Lesson 4

Lesson 4: Combining Transformations

 

Focus

 

Some artwork is just one image manipulated over and over in slightly different ways. This graphic was made using stretches, reflections, and translations. Is it important in what order the transformations were applied? If you wanted to use a coordinate grid to help with these transformations, does it matter where you put the grid?

 

 

In this lesson you will learn to use multiple transformations on a function. You will also learn to relate the graph of a transformed function to its equation.

 

Lesson Outcomes

 

At the end of this lesson you will be able to

• sketch the graph of the function for given values of a, b, h, and k, given the graph of the function y = f(x)
• write the equation of a function, given its graph, which is a translation and/or stretch of the graph of the function y = f(x)
  

Lesson Questions

 

You will investigate the following questions:

• How can multiple transformations be applied to a function?
• How can you write the equation of a transformed function given the original graph?

Assessment

 

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

• completion of the Lesson 4 Assignment (Download the Lesson 4 Assignment and save it in your course folder now.)
  
• course folder submissions from Try This and Share activities
  
• additions to Formula Sheet

Materials and Equipment

• graph paper

Discover

 

Try This 1

 

Projectile motion simulators like the one that follows are common. Open “Golf Range” and experiment with the effects of using different angles and initial speeds. Turn on “Show paths” to see the trajectory of your ball. Press to shoot and to swing again.

 

1. What general shape does the path always take?
2. How does changing the speed or angle affect the shape?
3. How can the path of a projectile be transformed into the path of another projectile?
4. The projectiles in the simulator all followed a parabolic trajectory, which can be modelled by y = x2 after various transformations.
   1. Open Projectile Equation Matcher to explore the equations of these parabolas. The function is given in red and its values of a, b, h, and k can be manipulated.
      
      
       
      
   2. Select projectile 1.
   3. Use the sliders to match the red function to the green projectile 1.
   4. Complete the following table.
      
      

      Projectile	Value of a	Value of b	Value of h	Value of k	Description of How f(x) = x2 Was Transformed
      1
      2
      3
      4

Save your responses in your course folder.

 

Share 1

 

With a partner or group, discuss the following question based on the table you created in Try This 1:

 

Is it possible for more than one equation to match a projectile path? Explain.

 

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

Explore

 

In Try This 1 you saw that it was possible to perform stretches, reflections, and translations across the same function. When multiple transformations are used, the order in which they are applied is significant. Complete the following investigation to see how important order is.

 

Hemera/Thinkstock

 

Try This 2

You have seen that multiple transformations can be applied to a function. By starting with the stretched and reflected equation y = af(bx), translations can be added by replacing y with y − k and x with x − h to give This is sometimes written as

 

You have also seen that switching the order of a translation and a stretch or a reflection affected the outcome of the transformation. To use an equation of the form apply the stretches and reflections before the translations.

 

 

The next example shows how stretches and translations can be applied to the same function.

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Connect

 

Lesson 4 Assignment

Lesson 4 Summary

 

It is possible to perform multiple transformations on a function. The order in which you apply the transformations is important in determining the resulting function. The form of the equation of the function described in the diagram is commonly used when transforming a function, as the different parameters are easily interpreted. Using this form implies that the function is stretched and reflected before it is translated.

 

 

Multiple Transformations reviews how some specific functions of the form behave when you change the different parameters.

 

 

In Lesson 5 you will begin to explore the idea of reversing a function.