1. 5-1 Slope October 25, 2010
What do you think of when you hear the word slope?
Are there different types of slopes?

2. FINDING THE SLOPE OF A L INE
Math Definition
The slope m of a nonvertical line is the number of units
the line rises or falls for each unit of horizontal change
from left to right.
Slope is the ratio of change in the y-coordinates (rise) to
the change in the x-coordinates (run) as you move from
one point to the other.
The number describes how steep the line is.

3. rise slope = = = run FINDING THE SLOPE OF A L INEx y -1
- 1 9 7 5 3 1
The slanted line at the right rises
3 units for each 2 units of horizontal
change from left to right. So, the
slope m of the line is . 3 2
(2, 5)
rise = = 3 units
rise
(0, 2)
run
run = = 2 units
Two points on a line are all that is needed to find its slope. =
slope = =

4. m = y2 - y1 = = x2 - x1 FINDING THE SLOPE OF A L INE and (x2, y2) is
The slope m of the nonvertical line
passing through the points (x1, y1)
and (x2, y2) is x y
(x2, y2)
(x1, y1)
(x1, y1)
(x2, y2)
rise change in y =
y2 - y1
(y2 - y1 )
Read y1 as “y sub one”
Read x1 as “x sub one”
run change in x
x2 - x1
(x2 - x1 )
m = =

5. m = m = = When you use the formula for the slope,
FINDING THE SLOPE OF A L INE
When you use the formula for the slope,
y2 - y1
m =
x2 - x1
m =
run change in x
rise change in y =
y2 - y1
x2 - x1
x1 - x2
y2 - y1
Subtraction order is different
INCORRECT
Subtraction order is the same
the numerator and denominator
must use the same subtraction order.
CORRECT
The order of subtraction is important. You can label either point as (x1, y1)
and the other point as (x2, y2). However, both the numerator and denominator
must use the same order.
numerator
y2 - y1
denominator
x2 - x1

6. Find the slope of the line that passes through (–3, –4) and (–2, –8).
Let and
Example 1-2a

7. Example 1-2b

8. Rise: difference of y-values Run: difference of x-values
A Line with a Zero Slope is Horizontal
Find the slope of a line passing through (-1, 2) and (3, 2).
line
(-1, 2)
(3, 2).
(-1, 2)
(3, 2).
(3, 2) x y -1 9 7 5 3 1
SOLUTION
(3, 2)
(x1, y1)
(-1, 2)
(x2, y2)
Let (x1, y1) = (-1, 2) and
(x2, y2) = (3, 2)
(-1, 2)
(x1, y1)
(3, 2)
(x2, y2)
Rise: difference of y-values
y2 - y1
m =
rise = = 0 units
2 - 2 =
x2 - x1
Run: difference of x-values
Substitute values.
run = 3 - ( -1) = 4 units
3 - (-1)
Simplify. = 4
Slope is zero. Line is horizontal. =

9. Find the slope of the line that passes through (–3, 4) and (4, 4).
Let and
Example 1-3a

10. Example 1-3b

11. Find the slope of the line that passes through (–2, –4) and (–2, 3).
Let and
Answer: Since division by zero is undefined, the slope is undefined.
These two points are on what kind of a line?
Example 1-4a

12. INTERPRETING SLOPE AS A RATE OF CHANGE
Slope as a Rate of Change – tells, on average, how a quantity is changing over time
You are parachuting. At time t = 0 seconds, you open your parachute at h = 2500 feet
above the ground. At t = 35 seconds, you are at h = 2115 feet.
t = 0 seconds,
h = 2500 feet
t = 35 seconds,
h = 2115 feet.
t = 0 seconds,
h = 2500 feet
t = 35 seconds,
h = 2115 feet. x y
2700
2500
2300
2100
1900
Height (feet)
Time (seconds)
a. What is your rate of change in height?
(0, 2500)
b. About when will you reach the ground?
(35, 2115)

13. Slope as a Rate of Change
SOLUTION
a. Use the formula for slope to find the rate of change. The change in time is
= 35 seconds. Subtract in the same order. The change in height is
= -385 feet.
Rate of
Change =
rate of change.
Change in Time
change in time
Change in Height
change in height
VERBAL
MODEL
LABELS
Rate of Change = m (ft/sec) m =
ALGEBRAIC
MODEL
Change in Height = (ft)
- 385
Change in Time = (sec) 35 =
-11
Your rate of change is -11 ft/sec. The negative value indicates you are falling.

14. Slope as a Rate of Change
b. Falling at a rate of -ll ft/sec, find the time it will take you to fall 2500 ft.
Distance
-2500 ft =
Distance
Time
Time =
Rate
-11 ft/sec
Rate
227 sec
You will reach the ground about 227 seconds after opening your parachute.

15. Concept Summary Slope: Rate of change SLOPE of a line
Four types of slopes:
Positive
Negative
Zero
Undefined (no slope)

16. Assignment
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