1. Section 3.6: Critical Points and Extrema
Objectives:
I can find the extrema (maximums and minimums) of a function.

2. Definitions Definition – Critical Point:
Definition – Absolute Max/Min:
Definition – Relative Max/Min:
Where the function changes directions
(where a line tangent to the curve is either horizontal or vertical)
The largest/smallest value on the entire graph (over the entire domain)
The largest/smallest value on a given interval (not necessarily over the entire domain)

3. Example 1:
Locate the extrema for the graph for g(x). Name and classify the extrema.
Absolute Maximum:
Absolute Minimum:
Relative Maximum (maxima):
Relative Minimum (minima):
None (arrows!)
None (arrows!)
(-3, 13)
(2, -10)

4. Example 2 (You Try It!):
Locate the extrema for the graph for h(x). Name and classify the extrema.
Relative Maximum: (-8, 5)
Relative Minimum: (7.5, -2.3 ish)
Relative Maximum (maxima): (0, 3)
Relative Minimum (minima): (-2.5ish, 2ish)

5. Example 3( add inc/dec):
Abs Max: none
Abs Min: none
Rel Max: (-2/3, 14.17)
Rel Min: (2, -33)
Inc: {x < -2/3}
Dec: {-2/3 < x < 2}
Inc: {x > 2}
Use a calculator to graph to determine and classify its extrema. Sketch a graph of the situation.

6. Example 4:
The function has critical points at x = 0 and x = 1. Classify each critical point and determine on which intervals it is increasing and decreasing. Sketch a graph of the situation.

8. Warmup Grab a “Foldable” packet (4 pages) Warmup
Cut off bottom (shaded) portion from each
Staple together on top left and right corners
Start warmup below
Warmup
Locate and classify the extrema of f(x) = 3x4 – 6x + 7
and write the intervals in which the function is increasing/decreasing.

9. _Finding Maximums and Minimums
Finding a(n)…
It means…
Example…
Absolute Maximum
Absolute Minimum
Relative Maximum (Maxima)
Relative Minimum (Minima)
Highest point on
entire domain
Lowest point on
entire domain
Highest point in
Local area
Lowest point in
Local area
_Finding Maximums and Minimums

10. Section 3.5: Continuity and End Behavior
Objectives:
Determine whether a function is continuous or discontinuous.
Identify the end behavior of functions.
Determine whether a function is increasing or decreasing on an interval.

11. Example 1(skip for now):
Determine whether the function f(x) = 3x2 + 7 is continuous at x = 1.
Does the function exist at the point?
f(1) = 3(1)2 + 7 = 10
Does the function have any domain restrictions that might cause issues?
Does the function approach ‘10’ from both sides? Yuppers.
CONTINUOUS
yup
nope
yuppers

12. Example 2 and 3 (slip for now):
Determine whether the function is continuous at x = 1.
Determine whether the function is continuous at x = -2.
Nope, domain restriction
Darn it….this one too…..
(even though your calc might trick you)

13. Example 4:
Find the intervals for which f(x) = 4x2 + 9 is increasing and/or decreasing, also determine its end behavior. Sketch a graph to illustrate.
Dec: x < 0
Inc: x > 0
Chillin’ when x = 0
End behavior:

14. Example 5:
Find the intervals for which is increasing and/or decreasing, also determine its end behavior. Sketch a graph to illustrate.
Inc: x < -.46
Dec: < x < .24
Inc: x > .24
End behavior:

15. Example 6:
Find the intervals for which is increasing and/or decreasing, also determine its end behavior. Sketch a graph to illustrate.
Dec: on the entire graph
{x: all real numbers}
End behavior:

17. Section 3.7: Graphs of Rational Functions (Day 1)
Objectives:
Graph rational functions.
Determine vertical, horizontal, and oblique asymptotes.

18. CONTINUOUS yup nope yuppers Example 1(from 3.6):
Determine whether the function f(x) = 3x2 + 7 is continuous at x = 1.
Does the function exist at the point?
f(1) = 3(1)2 + 7 = 10
Does the function have any domain restrictions that might cause issues?
Does the function approach ‘10’ from both sides? Yuppers.
CONTINUOUS
yup
nope
yuppers

19. Example 2 and Example 3(from 3.6):
Determine whether the function is continuous at x = 1.
Determine whether the function is continuous at x = -2.
Nope, domain restriction
Darn it….this one too…..
(even though your calc might trick you)

20. Definition – Vertical Asymptote:
Essential (Infinite) Discontinuity
An asymptote in the vertical direction
A vertical asymptote ;)
- Found from the denominators
domain restrictions

21. Example 1: V.A. : x = 4 Using answer the following:
What is the vertical asymptote?
What is the limit of the function near the asymptote?
V.A. : x = 4
“from the left”
“from the right”

22. Example 2: V.A. : x = 0 Using answer the following:
What is the vertical asymptote?
What is the limit of the function near the asymptote?
V.A. : x = 0
“from the left”
“from the right”

23. Example 3:
Discuss the discontinuities and end behavior for the following graphs:
Hole (removable) at (4, 6)
Vertical asymptote x = 0
End behav: as x goes to –infinity? + infinity?
Horiz. Asymptote y = 0
End behav: as x goes to –infinity? + infinity?
Vertical asymptotes x = 2 and -2
No discontinuities for this one.
Horiz. Asymptote y = 0
End behav: as x goes to –infinity? + infinity?
End behav: as x goes to –infinity? + infinity?

