Lesson 1

Lesson 1: Horizontal and Vertical Translations

Focus

A tile puzzle can be solved by sliding the tiles. In the puzzle shown, the goal is to rearrange the tiles to complete a picture of a horse. Imagine giving instructions to a friend who was playing the game. How would you describe what moves the friend should make? Suppose you put the tile puzzle onto a grid. Would you describe how a tile moves differently? Learning about transformations will give you a method to mathematically describe these movements.

Try your hand at Image Shuffle. After you launch the puzzle, you will need to right-click on the image and then select “Play.”

In this lesson you will learn how to translate the graph of a function of the form y − k = f(x − h). You will also learn how to determine the equation of the form y − k = f(x − h) from a graph.

Lesson Outcomes

At the end of this lesson you will be able to

• compare graphs of the form y − k = f(x − h) to the graph of y = f(x) and generalize a rule about h and k
  
• sketch a graph of a function of the form y − k = f(x − h) given the graph of y = f(x) and the values of h and k

Lesson Questions

You will investigate the following questions:

• How are the graphs of the functions y = f(x) and y − k = f(x − h) related?
• How can you graph the function y − k = f(x − h) given y = f(x)?

Assessment

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

• completion of the Lesson 1 Assignment (Download the Lesson 1 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
  
• work under Project Connection

Self-Check activities are for your own use. You can compare your answers to suggested answers to see if you are on track. If you are having difficulty with concepts or calculations, contact your teacher.

Remember that these questions and activities provide you with the practice and feedback that you need to successfully complete this course. You should complete all the questions and place your responses in your course folder. Your teacher may wish to view your work to check on your progress and to see if you need help.

Materials and Equipment

• graph paper

Time

Each lesson in Mathematics 30-1 Learn EveryWare is designed to be completed in approximately two hours. You may find that you require more or less time to complete individual lessons. It is important that you progress at your own pace, based on your individual learning requirements.

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

• graph a function given an equation of that function
• interpret and use function notation
• work with absolute values
• state the domain and range

Are You Ready?

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

1. Graph the function y = 3x2 − 2 using a table of values. State the domain and range of the function. Answer
2. Given f(x) = 3x + 1, determine the following values.
   
   1. f(3) Answer
   2. x if f(x) = 7  Answer
3.  
   1. Given f(x) = |2x − 5| determine f(1). Answer
   2. Graph the function f(x) = |2x − 5| using a table of values. 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 graphing a function given the equation in “Algebra: Graphing Lines 1.”

Use Graph to review how to complete a data table of values and draw a graph for a given function.

Review how to determine domain and range of a relation. Work through Concept Development Domain and Range, which offers information, practice questions, and feedback. To access this activity, choose Domain and Range from the first menu, and then select Domain and Range again from the second menu (on the orange screen).

To review function notation, watch “Functions Part 2.”

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

Open Transformations of a Polygon and complete the steps that follow.

1. Use the sliders on the side of the program to complete a table like the one that follows.

    

   Transformation	How the Polygon Is Affected
   Horizontal Translation
   Vertical Translation
   Horizontal Stretch
   Vertical Stretch
   Horizontal Reflection
   Vertical Reflection
   Rotation About a Point*

   *Rotations are not part of this course, but they are interesting.
2. Which transformations changed the size of the polygon?
3. Which transformations changed the location of the polygon?

Save your responses in your course folder.

Remember that it is important to place your completed Try This activities in your course folder. Your teacher may ask to see your completed Try This activities at any time. For more information on Try This activities, refer to the Course Introduction.

Share 1

After a discussion with your teacher, you will decide how to connect with other students for Share activities. Follow your teacher’s instructions to complete this Share activity. When exchanging ideas, make sure to include support for your response. Support could be in the form of words, graphs, and/or calculations. You should reply to other students’ responses and build on their ideas to come to a deeper understanding of the topic. Place a summary of your Share discussion in your course folder. For more information on Share discussions, refer to the Course Introduction.

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

1. Compare your descriptions of how the polygon is affected. What are the similarities and differences?
   
2. Do you think your results would change if you used a different shape? Why or why not?

  If required, place a summary of your discussion in your course folder.

Explore

In Try This 1 you investigated four different types of transformations. A transformation of a relation is a change made to the relation such that the graph of the relation is shifted or changed in shape. In this lesson you will learn more about translations, a specific type of transformation. A translation of a relation is a shift in the position of the graph without changing the graph’s shape or orientation.

Just as transformations can be applied to figures, such as the polygon from Try This 1 or the star in the diagram shown, transformations can also be performed on the graph of a function. In Try This 2 you will explore the transformation of the function f(x) = |x|.

In Try This 2 you may have noticed that the graph y = |x| + k is the graph y = |x| translated up k units. Similarly you may have seen that y = |x − h| is the graph of y = |x| translated h units to the right. These results can be generalized to any function f(x).

The graph of y = f(x) + k is the graph of y = f(x) translated up k units and the graph of y = f(x − h) is the graph of y = f(x) translated to the right h units. Often y = f(x) + k is written as y − k = f(x) to emphasize that the k-value causes a change in the y direction.

So far you have looked at the effects of h and k on functions of the form y − k = f(x) and y = f(x − h). Next you will look at combining these ideas to predict the effects of h and k on functions of the form y − k = f(x − h). Often this equation is rearranged to isolate y, giving y = f(x − h) + k. How do you think this function will respond to changing h and k?

Try This 3

Open Translations [y = f(x − h) + k].

Step 1: Select a cubic function using the button.

Step 2: Use the applet to complete a table like the one that follows. The equation of the graph is shown above the graph.

Function in Form y = f(x − h) + k	Value of h	Value of k	Effect on Graph	Diagram
y = x3
y = (x − 2)3 + 1
y = (x + 3)3 − 4

Step 3: Push the SET FCN button at the bottom right of the diagram and select the absolute-value function .

Step 4: Use this function to complete a table like the one that follows.

1. What effects do h and k have on a function that contains both h and k?
2. How do the effects of changing h and k compare for the cubic function? The absolute-value function?

Save your responses in your course folder.

Functions of the form y = f(x − h) + k can be graphed quickly from y = f(x) using what you have learned so far about h and k. Watch Graphing with h and k to see an example of how to graph such a function.

Connect

Lesson 1 Assignment

Lesson 1 Summary

A transformation is a change in the shape or position of a figure or relation. A translation is a type of transformation that causes a “slide” in the graph. The new graph is the same size, shape, and orientation as the original but in a different position.

In a graph of the form y − k = f(x − h), k is the vertical translation from y = f(x). If k > 0, the translation is upwards. If k < 0, the translation is downwards.

In a graph of the form y − k = f(x − h), h is the horizontal translation from y = f(x). If h > 0, the translation is to the right. If h < 0, the translation is to the left.

You may find it useful to keep a summary of the different transformations you learn in this module using a table similar to the following.

In the next lesson you will begin to look at reflections of a function.