The document discusses slope and rate of change, defining slope as rise over run and explaining how to calculate and classify slopes as positive, negative, zero, or undefined. It also covers how to determine if lines are parallel, perpendicular, or neither based on their slopes and provides examples of using slope to solve real-world problems involving rates of change like the grade of a road or average speed.

1. 2.2 Slope and Rate of Change

2. What is Slope?
• Slope is the slant of a line.
• Represented by the letter m
• The slope of a non-vertical line passing
through points (x1, y1) and (x2, y2) is:
= rise
run
• http://www.youtube.com/watch?v=IyfpM-ruafo

3. Example:
• Find the slope of the line passing through
the points (-1,-1) and (1,2).

4. Your Turn!
• Find the slope of the line passing through
the points (-3, 5) and (2, 1).

5. Classifying Lines by Slope
positive negative
m>0 m<0
zero undefined
m=0

6. Classifying Lines Without Graphing
• Calculate the slope, then decide if the line
rises, falls, is horizontal, or is vertical.
• Example: The line through ( 3, -4) and (1, -6).
• Example: The line through ( 2, -1) and (2, 5).

7. Comparing the Steepness of Lines
• For two lines with positive slopes, the
line with the larger slope is steeper.
• For two lines with negative slopes, the
line with the smaller slope (the more
negative) is steeper.
activity

8. Example:
• Tell which line is steeper.
Line 1: through (2,3) and (4,7)
Line 2: through (-1,2) and (4,5)

9. Your Turn!
• Tell which line is steeper.
Line 1: through (-1,-3) and (-3,-2)
Line 2: through (3,-4) and (0,-3)

10. What are Parallel and Perpendicular Lines?
• Parallel lines lie in the same plane and
never intersect.
• Perpendicular lines lie in the same plane
and intersect at a right angle.
• We can use slope to determine whether
two lines are parallel or perpendicular.

11. Parallel Lines
• Two non-vertical lines are parallel if and
only if they have the same slope.
m1 = m2

12. Perpendicular Lines
• Two non-vertical lines are perpendicular
if and only if their slopes are negative
reciprocals of each other.
m1 = - 1 or m1m2 = -1
m2

13. Example:
• Tell whether the lines are parallel,
perpendicular, or neither.
Line 1: through (-3,3) and (3,-1)
Line 2: through (-2,-3) and (2,3)

14. Your Turn!
• Tell whether the lines are parallel,
perpendicular, or neither.
Line 1: through (-2,-2) and (4,1)
Line 2: through (-3,-3) and (1,5)
stop

15. Using Slope In Real Life
• Example 1: For safety reasons, the distance
from the base of a ladder to the wall should
be at least one quarter of the height where
the ladder’s top hits the wall. For example, a
ladder that hits the wall at 12 feet should
have its base out 3 feet from the wall.
a. Find the maximum recommended slope for a
ladder.
b. Find the maximum distance a ladder’s base
should be from a wall if you need the ladder
to reach a height of 20 feet.

16. • Example 2:
• The slope, or “grade”, of a road is
expressed as a percent. For example, if a
road has a 3% grade, it rises 3 feet for
every 100 feet of horizontal distance.
a.Find the grade of a road that rises 75 feet
over a horizontal distance of 2000 feet.
b.Find the horizontal length x of a road with
a grade of 4% if the road rises 50 feet over
its length.

17. Your Turn!
• A water park slide drops 8 feet over a
horizontal distance of 24 feet. Find its
slope. Then find the drop over a 54 foot
section with the same slope.

18. Slope as a Rate of Change
• In real-life problems we often use slope
to describe and average rate of change.
• These rates use units like “miles per
hour” or “dollars per year”
• What is another unit used for a rate of
change?

19. Example:
• In the Mojave Desert in California,
temperatures can drop quickly from day to
night. Suppose the temperature drops
from 100 degrees at 2 pm to 68 degrees at
5 am. Find the average rate of change and
use it to determine the temperature at 10
pm.

20. Example 2:
• The number of U.S. cell phone
subscribers increased from 16 million in
1993 to 44 million in 1996. Find the
average rate of change and use it to
estimate the number of subscribers in
1997.

21. Your Turn!
• The average monthly cell phone bill
decreased from $61.48 in 1993 to $47.70
in 1996. Find the average rate of change
and use it to estimate the average
monthly bill in 1997.

22. Your Turn!
• You are driving on a road trip through
Europe. If at 9 am you are 420 km from
Rome and at 3 pm you are 108 km from
Rome, what is your average speed?