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

Interactive Linear Function Graph: Point-Slope Form

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

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:

Continue reading Polynomials and VEX Drive Motor Control

3 Ways to “Complete the Square”

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:

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

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

$\dfrac{y-10}{2}~=~x^2+6x$

$\dfrac{y-10}{2}+(\frac{6}{2})^2~=~x^2+6x+(\frac{6}{2})^2$

$\dfrac{y-10}{2}+9~=~x^2+6x+3^2$

$\dfrac{y-10}{2}+9~=~(x+3)^2$

$\dfrac{y-10}{2}~=~(x+3)^2-9$

$y-10~=~2(x+3)^2-18$

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

Second Approach

This approach keeps everything on the same side of the equation, and is therefore suitable for use when you either do not have an equation to work with, or do not wish to “mess up” the rest of the equation with your work to complete the square. It factors the leading coefficient out of all terms before proceeding:

Continue reading 3 Ways to “Complete the Square”

Function Dilations: How to recognize and analyze them

For approach I now prefer to this topic, which uses transformation equations, please follow this link: Function Transformations: Dilation

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 points on the solid line are shown, as are the coordinates of the two corresponding points on the dashed line, to help you verify that the dashed line is exactly twice as far from the x-axis as the same color point on the solid line.

Continue reading Function Dilations: How to recognize and analyze them

Function Translations: How to recognize and analyze them

For the approach I now prefer to this topic, using transformation equations, please follow this link: Function Transformations: Translation

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. A function can be translated either vertically, horizontally, or both. Other possible “transformations” of a function include dilation, reflection, and rotation.

Imagine a graph drawn on tracing paper or a transparency, then placed over a separate set of axes. If you move the graph left or right in the direction of the horizontal axis, without rotating it, you are “translating” the graph horizontally. Move the graph straight up or down in the direction of the vertical axis, and you are translating the graph vertically.

In the text that follows, we will explore how we know that the graph of a function like

$g(x)=x^2-6x+10\\*~\\*~~~~~~~=(x-3)^2+1$

which is the blue curve on the graph above, can be described as a translation of the graph of the green curve above:

$f(x)=x^2 ~~~~\text{translated right by 3, and up by 1}$

Describing $g(x)$ as a translation of a simpler-looking (and more familiar) function like $f(x)$ makes it easier to understand and predict its behavior, and can make it easier to describe the behavior of complex-looking functions. Before you dive into the explanations below, you may wish to play around a bit with the green sliders for “h” and “k” in this Geogebra Applet to get a feel for what horizontal and vertical translations look like as they take place (the “a” slider dilates the function, as discussed in my Function Dilations post).

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 combining like terms, expanding terms, and doing something that cancels itself out.

A number of students seem to find the introduction of quadratic equations frustrating. After spending much time learning about linear equations, and finally just getting to the point where everything seems to be starting to make sense and be “easy” again, all of a sudden the teacher starts in on a totally different and seemingly unrelated topic…

However, quadratics are not a totally different topic. Linear equations (where the highest power of a variable is the first power) and quadratic equations (where the highest power of a variable is the second power) are the least complex two members of the family of polynomial equations.

Keep Your Eye On The Variable

The following equations all have a similarity:

$y = |x - 8| + 5\\*~\\*y = 4(x - 6) - 7\\*~\\*(x - 3)(13x + 11) = 0\\*~\\*y = (x + 1)^2 - 9$

The similarity is that they all have expressions like (x- 6) or (13x + 11), which are often either translations or factors. Situations such as this occur with:
– Point-slope form of the equation of a line
– Vertex form of the equation of a quadratic
– Horizontal or vertical translation of any function
– Factors of a polynomial
– “Bounce point” of an absolute value function