Symmetry occurs when parts of an object are identical to other parts of the same object. In this lesson, you will explore three types of symmetry.

Try to get as far as you can with the tasks below. If you get stuck, here are some additional resources that discuss some of the topics below:

http://home.windstream.net/okrebs/page43.html for Parts 1 & 2

http://www.purplemath.com/modules/symmetry3.htm for all parts

http://ed526b.wikispaces.com/Applet+Collection Look at the Symmetry and Functions applet. This will help with all parts of the assignment.

Also, you may discuss the problems with each other.

**Part 1**

Using GeoGebra, type in f(x) = 4 – x^2 and g(x) = x^2 – 4.

What do you notice about the two graphs?

Now try f(x) = 1 – x^3 and g(x) = x^3 – 1. What do you notice?

What do you notice about the combined graphs as an object?

We say that the objects made by the combination of f(x) and g(x) is symmetric about the x-axis. Do you see why?

We call f and g reflections of each other in the x-axis.

Can you find a rule for creating the reflection in the x-axis, g(x), of a function, f(x)? Can you give an argument why the rule is true?

**Part 2**

Now compare f(x) = 1 – x^3 and g(x) = x^3 + 1. What do you notice?

How about f(x) = x^2 – x and g(x) = x^2 + x?

How about f(x) = x^3 – x and g(x) = x – x^3? What’s happening?

We say that the object made by the combination of f(x) and g(x) is symmetric about the y-axis. Do you see why?

We call f and g reflections of each other in the y-axis.

Can you see a rule for creating the reflection in the y-axis, g(x), of a function, f(x)?

Hint: For each value x, what does g(x) need to equal to be the reflection of f?

Can you give an argument why the rule is true?

Can you see a characteristic of functions that are the same as their reflection in the y-axis?

Hint: Such functions, which have y-axis symmetry, are called even functions.

Question: Why (with one exception) aren’t there functions with x-axis symmetry? What is the exception?

**Part 3**

Now compare f(x) = 1 – x^3 and g(x) = -x^3 – 1. What do you notice?

How about f(x) = x^2 – x and g(x) = -x^2 – x?

How about f(x) = x + 2 and g(x) = x – 2?

We say that the object made by the combination of f(x) and g(x) is symmetric about the origin.

We also call f and g 180 degree rotations about the origin of each other. Do you see why?

Can you see a rule for creating the rotation about the origin, g(x), of a function, f(x)?

Hint: For each value x, what does g(x) need to equal to be the rotation of f?

Can you give an argument why the rule is true?

Can you see a characteristic of functions that are the same as their reflection in the y-axis?

Hint: Such functions, are called odd functions.

**Assessment questions:**

1. What does it mean to be reflected in the x-axis?

2. What is the difficulty in finding functions that have x-axis symmetry?

3. Give two examples of polynomials that have y-axis symmetry.

4. Give two examples of polynomials that are symmetric about the origin.

5. If a graph is symmetric about both the x and y axes, does it have to be symmetric (by 180 degree rotation) about the origin?

6. Create (using GeoGebra) with your own inputs:

a graph that is symmetric about the x-axis,

a graph that is symmetric about the y-axis,

and a graph that is symmetric about the origin.

Ethan, excellent job once again, I love this task. I was thinking how I could make it better and I thought you could provide your students with a website that helps with the idea of symmetry. I found a couple that I will provide links for below.

http://home.windstream.net/okrebs/page43.html

http://www.purplemath.com/modules/symmetry3.htm

While I understand your idea is to have the students “discovery” the ideas on their own, it might not hurt to have a little scaffold for those that need a little help. You can always give bonus points to those that do not use the scaffolds.

Peter,

Once again, I like your suggestion and think it will improve the lesson, and thanks also for finding the websites. They are good ones. I have used purplemath before, it really should be on my diigo links. I will revise my plan.

Giving bonuses to those who DO use the scaffolds can work, as well. After all, it takes more work to look at more “stuff” and using scaffolds raises quality.

Ethan, this is an incredible lesson. I liked how you had the students study the relations of quadratic and cubed equations. This exhibits one of the values of math sophistication we are studying this week!