With the new school year upon us (in many parts of the world), now is a good time for teachers to consider what they plan on teaching to their students this year. Undoubtedly, many teachers will just stick to the curriculum prescribed to them by their school district. But, is that always the “best” way…

# Category: Main Concepts

## Stretching Graphs and Compressing Graphs

If you understand how to shift a curve horizontally or vertically, stretching and shrinking graphs isn’t much different. Once again, it’s only a small modification to the equation that causes the stretch or compression. Stretching and compressing graphs vertically is determined by the coefficient in front of the x (or more specifically, in front of…

## Horizontal Translation: How to Shift Graphs

Shifting graphs horizontally (also known as horizontal translation) is slightly different from vertical translation, but still pretty straight-forward. Perhaps it would be helpful to review my posting on vertical shifts of graphs. Recall from that section: “Picture all the complex stuff that is happening to x as being one chunk of the height component, and…

## Vertical Translation – Shifting Graphs Vertically

As I’ve discussed several basic concepts about functions and graphing to this point, now I want to build upon those lessons and explain some more complex concepts. In particular, I want to show you how you can modify a function in order to produce a defined change in its graph. Similarly, by the same idea,…

## Why Does m Represent Slope?

Few people know how to answer this question properly, which doesn’t help when dealing with inquisitive students! There actually isn’t a definitive answer to this question (as far as I can tell), and scholars are still looking for its first use! Unfortunately, that likely won’t cure any lingering curiosity. And, frustratingly for teachers, this bit…

## Graphing Piecewise Functions

In this post, I would like to elaborate a bit on the concept I presented in a previous lesson. Specifically, I’m going to go over graphing piecewise functions. There are a few conventions that are important to keep in mind when graphing these. Today I will explain these points and show you additional examples of…

## Complex Functions and Piecewise Functions

One of my previous posts introduced you to functions and how they work, but there are few other things worth discussing that I would like to add here. The first is more about function notations, then I will expand that into a demonstration of complex functions, and then wrap up this post by introducing the…

## Vertical Line Test Definition

Following up on my post where I introduced the concept of functions, here I would like to provide you with a quick and easy test that you can perform on any graph which will help you decide if you are looking at the graph of a function or not. This check is called the vertical…

## The Definitive Guide to Domain and Range

When working with functions and their graphs, one of the most common types of problems that you will encounter will be to identify their domain and range. This isn’t necessarily a difficult problem to solve, once you know what domains and ranges actually are! After all, these terms sound like they belong in geography or…