The document discusses increasing and decreasing functions and average rate of change, providing examples of finding the average rate of change between two points on a function by taking the slope of the secant line and sketches of graphs verifying solutions. It also provides homework problems asking to find the average rate of change for a given function between intervals.

1. 2.3 Increasing &
Decreasing Functions;
Average Rate of Change
Psalm 23:4 Even though I walk through the valley of
the shadow of death, I will fear no evil, for you are
with me; your rod and your staff, they comfort me.

2. f is increasing on an
interval, I, if f (x) < f (y)
whenever x < y in I

3. f is increasing on an
interval, I, if f (x) < f (y)
whenever x < y in I
With a partner, sketch the graph and write
the statement defining a decreasing function.

4. f is increasing on an
interval, I, if f (x) < f (y)
whenever x < y in I
With a partner, sketch the graph and write
the statement defining a decreasing function.
Can we get a volunteer to put their work on
the board?

5. Indicate the intervals
where f is increasing,
decreasing or constant.

6. Indicate the intervals
where f is increasing,
decreasing or constant.
Increasing: ( −5,−2 ), ( 4,5 )

7. Indicate the intervals
where f is increasing,
decreasing or constant.
Increasing: ( −5,−2 ), ( 4,5 )
Decreasing: ( −2,2 )

8. Indicate the intervals
where f is increasing,
decreasing or constant.
Increasing: ( −5,−2 ), ( 4,5 )
Decreasing: ( −2,2 )
Constant: ( 2, 4 )

9. Indicate the intervals
where f is increasing,
decreasing or constant.
Increasing: ( −5,−2 ), ( 4,5 )
Decreasing: ( −2,2 )
Constant: ( 2, 4 )
Why are these open intervals?

10. One of my all-time favorite places to eat
breakfast is at Perkins, just north of Madison, on
Hwy. 151, I believe. From Martin Luther, it takes
me 85 minutes to get there. What is my average
speed for the trip?

11. One of my all-time favorite places to eat
breakfast is at Perkins, just north of Madison, on
Hwy. 151, I believe. From Martin Luther, it takes
me 85 minutes to get there. What is my average
speed for the trip?
What information do we need?

12. One of my all-time favorite places to eat
breakfast is at Perkins, just north of Madison, on
Hwy. 151, I believe. From Martin Luther, it takes
me 85 minutes to get there. What is my average
speed for the trip?
What information do we need?
Go get it ...

14. I got 78.3 miles. Did you get that as well?
Could it be different ... and why ... and what must
we keep in mind?

15. I got 78.3 miles. Did you get that as well?
Could it be different ... and why ... and what must
we keep in mind?
d
Given: d = rt ∴ r =
t

16. I got 78.3 miles. Did you get that as well?
Could it be different ... and why ... and what must
we keep in mind?
d
Given: d = rt ∴ r =
t
d = 78.3 miles
t = 85 minutes = 1.417 hours

17. I got 78.3 miles. Did you get that as well?
Could it be different ... and why ... and what must
we keep in mind?
d
Given: d = rt ∴ r =
t
d = 78.3 miles
t = 85 minutes = 1.417 hours
78.3 miles
∴ r= ≈ 55.3
1.417 hour

18. I got 78.3 miles. Did you get that as well?
Could it be different ... and why ... and what must
we keep in mind?
d
Given: d = rt ∴ r =
t
d = 78.3 miles
t = 85 minutes = 1.417 hours
78.3 miles
∴ r= ≈ 55.3
1.417 hour

19. Let’s write the modeling equation:

20. Let’s write the modeling equation:
d = 55.3t

21. Let’s write the modeling equation:
d = 55.3t
Let’s sketch a graph ...

22. Let’s write the modeling equation:
d = 55.3t
Let’s sketch a graph ...
Notice the equation is in the form:
y = mx + b

23. Let’s write the modeling equation:
d = 55.3t
Let’s sketch a graph ...
Notice the equation is in the form:
y = mx + b
The Average Rate of Change is the slope!

24. Consider this graph:

25. Consider this graph:
The Average Rate of
Change of the
function y = f (x)
between xP and xQ
is equal to the slope
of the secant line.
Vy yQ − yP
mPQ = =
Vx xQ − xP

26. Given g(x) = x + 7 find the average rate of
2
change between
a) x=2 and x=7

27. Given g(x) = x + 7 find the average rate of
2
change between
a) x=2 and x=7
f (7) − f (2)
a) m=
7−2

28. Given g(x) = x + 7 find the average rate of
2
change between
a) x=2 and x=7
f (7) − f (2)
a) m=
7−2
2 2
(7 + 7) − (2 + 7)
m=
7−2
m=9

29. Given g(x) = x + 7 find the average rate of
2
change between
a) x=2 and x=7
f (7) − f (2)
a) m= graph and verify ...
7−2
2 2
(7 + 7) − (2 + 7)
m=
7−2
m=9

30. Given g(x) = x + 7 find the average rate of
2
change between
a) x=2 and x=7
f (7) − f (2)
a) m= graph and verify ...
7−2
secant line is:
2 2
(7 + 7) − (2 + 7)
m=
7−2
(y − 11) = 9(x − 2)
m=9
y = 9(x − 2) + 11

31. Given g(x) = x + 7 find the average rate of
2
change between
b) x=-3 and x=-1

32. Given g(x) = x + 7 find the average rate of
2
change between
b) x=-3 and x=-1
f (−3) − f (−1)
b) m=
(−3) − (−1)
((−3)2 + 7) − ((−1)2 + 7)
m=
−3 + 1
m = −4

33. Given g(x) = x + 7 find the average rate of
2
change between
b) x=-3 and x=-1
f (−3) − f (−1)
b) m= graph and verify ...
(−3) − (−1)
((−3)2 + 7) − ((−1)2 + 7)
m=
−3 + 1
m = −4

34. Given g(x) = x + 7 find the average rate of
2
change between
b) x=-3 and x=-1
f (−3) − f (−1)
b) m= graph and verify ...
(−3) − (−1)
((−3)2 + 7) − ((−1)2 + 7) secant line is:
m=
−3 + 1
y = −4(x + 1) + 6
m = −4

35. Be sure to carefully go through the
examples in your textbook. They are
really good!!
HW #4
“Great discoveries and achievements invariably
involves the cooperation of many minds.”
Alexander Graham Bell