Question B 04
The daily temperature in Vancouver during the month of March varies from an average low of C to an average high of C. It is coolest just before sunrise which is roughly 7 AM. Construct a function (using cosine) that provides a good description of the temperature throughout a day in March where is measured in hours from 12 AM.
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
What is the general form of a cosine function? How do we stretch or shift a cosine (or sine) function?
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
The general expression of a cosine function is Suppose , are both positive. Here the lowest temperature is C and highest is C, then
so and . The value is the period of this function. As it describes the temperature changes throughout a day which lasts for 24 hours, we have From the problem statement we know the lowest temperature is obtained at 7:00, by symmetry the highest is around 19:00. Then to shift the maxima from to we subtract by 19. So and the final function is
Note the expression of the solution is not unique, it can be written in infinitely many ways, such as