## Interactive Graphs for Linear, Quadratic, Rational, and Trig Functions Moved to GeoGebraTube

Some may have had trouble using my GeoGebra applets in their browsers. I have moved all of them to GeoGebraTube, which will hopefully fix the problem. You may search for them by typing “MathMaine” into the GeoGebraTube search box.

Links to all updated interactive graph applets are below. Comments and suggestions are always welcome!

Linear Functions

GeoGebraBook: Exploring Linear Functions, which contains:

Interactive Linear Function Graph: Slope-Intercept Form

## Polynomials and VEX Drive Motor Control

VEX Robots can be more competitive when they have addressed several drive motor control challenges:

1. Stopping a motor completely when the joystick is released. Joysticks often do not output a value of  “zero” when released, which can cause motors to continue turning slowly instead of stopping.
2. Starting to move gradually, not suddenly, after being stopped. When a robot is carrying game objects more than 12 inches or so above the playing field, a sudden start can cause the robot to tip over.
3. Having motor speeds be less sensitive to small joystick movements at slow speeds. Divers seeking to position the robot precisely during competition need “finer” control over slow motor speeds than fast motor speeds.

These challenges can be solved using one or more “if” statements in the code controlling the robot, however using a single polynomial function can often solve all of these challenges in one step. A graph can help illustrate the challenges and their solution:

## Completing the Square Procedures

I have seen three approaches to “Completing the Square”, as shown below. Each successfully converts a quadratic equation into vertex form.  Which do you prefer, and why?

### First Approach

This approach can only be used when you are working with an equation. It moves all terms that are not part of a perfect square to the other side of the equation to get them out of the way:

$y=2x^2+12x+10$

$y-10=2x^2+12x$

$\frac{1}{2} y-5=x^2+6x$

$\frac{1}{2} y-5+(\frac{6}{2})^2=x^2+6x+(\frac{6}{2})^2$

$\frac{1}{2}y-5+9=x^2+6x+3^2$

$\frac{1}{2}y+4=(x+3)^2$

$\frac{1}{2}y=(x+3)^2-4$

$y=2(x+3)^2-8$

## Function Dilations: How to recognize and analyze them

This post explores one type of function transformation: “dilation”. If you are not familiar with “translation”, which is a simpler type of transformation, you may wish to read Function Translations: How to recognize and analyze them first.

A function has been “dilated” (note the spelling… it is not spelled or pronounced “dialated”) when it has been stretched away from an axis or compressed toward an axis.

Imagine a graph that has been drawn on elastic graph paper, and fastened to a solid surface along one of the axes. Now grasp the elastic paper with both hands, one hand on each side of the axis that is fixed to the surface, and pull both sides of the paper away from the axis. Doing so “dilates” the graph, causing all points to move away from the axis to a multiple of their original distance from the axis. As an example of this, consider the following graph:

The graph above shows a function before and after a vertical dilation. The coordinates of two Continue reading Function Dilations: How to recognize and analyze them

## Function Translations: How to recognize and analyze them

A function has been “translated” when it has been moved in a way that does not change its shape or rotate it in any way. Such changes are a subset of the possible “transformations” of a function, and can be accomplished through vertical translation, horizontal translation, or both.

Imagine a graph that has been drawn on tracing paper that was loosely lain over a printed set of axes. If you move the tracing paper left, right, up, or down some distance, without rotating it in any way, you are “translating” the graph.

It is very useful to be able to interpret a function as a translation of a “parent function”. Understanding the behavior of

$g(x)=(x-3)^2+1$

is much easier if you think of it as the graph of a simpler looking function that you should already be very familiar with

$f(x)=x^2$

shifted right by 3, and up by 1.

This approach helps us understand the behavior of Continue reading Function Translations: How to recognize and analyze them

## GeoGebra Applets That Help Understand Equation Behavior

The greatest value of GeoGebra, or Geometer’s Sketchpad, or other such packages, in my eyes is their ability to help students dynamically visualize the effect each constant has on the graph of an equation. I find them an invaluable “thinking aid” as I ponder a new equation form, and they help me formulate my own answers to questions such as “why does it do that?”

Check out applets that help students explore the relationship between function parameters and their graphs:

Linear functions:

GeoGebraBook: Exploring Linear Functions, which contains

GeoGebraBook of Quadratic Applets, which contains

Exponential and Logarithmic functions:

GeoGebraBook of Exponential and Logarithmic Applets, which contains

Rational functions:

Trigonometric functions:

GeoGebraBook for Exploring Trig Functions, which contains

Unit Circle Symmetries:

GeoGebraBook of Unit Circle Symmetry applets, which contains:

## Summary: Algebra

When faced with an algebraic expression or equation, there are only two types of things you can do to it without changing the quantitative relationship that it describes.

### Re-write one or more terms in an equivalent form

This can be done to any expression (no equal sign) or equation (with an equal sign) at any time.

There are three common ways in which this is done: by Continue reading Summary: Algebra