Lesson 2

Lesson 2: Reflections

Focus

A portrait is an interesting way to celebrate people in your life. A strong portrait captivates viewers, causing them to wonder about the person depicted, almost like a visual biography. Proportions of a head will vary from person to person and change slightly with age, but there are some basic principles in creating portraits.

Look at the following diagram of half a face. Suppose you wanted to complete this picture. Could you do it? How would you decide where the other eye should go? Where would you start to draw an ear? Would your strategy work for any half picture?

Lesson Outcomes

At the end of this lesson you will be able to

• determine the relationship between the coordinates of a point and the coordinates of that point reflected across the x-axis or y-axis
  
• sketch the graph of the functions y = f(−x) and y = −f(x) given the graph of y = f(x)
  
• determine the equation of a function given its graph, which is a reflection of the graph of y = f(x) through the x-axis or the y-axis

Lesson Questions

You will investigate the following questions:

• How is a point related to its reflection?
• How are the graphs of the functions y = f(x), y = f(−x), and y = −f(x) related?
• How can you graph the function y = f(−x) or y = −f(x) given y = f(x)?

Assessment

Your assessment may be based on a combination of the following tasks:

• completion of the Lesson 2 Assignment (Download the Lesson 2 Assignment and save it in your course folder now.)
  
• course folder submissions from Try This and Share activities
  
• additions to Glossary Terms and Formula Sheet
  

Materials and Equipment

• graph paper

Launch

Do you have the background knowledge and skills you need to complete this lesson successfully? Launch will help you find out.

Before beginning this lesson you should be able to determine whether or not a relation is a function.

Are You Ready?

Complete the following questions. If you experience difficulty and need help, visit Refresher or contact your teacher.

Determine whether or not the following relations are functions. Explain your reasoning.

1. Set A = {(1, 3), (2, 9), (7, 5), (−2, 8), (−1, 4), (2, −1)} Answer
2. Set B = {(1, 2), (2, 9), (−3, 4), (3, 4), (0, 5), (4, −7)} Answer
   
3. 
   
   Answer
4. 
   
   Answer

If you answered the Are You Ready? questions without difficulty, move to Discover.

If you found the Are You Ready? questions difficult, complete Refresher.

Refresher

Review the difference between relations that are functions and relations that are not functions in “Introduction to Functions.”

Go back to the Are You Ready? section and try the questions again. If you are still having difficulty, contact your teacher.

Discover

Try This 1

On a piece of graph paper, plot the points (1, 3), (2, 2), (3, 4), and (5, 2) as shown in the diagram.

1. Imagine using the y-axis as a line of reflection, or a line of symmetry, to reflect the points across the y-axis. Sketch the new image on your graph.
2. Reflect your original points across the x-axis and sketch the new image.
3. Use the diagrams you created to complete a table like the following.
   
   

   Coordinates of Original Point	Coordinates After Reflecting Across the y-axis	Coordinates After Reflecting Across the x-axis
   (1, 3)
   (2, 2)
   (3, 4)
   (5, 2)

4. What relationship do you notice between an original point and that point reflected across the
   1. y-axis?
   2. x-axis?
5. Was the original figure a function? Are the reflected figures functions?
6. Use Reflection Interactive to check your table from question 3.
   
   
    
   
   
7. The original relation had all points located in the first quadrant. Drag each point of the blue figure so the figure has points in multiple quadrants. Use Reflection Interactive to repeat questions 1 to 5.

Save your responses in your course folder.

Share 1

With a partner or group, discuss the following questions based on the information from Try This 1.

1. Did you find the same relationship in question 4?
2. Did reflecting a function create a function? Will this always be true? Explain.
3. If you reflected a relation that was not a function, would the reflection create a relation that is not a function?

Explore

A reflection is a transformation that produces a mirror image of the original figure. The “mirror” line is called the line of reflection. Although any line can be a line of reflection, you will mainly use the x-axis and y-axis in this lesson.

In Try This 1 you may have noticed that reflecting a point across an axis just changed the sign of one of the coordinates. Look ahead to see how this pattern can be formalized.

Try This 2

1. Using the information from Try This 1, describe what happens to the coordinates of an individual point as the point is reflected across an axis. Determine a mapping that will represent this.
   
   

   Line of Reflection	Description	Mapping
   y-axis		(x, y) → (__, __)
   x-axis		(x, y) → (__, __)

2. An invariant point is a point on the graph that remains unchanged after a transformation has been applied to it. What point(s) would you expect to be invariant when reflecting across the
   1. x-axis?
   2. y-axis?

Save your responses in your course folder.

Mapping a Reflection

The mapping of a reflection across the y-axis can be written (x, y) → (−x, y). Similarly, the mapping for a reflection across the x-axis is (x, y) → (x, −y).

In Try This 3 you may have found that when y = f(x) is reflected across the y-axis, the graph of y = f(−x) is produced. Also, when y = f(x) is reflected across the x-axis, the graph of y = −f(x) is produced.

Connect

Lesson 2 Assignment

Lesson 2 Summary

A reflection is a transformation that creates a mirror image of the original. A reflection in the x-axis changes all y-coordinates to –y. A reflection in the y-axis changes all x-coordinates to –x. A point that does not change during a transformation is called an invariant.

Now is a good time to update your summary table for the reflections you learned in this lesson.

In the next section you will begin to look at stretching a function.