(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 »
A common topic in algebra courses is how to transform functions and their graphs. In the series starting today, we’ll start with the basics of how and why a graph is moved or stretched, then combine transformations and look at various special cases and other transformations, ending up with graphing trigonometric functions.
In the Ask Dr. Math service, new volunteers went through a period of training (“internship”), first demonstrating their ability to write effectively about math with some example problems, and then answering actual questions under supervision, discussing their answers with a mentor online before they were actually sent out. Once our reliability was confirmed, we were …
While I was setting up The Math Doctors site a year ago, I ran across the following email I received in 1998 inviting people to become Math Doctors. It illustrates well the ethos of the team: In the fall of 1994 the Math Forum at Swarthmore College (then the Geometry Forum) started an email program …
I previously wrote about finding the range of various kinds of functions. The examples there were relatively easy. A recent question raised the level of difficulty, bringing up some interesting issues.
(An archive question of the week) Last time we looked at a formula for approximating the mode of grouped data, which works well for normal distributions, though I have never seen an actual proof, or a statement of conditions under which it is appropriate. We have also received questions about a much more well-known, and …
The mode of a list of data values is simply the most common value (or values … if any). When data is grouped (binned) as in a histogram, we normally talk only about the modal class (the class, or group, with the greatest frequency), because we don’t know the individual values. But some sources teach …
(A new question of the week) It’s surprising how many questions we get that end up being about problems that are poorly worded or simply wrong. But these can be as illuminating as good problems, by showing ways to catch the error. This is one simple in itself, but will lead us into a common …