The main topic of Precalculus I is Functions. Specifically we study linear, quadratic, power, polynomial, exponential and logarithmic functions. To understand a function we have to understand its uses, its behavior, its algebra, its geometry and its rate-of-change behavior. Some questions that arise when we look more closely at these aspects of functions include:
Uses (modeling):
- What things in the real world can we model using a given function?
- What sorts of functions do we typically use in modeling the real world, and how do we use them?
Behavior:
- What real numbers can we 'feed' into our function, and what sorts of number does it give us back?
- What are the patterns of the numbers we get from the function as we 'feed' it increasing numbers in its domain, and how can we understand why these patterns emerge?
Algebra:
- How can we symbolically represent a given function?
- How can we symbolically manipulate the representation of the function to analyze and explain its behavior?
- How can we combine simple functions to get new functions?
Geometry:
- How can we graphically represent a given function? What does the geometry of the graph tell us?
- How can the graphs of function families be understood in terms of geometric transformations of fundamental functions?
Rate-of-change behavior
- As we 'feed' into a function numbers which are increasing at a given rate, how can we analyze and predict the rates at which the numbers the function gives us back are increasing or decreasing? (Analysis by numerical and graphical means is within the scope of a precalculus course; algebraic analysis of rates is the central subject of calculus).
The following are important questions:
To try to understand these questions we might think about the following questions:
These are serious questions, and a there is a wide variety of opinions on the answers, even among the most knowledgeable.
The next set of questions is addressed to the typical student entering a precalculus class, who knows a significant amount of mathematics but does not yet have the knowledge or experience to answer the previous questions:
You have a good understanding of many mathematical ideas.
I'm currently thinking about building a fishing pond. I don't know how big to build it, or how deep.
That's just a fishing pond. If I want to make business decisions, think intelligently about environmental policy, design a vehicle, understand strength training, or any of myriad other things, I can do so much more effectively if I understand the relevant mathematics.
"