5.1 Making Connections
Let's take a moment to see where we've been so that we might be able to think of some new questions that we can ask. We've now studied two different types of functions, linear functions, which look like:
and we've also studied quadratic functions, which look like:
Can we find a pattern? One thing that you might notice is that quadratics have one more x than linear functions. Since we saw that quadratics behave differently than linear functions, a natural question would be to investigate creatures that look like: x3, x4, x5, ... In fact, we'll do this in a future chapter, when we study polynomials.
For now, we'll focus on the exponents themselves and their relationships. Since it looks like each of the exponents is a counting number, we could ask what it would mean if the exponent were an integer, rational number, or irrational number. And, since we've developed tools to solve equations involving linear and quadratic creatures, it would natural for us to ask if we could develop tools to solve exponential equations as well.
In this chapter we're going to begin to study exponential relationships; these relationships begin to arise when we multiply specific numbers multiple times.