Integrated Math 2 Section 6-2

Section 6-2
Slope of a Line

Thursday, November 12, 2009

Essential Questions
How do you ﬁnd the slope of a line?
How do you identify horizontal and
vertical lines?

Where you’ll see it:
Business, science, transportation


Vocabulary
1. Slope:


Vocabulary
1. Slope: The ratio of vertical distance change to
horizontal distance change


Vocabulary

Let’s try again.


Vocabulary

Let’s try again.

1. Slope:


Vocabulary

Let’s try again.

1. Slope: How steep a line is, measured in “rise over run”


Vocabulary

Let’s try again.


Formula:


Vocabulary

Let’s try again.


Formula:
y 2 − y1
m= , for points ( x 1 , y 1 ) and ( x 2 , y 2 )
x 2 − x1

MATH CALISTHENICS!


Example 1
Graph the line the goes through the given points,
then ﬁnd the slope of the line.

C = (−4,0)
D = (4, 4)


Example 1

C = (−4,0) C
D = (4, 4)


Example 1

D

C = (−4,0) C
D = (4, 4)


Example 1

y 2 − y1
D m=
x 2 − x1
C = (−4,0) C
D = (4, 4)


Example 1

y 2 − y1
D m=
x 2 − x1
C = (−4,0) C 4−0
=
D = (4, 4) 4 − (−4)


Example 1

y 2 − y1
D m=
x 2 − x1
C = (−4,0) C 4−0
=
D = (4, 4) 4 − (−4)
4
=
8


Example 1

y 2 − y1
D m=
x 2 − x1
C = (−4,0) C 4−0
=
D = (4, 4) 4 − (−4)
4 1
= =
8 2


Example 1

y 2 − y1
D m=
x 2 − x1
C = (−4,0) C 4−0
=
D = (4, 4) 4 − (−4)
4 1
= =
8 2

Here, the slope tells us “Up 1, Right 2”

Example 2
Find the slope for the line containing the following:

a. (9, -2), (3, -2) b. (3, 12), (3, -4)


Example 2

a. (9, -2), (3, -2) b. (3, 12), (3, -4)
y 2 − y1
m=
x 2 − x1


Example 2

a. (9, -2), (3, -2) b. (3, 12), (3, -4)
y 2 − y1
m=
x 2 − x1
−2 − (−2)
=
3−9


Example 2

a. (9, -2), (3, -2) b. (3, 12), (3, -4)
y 2 − y1
m=
x 2 − x1
−2 − (−2)
=
3−9
0
=
−6


Example 2

a. (9, -2), (3, -2) b. (3, 12), (3, -4)
y 2 − y1
m=
x 2 − x1
−2 − (−2)
=
3−9
0
= =0
−6


Example 2

a. (9, -2), (3, -2) b. (3, 12), (3, -4)
y 2 − y1
m=
x 2 − x1
−2 − (−2)
=
3−9
0
= =0
−6
Horizontal

Example 2

a. (9, -2), (3, -2) b. (3, 12), (3, -4)
y 2 − y1 y 2 − y1
m= m=
x 2 − x1 x 2 − x1
−2 − (−2)
=
3−9
0
= =0
−6
Horizontal

Example 2

a. (9, -2), (3, -2) b. (3, 12), (3, -4)
y 2 − y1 y 2 − y1
m= m=
x 2 − x1 x 2 − x1
−2 − (−2) −4 − 12
= =
3−9 3−3
0
= =0
−6
Horizontal

Example 2

a. (9, -2), (3, -2) b. (3, 12), (3, -4)
y 2 − y1 y 2 − y1
m= m=
x 2 − x1 x 2 − x1
−2 − (−2) −4 − 12
= =
3−9 3−3
0 −16
= =0 =
−6 0
Horizontal

Example 2

a. (9, -2), (3, -2) b. (3, 12), (3, -4)
y 2 − y1 y 2 − y1
m= m=
x 2 − x1 x 2 − x1
−2 − (−2) −4 − 12
= =
3−9 3−3
0 −16
= =0 = Undeﬁned
−6 0
Horizontal

Example 2

a. (9, -2), (3, -2) b. (3, 12), (3, -4)
y 2 − y1 y 2 − y1
m= m=
x 2 − x1 x 2 − x1
−2 − (−2) −4 − 12
= =
3−9 3−3
0 −16
= =0 = Undeﬁned
−6 0
Horizontal Vertical

Horizontal vs. Vertical


Horizontal lines have slopes of


Horizontal lines have slopes of zero


(Think “horizon”)



Vertical lines have a slope that is



Vertical lines have a slope that is undeﬁned



Vertical lines have a slope that is undeﬁned
(It’s neither uphill, downhill, or level)


Example 3
Graph the line that passes through P = (-1, 1) and
has a slope of -2.


Example 3
has a slope of -2.

−2
−2 =
1


Example 3
has a slope of -2.

−2
−2 =
1

Down 2, right 1


Example 3
has a slope of -2.

−2
−2 =
1
P
Down 2, right 1


Example 4
a. Find the slope of AB and CD for the given points.
A = (0, −1), B = (2, 2), C = (−3,1), D = (−1, 4)


Example 4
A = (0, −1), B = (2, 2), C = (−3,1), D = (−1, 4)

y 2 − y1
m (AB ) =
x 2 − x1


Example 4
A = (0, −1), B = (2, 2), C = (−3,1), D = (−1, 4)

y 2 − y 1 2 − (−1)
m (AB ) = =
x 2 − x1 2−0


Example 4
A = (0, −1), B = (2, 2), C = (−3,1), D = (−1, 4)

y 2 − y 1 2 − (−1) 3
m (AB ) = = =
x 2 − x1 2−0 2


Example 4
A = (0, −1), B = (2, 2), C = (−3,1), D = (−1, 4)

y 2 − y 1 2 − (−1) 3
m (AB ) = = =
x 2 − x1 2−0 2

y 2 − y1
m (CD ) =
x 2 − x1


Example 4
A = (0, −1), B = (2, 2), C = (−3,1), D = (−1, 4)

y 2 − y 1 2 − (−1) 3
m (AB ) = = =
x 2 − x1 2−0 2

y 2 − y1 4 −1
m (CD ) = =
x 2 − x 1 −1− (−3)


Example 4
A = (0, −1), B = (2, 2), C = (−3,1), D = (−1, 4)

y 2 − y 1 2 − (−1) 3
m (AB ) = = =
x 2 − x1 2−0 2

y 2 − y1 4 −1 3
m (CD ) = = =
x 2 − x 1 −1− (−3) 2


Example 4
b. Graph the t wo lines. What do you notice?
A = (0, −1), B = (2, 2), C = (−3,1), D = (−1, 4)


Example 4
A = (0, −1), B = (2, 2), C = (−3,1), D = (−1, 4)

A


Example 4
A = (0, −1), B = (2, 2), C = (−3,1), D = (−1, 4)

B

A


Example 4
A = (0, −1), B = (2, 2), C = (−3,1), D = (−1, 4)

B
C
A


Example 4
A = (0, −1), B = (2, 2), C = (−3,1), D = (−1, 4)

D
B
C
A


Example 4
A = (0, −1), B = (2, 2), C = (−3,1), D = (−1, 4)

D
B The lines are parallel.
C
A


Example 4
A = (0, −1), B = (2, 2), C = (−3,1), D = (−1, 4)

D
B The lines are parallel.
C
A They have the same slope.


Homework


Homework

p. 250 #1-35 odd

“The power of imagination makes us inﬁnite.” - John Muir

Integrated Math 2 Section 6-2

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