(A new question of the week) Some topics are hard to find information about at a basic level, because they are usually dealt with in advanced math courses, and yet the basic ideas can be understood without all the trappings. That is the case for the Affine Tangent Cone, which involves tangents to an algebraic …
(A new question of the week) A question in September, about graphing a horizontally-stretched cosine function, led to a long conversation. Between a typo in the problem and some inside-out thinking, this surprisingly non-routine problem led to some good mind-stretching! I have edited this down considerably by removing distractions from the main ideas, but it …
A Mind-Stretching Exercise with a Stretched Cosine Read More »
(A new question of the week) Having just discussed several mathematical topics that lie behind the various graphs we have seen in the news lately, I want to depart from our usual style and answer my own current questions. We’ll look at several graphs of COVID-19’s growth and think about what we can learn from …
We’ve been looking at the math underlying some of the graphs associated with the COVID-19 pandemic, starting with exponential growth, and then logistic growth. I want to look in more detail at a feature I mentioned in the first post, viewing a graph logarithmically. This is a powerful technique that goes far beyond a button …
The term “exponential” has gone viral, so to speak. Do we all know what it means? In the next few posts I’ll look at answers we’ve given to questions about exponential growth and related concepts, some of them about the spread of diseases or rumors. (Disclaimer: I will be writing about the basic math, not …
I’ll close out our look at transformations of functions with some trigonometric graphs. These are the best example of combined transformations, and involve some special tricks as well. We’ll start with an early question that gives an overview of the process, then focus in on important details.
Having looked at all the usual transformations of a function and its graph, there are two more situations I want to look at. The first of these are the horizontal and vertical transformations introduced by absolute values (with a bonus thrown in: square roots); then next time we’ll conclude this series with a look at …
(An archive question of the week) We’ve looked at the basic transformations of a function and how they affect its graph, then at how they combine, and then how they can interact with specific functions. Now let’s look at one problem from beginning to end, looking at a graph and finding the function that goes …
The last two posts were about transformations of functions (shift, stretch, reflect) and their effect on a graph, first individually and then in combination. The next thing to look at will be how to determine the transformations when you are given a graph; but before we take that challenge in general, we need to see …
Last time we looked at questions about how to shift, stretch, or flip a graph by changing the equation of a function. All our examples involved only a single transformation. Now we can look at cases where two or more transformations are combined. As we do this, we will develop a deeper understanding of how …
Combining Function Transformations: Order Matters Read More »