3.3 – Properties of Functions
Precal
Review increasing and decreasing:
• Increasing function – up when going right
• Decreasing function – down when going right
• Constant – neither increasing nor decreasing
(horizontal)
Determine the parts of the graph where the function is
increasing, decreasing, and/or constant
• Increasing:
4 x  0
• Decreasing:
 6  x  4; 3  x  6
• Constant:
0 x3
Local Extrema
• Extrema is the plural of extreme
• This refers to where the graph reaches peaks
and valleys
• We call the “peaks” local maximums
• We call the “valleys” local minimums
What is the local maximum of this
function?
• Point A is a local
maximum because the
graph changes from
increasing to decreasing
at that point
• It is only a LOCAL
maximum instead of a
global maximum
because there are
points on the graph
higher (like point D)
What is the local minimum of this
function?
• Point C is a local
minimum because the
graph changes from
decreasing to increasing
at that point
• It is only a LOCAL
minimum instead of a
global minimum
because there are
points on the graph
lower (like point F)
Identify the local extrema of the graph
• Local Minimums:
• C, F, H
• Local Maximums:
• A, D, G
Using the calculator for max’s and
min’s
f ( x)  .2 x  1.8x  4.8x  3.2
3
2
f ( x)  0.25x  0.3x  0.9 x  3
4
3
2
Partner Activity
• In a little bit you will follow these instructions:
– Find a partner
– One partner come up and grab a marker
– Both partners find a spot at the board
– Be prepared to graph some functions
Partner Roles
• The partner who got the marker is the
“player”
• The partner without the marker is the “coach”
• When I give you the first problem, the coach is
going to tell the player how to graph it
• Players cannot draw anything unless the coach
tells them to do so
• Coaches cannot have the marker and draw
The “Big Ten”
• You are going to graph the ten most important
base graphs of functions to remember
• This is a part of section 3.4 (I have a handout
for you on these graphs that you can use as
notes)
Functions to graph (1)
• Graph f(x) = 1
• Is there any symmetry to this graph?
– Can you reflect it over anything?
Functions to graph (2)
• Graph f(x) = x
• Is there any symmetry to this graph?
– Can you reflect it over anything?
Switch roles
• Give the marker to the other partner
• The original “player” is now the “coach and
vice versa
Functions to graph (3)
• Graph f(x) = x2
• Is there any symmetry to this graph?
– Can you reflect it over anything?
Functions to graph (4)
• Graph f(x) = x3
• If the coach needs the help of a calculator,
that is okay
• Is there any symmetry to this graph?
– Can you reflect it over anything?
Do you notice the pattern of
symmetry?
• A function with an odd power reflects over
the origin
• A function with an even power reflects over
the y-axis
• Go write the red part of this slide in your
notes for 3.3, then go back to the board
• Switch player-coach roles again
Functions to graph (5)
• Graph f ( x)  x
• If the coach needs the help of a calculator,
that is okay
Functions to graph (6)
3
f
(
x
)

x
• Graph
• If the coach needs the help of a calculator,
that is okay
Switch roles
Functions to graph (7)
1
• Graph f ( x ) 
x
• If the coach needs the help of a calculator,
that is okay
Functions to graph (8)
• Graph f ( x)  x
• If the coach needs the help of a calculator,
that is okay
Switch roles
Functions to graph (9)
• Graph f ( x)  sin( x)
• If the coach needs the help of a calculator,
that is okay
Functions to graph (10)
• Graph f ( x)  cos( x)
• If the coach needs the help of a calculator,
that is okay
• This is the last one, so return the marker and
head back to your seats when you are finished
Is this function odd, even, or neither?
• Even (reflects over the
y-axis)
Is this function odd, even, or neither?
• Neither – it is not a
function, even though it
reflects over the x-axis
Is this function odd, even, or neither?
• Odd – it reflects over
the origin