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:

• Decreasing:

• Constant:

04 x

63;46 xx

30 x

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

39.03.025.0)( 234 xxxxf

2.38.48.12.)( 23 xxxxf

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• If the coach needs the help of a calculator,

that is okay

xxf )(

Functions to graph (6)

• Graph• If the coach needs the help of a calculator,

that is okay

3)( xxf

Switch roles

Functions to graph (7)

• Graph• If the coach needs the help of a calculator,

that is okay

xxf

1)(

Functions to graph (8)

• Graph• If the coach needs the help of a calculator,

that is okay

xxf )(

Switch roles

Functions to graph (9)

• Graph• If the coach needs the help of a calculator,

that is okay

)sin()( xxf

Functions to graph (10)

• Graph• 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

)cos()( xxf

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