24. Definitions Comes from the end behavior (Limit!!!!)
Definition – Horizontal Asymptote:
Definition – Removable Discontinuity:
Definition – Oblique Asymptot:
Comes from the end behavior (Limit!!!!)
Just a hole in the graph  (factor to find)
When the asymptote is a diagonal line…stay tuned for this…

25. Revisit Example 3(from 3.6):
Determine whether the function is continuous at x = -2.

26. Horizontal Asymptotes
Option 1: Same over Same Option 2: Bigger over Smaller Option 3: Smaller over Bigger

27. Warm-up: Match up the Function, its graph, and the type of discontinuity

28. Foldable

29. End Behavior (Horz Asym)
Exponents
How to find Limits…
Same Power on top and bottom
(Horizontal Asymptote)
Lower power on top
Higher power on top
(Oblique Asymptote)
End Behavior (Horz Asym)
Cut the colored area off each sheet and staple together at the top.

30. Types of Discontinuities
Equation
Graph
Removable (Hole/Point)
Essential/Infinite
(Vertical Asymptote)
Jump
(Piecewise!)
Add notes on angles supp/comp to congruent angles are congruent….Cut off gray area to create foldable.
Types of Discontinuities

31. Example 4: Determine the asymptotes and limits for
Vertical asymptote x = 2
Horiz. Asymptote y = 3
End behavior:

32. Example 5: Determine the asymptotes for
Vertical asymptotes x = -5 and x = -1
(x+5)(x+1)
Horiz. Asymptote y = 0
End behavior:

33. Example 6: Determine the asymptotes for Vertical asymptotes x = -2
Horiz. Asymptote none
End behavior:

34. Watch-me!!!!

35. What now…
1. FINISH QUIZ CORRECTIVES 2. PICK UP Horizontal Asym Worksheet. 3. Do 3.7 *Day 1 HW

36. Warm-Up Grab the matching sheet and fill out.

37. Horizontal asym = 0
x-int: plug in y = 0
You get an error…therefore…
Y-int: plug in x = 0
Factor
Domain Restr.
Asym? Hole?
Hor. Asym
Intercepts
Shifted/Graph
Limits 

38. Horizontal asym.
(there is none for this problem)
This is just a line ;)
x-int: plug in y = 0
Y-int: plug in x = 0

39. Horizontal asym.
x-int: plug in y = 0
Y-int: plug in x = 0

41. Warm Up
Compare the graphs below. Include discussions of Critical points, extrema, increasing and decreasing intervals, holes, asymptotes, etc.
Also, write ALL the limits of the functions! ALL.

42. Warm Up Compare the graphs below. Include discussions
of Critical points, extrema, increasing and
decreasing intervals, holes, asymptotes, etc.
Continuous
Removable (Hole)
2 Essentials (V.A.’s)
This equation must have a domain restriction that DOESN’T cancel out…
This equation must have a domain restriction that cancels out…

43. Warm Up
Compare the graphs below. Include discussions of Critical points, extrema, increasing and decreasing intervals, holes, asymptotes, etc.
Also, write ALL the limits of the functions! ALL.

44. Section 3.8: Direct, Inverse, and Joint Variation
Objectives:
Solve problems involving direct, inverse, and joint variation.

45. Definitions Definition – Direct Variation:
Definition – Constant of Variation:
When two variables are related to one another through the
Multiplication of a constant (a number).
The constant (number) from above.
(most of the time you will have to find it…)

46. Example 1: Suppose y varies directly as x and y = 45 when x = 2.5
Find the constant of variation and write an equation.
Use the equation to find the value of y when x = 4.

47. Example 2:
When an object such as a car is accelerating, twice the distance (d) it travels varies
directly with the square of the time (t). One car accelerating for 4 minutes travels 1440 feet.
Write an equation of direct variation relating travel distance to time elapsed. Then sketch a graph of the equation.
Use the equation to find the distance traveled by the car in 8 minutes.

48. Example 3:
If y varies directly as the square of x and y = 30, when x = 4, find x when y = 270.

49. Definitions Definition – Inverse Proportion:
When two variables are related to one another through division.
There is still a constant of variation
Notice: the x is on the bottom!

50. Example 4:
If y varies inversely as x and y = 14, when x = 3, find x when y = 30.

51. Definition – Joint Variation:
When more than two variables are related to one another through
Multiplication….There is still a constant of variation

52. Example 5:
In physics, the work (W) done in charging a capacitor varies jointly as the charge (q) and the voltage (V). Find the equation of joint variation if a capacitor with a charge of coulomb and a voltage of 100 volts performs 0.20 joule of work.