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:

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

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) = x 2 Is there any symmetry to this graph? – Can you reflect it over anything?

Functions to graph (4) Graph f(x) = x 3 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

Functions to graph (6) Graph If the coach needs the help of a calculator, that is okay

Switch roles

Functions to graph (7) Graph If the coach needs the help of a calculator, that is okay

Functions to graph (8) Graph If the coach needs the help of a calculator, that is okay

Switch roles

Functions to graph (9) Graph If the coach needs the help of a calculator, that is okay

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

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