Lesson 3

Lesson 3: Stretches

Focus

Have you ever been to an amusement park with those crazy mirrors that make you look funny? When the mirror curves inward (is concave), it makes a person look elongated and thinner. The convex mirror bulges outward and results in a short and wide stature. Similar kinds of distortions can take place as you work with transformations.

Look at the photos. The lower right photo was distorted to produce the other two. What is similar about the three photos? How could you create either distortion without using a computer? When is it useful to distort a photo in these ways?

In this lesson you will learn to stretch a function horizontally and vertically. You will also learn to relate a graph to the corresponding equation of the form y = af(bx).

Lesson Outcomes

At the end of this lesson you will be able to

• compare graphs of the form y = af(bx) to the graph y = f(x) and generalize a rule about a and b
  
• sketch a graph of a function of the form y = af(bx) given the graph of y = f(x) and the values of a and b

Lesson Questions

You will investigate the following questions:

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

Assessment

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

• completion of the Lesson 3 Assignment (Download the Lesson 3 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

Discover

Try This 1

1. On a piece of graph paper, plot the following function. You can use Grid Paper Template, available in the Toolkit, if you like.
   
   
    
   
2. Use the original function from question 1 to draw each mapping from the following table. Use a different colour for each mapping or drawing to make your graph(s) easier to read.
   
   

   Mapping	Image Points	Description of How the Graph Transformed	Diagram of Transformation
   (x, y) → (3x, y)
   (x, y) → (x, y)
   (x, y) → (x, 4y)
   (x, y) → (x, y)
   (x, y) → (−2x, y)
   (x, y) → (x, −3y)

3.  
   1. What effect does multiplying all x-coordinates by a positive constant have on the figure?
   2. What effect does multiplying all y-coordinates by a positive constant have on the figure?
4. Repeat questions 1 and 2 by using the following function.
   
   
    
   
5. Revisit question 3. Are your answers still the same?

Save your work in your course folder.

Share 1

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

1. How does the graph change when the x-coordinate is multiplied by a value
   1. greater than 1?
   2. between 0 and 1?
   3. that is negative?
2. How would your answers from question 1 change for a y-coordinate?

If required, save a record of your discussion in your course folder.

Explore

Did You Know?

When people hear the word stretch, they often think of something getting larger. In this course, stretching refers to something getting larger or smaller. One reason for using this word is to avoid confusion when using the term compress. Although “compress by a factor of ” seems like it should mean the object will become half its size, it actually means the object will double in size. Think about why this is true.

In Try This 1 you explored what happened to the shape of a graph when you multiplied all of the x-coordinates of a function by a constant or when you multiplied all of the y-coordinates by a constant. The change in shape seen is called a stretch. In this lesson you will focus on stretches about the x-axis or the y-axis. When you're stretching about an axis, it is useful to think of a function in terms of the equation y = af(bx). In Try This 2 you will explore the outcomes of changing a and b for a function of this form.

Try This 2

Open Stretching/Reflecting y = af(bx).

1. Select the square root function button .
   
   

   Notice the point highlighted on the function that follows. This highlighted point shows where a particular point moves as you adjust a and b. The original coordinates and transformed coordinates of this point are listed at the bottom of Stretching/Reflecting y = af(bx). Make sure you pay attention to this information during Try This 2.
   
   

    

   

2. Drag the red point along the graph to (4, 2); then use the applet to complete a table like the one that follows.
   
   

   Value of a	Value of b	Coordinates of Original Point	Coordinates of Transformed Point	Description/Diagram of How the Point Changed	Description/Diagram of How the Function Changed
   1	1	(4, 2)	(4, 2)	no change	no change
   3	1
   0.7	1
   −2	1
   1	4
   1	0.5
   1	−2
   Student Choice
   Student Choice

3. For positive a- and b-values, what effect does each of the following have on a function?
   1. increasing a
   2. decreasing a
   3. increasing b
   4. decreasing b
4. Press the “SET FCN” button on the applet and experiment with a different function. Do you still agree with your responses from question 3?

Save your work in your course folder.

Share 2

With a partner or group, discuss the following questions based on the table you created in Try This 2.

1. What effect does a negative a or b have on the function? Where have you seen this effect before?
2.  
   1. What mathematical relationship is there between a and how an individual point on the function y = af(bx) moves?
   2. What mathematical relationship is there between b and how an individual point on the function y = af(bx) moves?
   3. Where would you expect to find an invariant point for each type of stretch?

In Try This 2 you may have noticed that the function y = af(bx) is the function y = f(x) stretched about the y-axis by a factor of and stretched about the x-axis by a factor of a.

As in previous lessons, the process of using an equation of the form y = af(bx) to graph a function can be reversed to determine the equation of a function given a graph.

Try This 3

Janila is working on her transformation project and has drawn an original function, y = f(x), and transformed the function to produce y = g(x) and y = h(x). Determine an equation to represent g(x) and h(x) in terms of f(x).

Save your work in your course folder.

Connect

Lesson 3 Assignment

Lesson 3 Summary

When you multiply all of the x-coordinates or y-coordinates of a function by a constant, the change in shape is called a stretch. When you're stretching about the x-axis or the y-axis, it is useful to think of a function in terms of the equation y = af(bx). When using an equation of this form, a corresponds to a vertical stretch about the x-axis by a factor of a. And b corresponds to a horizontal stretch about the y-axis by a factor of .

Now is a good time to update your summary table for the stretches you learned in this lesson. In Lesson 4 you will begin to combine the transformations learned so far.