1. 2.2 Slope and Rate of Change

2. In this lesson you will:
Find slopes of lines and classify parallel and perpendicular lines.
Use slope to solve real-life problems.

3. The Slope of a Non-vertical line
Is the ratio of vertical change (the rise)
To the horizontal change (the run). x y
“Change in y”
“Change in x”

4. The Slope Formula: m= y x - rise run x y1 2 x -
rise
run 1 2 3 4 5 6 7 8 9 10 x y
Find the slope for the segments at the right:
Segment AB:
A(3,2), B(7,5) B x 2 y 2 x 1 y 1
m = - AB 2 5 = -3 -4 = 3 4 3 7 A C D
Segment CD:
C(8,2), D(10,1) x 2 y 2 x 1 y 1
m = - cd 2 1 = 1 -2 8 10

5. Using the slope formula below, find the slope for the lines in the coordinate plane:y 1 2 x -
Line a:
(1,5) (-1,1) x 2 1 y 2 5 x 1 -1 y 1
m = - a 4 = 4 2 4 2 6 -2 -4 -6 8 10 -8
-10 x y = =2 d
1+1 a
(-8, 7)
Line b:
(3,2) (10,2)
(1, 5) x 2 3 y 2 x 1 10 y 1 2
m = - b = -7 b
(-1, 1) =0
(3, 2)
(10, 2)
Line c:
(6,-8) (2,-5)
(2, -5)
(-4, -4) y 2 -8 x 2 6 y 1 -5 x 1 2 c
m = - c
-8+5 = -3 4
(6, -8)
= - 3 4 = 4
Line d:
(-8,7) (-4,-4) x 2 -8 y 2 7 y 1 -4 x 1 -4
m = - d
7+4 = 11 -4
= - 11 4 =
-8+4

6. Or another way to determine slope is using the “rise” over the “run” method. Find the slope for the lines in the coordinate plane: m= y 1 2 x - + -
rise =
run +
Line a: 4 2 6 -2 -4 -6 8 10 -8
-10 x y - -4 = 4 2
m = a =2 d -2 a 4 + 2 -
Line b: - 11 - 4
m = b =0 +7 b 7 +
Line c: +3
= - 3 4
m = c c -4 3 + 4 -
Line d:
-11
= - 11 4
m = d +4

7. + + + + FALLING TO THE RIGHT y - - - x - c Slope for c: + m = = - -
When we move to the right from one point to the other, we go down the right and the slope is negative!

8. RISING to the RIGHT x y a Slope for a: + m = =+ +
When we move to the left from one point to the other, we go up and the slope is always positive!

9. Compare the slopes of these lines:

10. Summary + - y y a b + x x Slope for b: Slope for a: + m = =0 m = =+ ++
m = b =0
m = a =+ + +
Slope of a horizontal line is always 0.
Slope of a line that rises to the right is always POSITIVE! x y x y d c + + -
Slope for c:
Slope for d: +
m = c
= - + -
m = d
= not defined!
The slope of a line that falls to the right is always NEGATIVE!
The slope of a vertical line IS NEVER DEFINED!

11. + + - + PARALLEL and PERPENDICULAR x y PARALLEL PERPENDICULAR x y a c4 2 6 -2 -4 -6 8 10 -8
-10 x y
PARALLEL
PERPENDICULAR 4 2 6 -2 -4 -6 8 10 -8
-10 x y a c b 2 + d 2 + 2 + 2 + 2 + 2 + 2 - 2 + +2 +2
Line a:
m = a =1
Line c:
m = c
=-1
m = d m c
(-1)(1)
=-1 +2 -2 +2 +2
Line b:
m = b =1
Line d:
m = d =1 +2 +2
The slope product of perpendicular lines is -1
Parallel Lines have the same slope!
m = a m b
m = d m c -1

12. Prove that segments AB and BC are perpendicular:
Using the slope formula: 1 2 3 4 5 6 7 8 9 10 x y m= y 1 2 x -
Segment AB:
A(2,1), B(6,4) B x 2 y 2 x 1 y 1
m = - AB 1 4 = -3 -4 = 3 4 2 6 A C
Segment BC:
B(6,4), C(9,0) x 2 y 2 x 1 y 1
m = - BC 4 = 4 -3 6 9
Is the product of the slopes -1?
The product of the slopes is -1. So, They are perpendicular. 3 4 = 4 -3 12
-12
= -1

13. Slope as a Rate of Change
In the Mojave Desert in California, temperatures can drop quickly from day to night. Suppose the temperature drops from 100 degree at 2 p.m. to 68 degrees at 5 a.m.
What is the average rate of change in temperature. Use this to estimate the temperature at 10 p.m.
Solution:
Change in temperature
Average rate of change n Temp. =
Change in time
Because 10 pm is 8 hours after 2pm., the temp. changed 8(-2)=-16 degrees. That means the temperature at 10 pm was about or 84 degrees.

14. Homework odd, 49, 52