Month: June 2023

Exponential Growth: Surprisingly Flexible

Two recent questions from the same student involve exponential functions: We can express different kinds of growth all using one base, called e; or we can use different bases (and ignore horizontal scaling transformations). And we can use different transformation to obtain the same graph. This relates to some important properties of exponential functions.

Polynomials: A Matter of Degrees

Last time we examined why polynomials are defined as they are. This time, let’s look at some tricky aspects of the concept of “degree”, mostly involving something being zero.

Polynomials: Why Are Terms What They Are?

A question last week (Hi, Zahraa!) led me to dig  up some old discussions of how we define a polynomial (or monomial, or term) and, specifically, why the exponents have to be non-negative integers. Why can we only multiply, and not divide by, variables? Since we’ve been looking at polynomials, let’s continue.

The Shape of a Polynomial at its Zeros

Last week’s discussion about zeros of a polynomial, and other conversations, have reminded me of a past discussion of the shape of the graph of a polynomial near its zeros. Let’s take a look, starting with some other questions that nicely lead up to it.