1. Introduction to Functions
Opening Routine
Which of these functions is a discrete function and which one is a continuous function?

2. Topic III: Introduction to Functions

3. Introduction to Functions
Increasing , Decreasing and Constant Functions.
Relative Maxima and Minima
Objective: For a function that models a relationship between two quantities, identify intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums
Essential Question: What would the graph of a function look like if the function is always increasing, or always decreasing?

4. Introduction to Functions
Increasing , Decreasing and Constant Functions.
Relative Maxima and Minima
Vocabulary
Function: A function is a set of ordered pairs in which each x-element has only ONE y-element associated with it.
Functional Notation: Functions are referred to by the notation f(x), which is read "f of x" or “f as a function of x” (The parentheses do not mean "multiplication“).
For example, since y = 3x + 7 is a function, it may also be written as f(x) = 3x + 7.

5. Introduction to Functions
Increasing , Decreasing and Constant Functions.
Relative Maxima and Minima
Vocabulary
Continuous graph: Graph of a function that has no
breaks of holes in given interval.
Discrete graph: Graph with distinct and separate values. This means that the values of the functions are not connected with each other. Also when the graph has breaks and holes in a given interval.

6. Introduction to Functions
Increasing , Decreasing and Constant Functions.
Relative Maxima and Minima
Vocabulary
Increasing functions: When the graph, in a given interval, has a positive slope, or the graph goes up.
Decreasing functions: When the graph, in a given interval, has a negative slope, or the graph goes down.
Constant function: Is a function whose (output) value is the same for every input value. The slope is 0 and the graph is an horizontal line.

7. Introduction to Functions
Increasing , Decreasing and Constant Functions.
Relative Maxima and Minima
Vocabulary
Relative maxima: Are points where the graph change from increasing to decreasing in a given interval, but they are not global maximum.
Relative minima: Are points where the graph change from decreasing to increasing in a given interval, but they are not global minimum.

8. Introduction to Functions
Increasing , Decreasing and Constant Functions.
Relative Maxima and Minima
A discrete function is a function
with distinct and separate values.
This means that the values of the
functions are not connected with
each other. Discrete functions are
used for things that can be counted.
For example, the number of
televisions or the number of
puppies born.

9. Introduction to Functions
Increasing , Decreasing and Constant Functions.
Relative Maxima and Minima
Continuous functions, on the
other hand, connect all the dots,
and the function can be any
value within a certain interval.
This continuous function gives
you values from 0 all the way
to positive infinity. It doesn't
have any breaks or holes within
this interval.

10. Introduction to Functions
Increasing , Decreasing and Constant Functions.
Relative Maxima and Minima
Increasing Function

11. Introduction to Functions
Increasing , Decreasing and Constant Functions.
Relative Maxima and Minima
Decreasing Functions

12. Introduction to Functions
Increasing , Decreasing and Constant Functions.
Relative Maxima and Minima
Constant Functions

13. Introduction to Functions
Increasing , Decreasing and Constant Functions.
Relative Maxima and Minima

14. Introduction to Functions
Increasing , Decreasing and Constant Functions.
Relative Maxima and Minima
Video

15. Introduction to Functions
Increasing , Decreasing and Constant Functions.
Relative Maxima and Minima
A function f has a relative
maximum at x0 if f(x0)  f(x)
for all x in some open
interval containing x0.
The function has a relative
minimum at x0 if f(x0)  f(x)

16. Introduction to Functions
Increasing , Decreasing and Constant Functions.
Relative Maxima and Minima
Guided Practice - WE DO
Consider the function
f (x) = 2x3 + 9x
a) Find all values of x for
which f is increasing.
b) Find all values of x for
which f is decreasing.
c) Find all values of x for which
f has a relative maximum.
d) Find all values of x for which
f has a relative minimum.
e) Sketch the graph of f.

17. Introduction to Functions
Increasing , Decreasing and Constant Functions.
Relative Maxima and Minima
Independent Practice - YOU DO
Worksheet “Increasing, Decreasing and Constant functions. Relative Maxima and Minima”
Exercises 1 - 6

18. Introduction to Functions
Increasing , Decreasing and Constant Functions.
Relative Maxima and Minima
Homework
Complete Worksheet “Increasing, Decreasing and Constant functions. Relative Maxima and Minima”
Exercises 1 - 6

19. Introduction to Functions
Increasing , Decreasing and Constant Functions.
Relative Maxima and Minima
Closure
Essential Question: What would the graph of a function look like if the function is always increasing, or always decreasing?