1. Increasing vs. Decreasing, Extrema, Boundedness, and Continuity
Unit 2 – Day 2
Increasing vs. Decreasing, Extrema, Boundedness, and Continuity

2. Definitions
A function is said to be increasing if … an increase in x-values results in a POSITIVE change in y-values
A function is said to be decreasing if … an increase in x-values results in a NEGATIVE change in y-values
A function is said to be constant if … an increase in x-values results in ZERO change in y-values

3. Extrema Maximums and Minimums 2 Types: “local” or “absolute”
The # of possible extrema is determined by: degree – 1

4. LOCAL (Relative) Extrema
Highest and Lowest points on the graph compared to other points nearby
Points in which the curve changes direction
Peaks and Valleys
EX:

5. ABSOLUTE (Global) Extrema
THE highest point or THE lowest point
Sometimes, they do not exist (and that is OKAY!)
EX:

6. Steps for Finding Extrema in the Calculator
Type function in y=
GRAPH to determine the types of extrema
2nd CALC
3 … for minimum
4 … for maximum
LB ENTER RB ENTER ENTER
Repeat for all extrema (Remember: degree – 1)
Identify Local vs. Absolute

7. Boundedness
A function is bounded above if there exists an absolute maximum
A function is bounded below if there exists an absolute minimum
A function is not bounded if there are no absolute extrema
EXs:

8. Proving Continuity
In order for a function f(x) to be continuous at a point x=c the following three conditions must be met:
f (c ) must exist
exists, which means

9. 3 Types of Discontinuites
Removable – Continuous everywhere except at the HOLE
Jump – (NOT removable) Space is skipped vertically
Infinite – (NOT removable) Graph approaches a vertical asymptote
EXs:

10. Example
Is the function continuous at x=2 ?

11. MORE Examples
Without using your graphing calculator, state whether the function is continuous at x=2. If not, what type of discontinuity is at x=2?

12. Ticket Out The Door Given 𝒇 𝒙 = 𝒙 𝟐 +𝟒𝒙−𝟐𝟏 𝟐 𝒙 𝟑 + 𝒙 𝟐 −𝟏𝟔𝒙−𝟏𝟓
Is f(x) continuous? Yes or No
If discontinuous, state what type of discontinuity exists & where: ____________________________________________________
Are there any extrema? Yes or No
If so, list them: _______________________________________