Learning Resources
Lesson
Investigation 3: A New Relationship
You have previously worked with functions that model periodic motion. The sine and cosine functions were examined in great detail in Mathematical Modeling, Book 2. You have done little work, however, with the tangent function. These Investigations are designed to show the relationship between the tangent function and the sine and cosine functions.
The following supplies are needed for Investigation 3.
- compass
- protractor
- ruler
- graph paper
- cardboard
- a few straws
- pushpins or thumbtacks
Carry out the Investigation as outlined on pages 220 & 221 of your text. You can easily model the Ferris wheel ride by using a pushpin to attach the centre of a straw to graph paper backed with cardboard. You can move the straw to represent the three diameters at various locations. Having your own model will allow you to visualize the movement as opposed to simply drawing lines as the wheel moves.
Whether you make a model of the Ferris wheel or use a drawing, your creation must be accurate. Accurate models or drawings are crucial for accurate results.
Some guidance is provided for several steps of the Investigation. Use them only if and when you are experiencing difficulty.
Once the Investigation is completed, answer the accompanying Investigation Questions.
Investigation 4: Models With Undefined Behaviour
If your on-line teacher permitted the use of pairs or small groups for Investigation 3, it is important that you use the same set-up for this Investigation. This ensures that all of you will be using data that you have gathered.
You may proceed with the Investigation and the accompanying questions as outlined on pages 222 & 223 of your text. Some hints and suggestions are provided for each step if you are experiencing difficulty.
The key ideas from Investigation 4 are reviewed below.
Activity
C.Y.U. pages 223 & 224 #'s 38 - 44
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.
Test Yourself
- If the tangent function was graphed over the interval [0, 4
], what would be the equations of the asymptotes? - Find the equation of a function that would be the result of the tangent function undergoing the following transformations.
- a horizontal translation of
- a vertical translation of 5 units up
- a horizontal stretch resulting in a period of 4
Solutions
1. Remember that the tangent function is undefined wherever cosx is zero. Thus, the equations of the asymptotes are 
.
2. 