Learning Resources
Lesson
Investigation 9: Transformations
To complete this activity graphing technology is necessary - the text book illustration utilizes the TI-83 which you should use if available. As you progress through the investigation you will discover how changes in the exponential equation affect the shape and orientation of the curve. You will connect these transformations to those learned earlier. Click here for a short review.
Following the Procedure, Step A and Step B are applications entered into your graphing calculator. Note that Step B is similar to the results in Investigation 8 in Section 2. Step C builds on Investigation 7 from Section 2 and forces you to identify the type of transformation being illustrated. In Step D a new transformation will be identified. Make note of it.
It is very important that you identify the mapping rules for each of these transformations in the various steps since these are used in the graphing of the functions. Go to the next page before continuing with the Focus.
Transformation of functions is important in graphing as you already know. The only new item in this investigation is found in Step D, a reflection in the y-axis.
For functions of the form 
Base of Function |
Reflection in y-axis |
Growth or Decay |
b > 1 |
No |
Growth |
b > 1 |
Yes |
Decay |
0 < b < 1 |
No |
Decay |
0 < b < 1 |
Yes |
Growth |
For example, 
is a reflection of 
in the y-axis and is a growth curve.
The writing of the the mapping rules from the transformational form of the function is used in graphing and is a very useful skill.
Example
State the transformations of and write the mapping rule for 
. Is this a growth or decay curve? Name its horizontal asymptote.
Solution
- Reflection in y-axis (negative exponent)
- horizontal stretch of 2
- vertical translation of 1
- This is a growth curve ( 0 < b < 1 with a reflection in the y-axis)
- horizontal asymptote is y = 1
Go to Activities Section and complete !
Focus D: Graphing Using Transformations
You will now use the transformations to graph exponential functions. If you have forgotten the procedures, check the quick review in the investigation above.
Read and study the material in your textbook which illustrates what is referred to as Method 1: Mapping Rule and Table of Values. For another example click here.
Activity
Check Your Understanding p.144. Complete 1, 2 and 3
Think About p.144
Focus Questions p.145 - 146. Complete 4
When you have completed these questions, ask your on-site teacher to get the solutions for you from the Teacher's Resource Binder and check them against your answers. After you do this, if there is something you had trouble with and still do not understand, contact your on-line teacher for help.
After you do the assigned activities, continue on to the Test Yourself section below for a quick quiz on this lesson.
Test Yourself
Change the equation
to transformational form. Write the mapping notation and sketch a graph from -3 to 3. State the horizontal asymptote, horizontal and vertical translation, vertical stretch, y-intercept, and range.
Click here for a solution.