−1), B = (2,2), C = (−3,1), D = (−1,4)b. Graph the two lines. What do you notice?
A
BC
D
Thursday, November 12, 2009
![Page 72: Integrated Math 2 Section 6-2](https://reader030.fdocuments.in/reader030/viewer/2022013011/55895c11d8b42a543f8b4585/html5/thumbnails/72.jpg)
Example 4
A = (0,−1), B = (2,2), C = (−3,1), D = (−1,4)b. Graph the two lines. What do you notice?
A
BC
D
The lines are parallel.
Thursday, November 12, 2009
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Example 4
A = (0,−1), B = (2,2), C = (−3,1), D = (−1,4)b. Graph the two lines. What do you notice?
A
BC
D
The lines are parallel.
They have the same slope.
Thursday, November 12, 2009
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Homework
Thursday, November 12, 2009
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Homework
“The power of imagination makes us infinite.” - John Muir
p. 250 #1-35 odd
Thursday, November 12, 2009