Learning Resources

Lesson

The completion of Investigation 8 may require you to seek help from your in school science teacher and the use of materials from the science lab. Care must be taken since boiling water is involved. A regular glass container is not recommended since it may crack with high heat.  A flask from the science lab is more appropriate. 

It is recommended you work in a group of three, if possible. One can measure the time, another read the thermometer, and the third record the data. Alternately, you may choose to do the experiment at home.

For Step A, it is important to record the room temperature since the water will not cool below this temperature.

For Step B, the Styrofoam is used to keep the thermometer away from the sides of the flask, since you want to record the changes in the water temperature only. Make sure that the Styrofoam does not completely cover the glass bulb of the thermometer.

For Step C, enough boiling water should be used to cover approximately 2 cm of the thermometer - too much water will require a lengthy cooling time. Record your data in a table.

If for some reason you are unable to perform the experiment (although you will miss a big aspect of the Investigation if you don't) you can obtain sample data to complete the Investigation Questions by clicking here.

Investigation Questions

Question 42 is getting at the fact that the horizontal asymptote for the graph of this set of data is not the x-axis (y = 0). For this experiment, the horizontal asymptote will be whatever the room temperature is, at the time of the experiment. To "slide" the graph of y = ab^x up or down so that its horizontal asymptote changes is a translation of the original data. Think in terms of the work you have already done on the transformational form of equations.

Now complete Investigation 8 and the Investigation Questions which follow it.

In Investigation 8 and the questions which followed it,  you should have discovered that the horizontal asymptote for exponential functions is not always 
y = 0 (i.e. the x-axis). In fact, y = 0 is the asymptote only when the function has the form .

When the graph approaches a different value, the horizontal asymptote changes. This was the case in the experiment completed for Investigation 8 where the value approached was the room temperature.

Changing the horizontal asymptote, changes the equation from to , or in its transformational form to . In this form
y = d is the horizontal asymptote.

The interactive below looks at this effect by varying the value of d for a particular exponential function:

From the above demonstration you should be able to see that the horizontal asymptote changes as the graph changes. Notice the relationship between the horizontal asymptote and the equation of the function.

Activity

1. Complete Investigation 8 on page 138 in your text.
2. Complete the Investigation Questions 40 - 42 on pages 139 - 140.
3. Do the CYU Questions 43 - 46 on pages 140 & 141.

Test Yourself

Given the function 

1. Is it a growth or decay curve?
2. Write the equation of its horizontal asymptote.
3. What is its y-intercept?
4. Write it in its transformational form.
5. Sketch the curve without using technology.

Solution