1. Section 1.5 More on Slope

2. Intro
A best guess at the future of our nation indicates that the numbers of men and women living alone will increase each year. Figure 1.46 show that in 2005, 12.7 million men and women lived alone, an increase over the numbers displayed in the graph for 1990.
Can you tell that the graph for men has a greater slope than the graph for women?
What does this indicate for the years 1990 through 2005?

3. Slope and Parallel Lines
If two nonvertical lines are parallel, then they have the same slope.
If two distinct nonvertical lines have the same slope, then they are parallel.
Two distinct vertical lines, both undefined slopes, are parallel.
y=2x+7

4. Parallel Lines

5. Example 1
Write the equation in slope intercept form for a line that is parallel to 3x - 4y = 12 and passing through (5, 2).
Point-Slope Form
Slope-Intercept Form

6. Slope and Perpendicular Lines
If two nonvertical lines are perpendicular, then the product of their slopes is -1.
(i.e. The slopes are opposite reciprocals of each other.)
If the product of the two lines is -1, then the lines are perpendicular.
A horizontal line having slope of zero is perpendicular to a vertical line having undefined slope.

7. Perpendicular Lines
 -6

8. Example 2
Write the equation in slope intercept form for a line perpendicular to 3x - 4y = 12 and passing through (5, 2).
Point-Slope Form
Slope-Intercept Form
Page Problems 1-12

9. Definition- Slope is defined as the ratio of a change in y to a corresponding change in x.

10. Interpreting a real life situation
The line graphs the percent of US adults who smoke cigarettes x years after 1997.
a. Find the slope of the line segment from 1997 to 2007
Percent Adults
(0,24.7)
(10,19.5)
b. What does this slope represent?
The percent of US adult cigarette smokers is decreasing by .52 percent each year. The change is consistent each year.
X years after 1997
Page 212 problems 13-18

11. Basically, you are finding the slope of two points found on the curve
Basically, you are finding the slope of two points found on the curve. The linear line passing through the two points is called a Secant line.

12. The average rate of change of a function.
If the graph of a function is not a straight line, the average rate of change between any two points is the slope of the line containing the two points. This line is called a secant line.
Secant line

13. The slope of this line between the points (1,3.83) and (5,7.83) is

14. The slope of this line between the points (1,3.83) and (4,7.34) is
Continuation of same problem

15. The slope of this line between the points (1,3.83) and (3,6.5) is
Continuation of same problem

16. Let’s look at the different slopes from the point (1, 3.83).x y
Slope of the secant line 3
6.5
1.34 4
7.34
1.17 5
7.83 1
Notice how the slope changes depending upon the point that you choose because this function is a curve, not a line. So the average rate of change varies depending upon which points you may choose.
Continuation of same problem

17. Find the average rate of change of f(x)= x3

18. Find the average rate of change of f(x) = 3x - 1 from
Example 5
Find the average rate of change of f(x) = 3x - 1 from
x1 = 0 to x2 = 1 x1 =1 to x2 =2 x1 = 2 to x2 = 3
Page 212 Problems 19-20

20. Average rate of change and the difference quotient
Suppose x1= x and x2= x + h, then
Do you recognize the difference quotient that we studied in section 1.3?
You will study more about the difference quotient in future math classes.

21. Average Rate of Change Application
Example 6
When a person receives a drug injection, the concentration of the drug in the blood is a function of the hours elapsed after the injection. X represents the hours after the injection and f(x) represents the drug’s concentration in milligrams per 100 milliliters.
Find the average rate of change of the drug’s concentration between the 1st and 4th hours.
What does this value mean in terms of the drug’s concentration?
concentration
This means that from the 1st hour to the 4th hour the drug has decreased its concentration 1.08 milligrams per 100 milliliters.
(1, 3.96)
(4, .72)
hours

22. Average Rate of Change Application
Example 7
Sometimes it takes a while for a drug to diffuse sufficiently to affect the desired organ. The curve below is a close approximation of the concentration in that organ compared to the time after the drug was taken. (a.) What is the average rate of change from the time the drug was taken until the first hour? (b.) What was the average rate of change from the second hour until the fourth? (c.) What is the interpretation of each answer?
concentration
hours cncntr
This means that the medication starts to work very quickly, then very, very slowly starts to lose concentration.
Page Problems 29-32
hours

23. Review Time
(a)
(b)
(c)
(d)
(d)

24. (a)
(b)
(c)
(d)
(c)

25. (a)
(b)
(c)
(d)
(a)