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

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

Second Approach

Continue reading Completing the Square Procedures

Function Dilations: How to recognize and analyze them

This posting assumes you have read Function Translation: a Transformation. Translations represent some combination of vertical and/or horizontal shifts of a function’s graph, but these are only two of the possible transformations of a function.

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 paper attached a solid surface along one axis. Now grasp the elastic paper with both hands, one hand on each side of the attached axis, and pull both sides of the graph away from the axis… you are “dilating” the graph, and causing all points to move away from the axis to a multiple of their original distance from the axis.

Vertical Dilation (no Translation)

The above graph shows what a function might look like both before and after a vertical dilation. Note that the points on the dashed line are each exactly twice as far from the “x” axis as the same color point on the solid line. This is also true at the origin; since 2 \cdot 0=0, any point that is “on” the “x” axis will not move when dilated vertically. Therefore, there are two ways of describing the relationship between the two graphs. Either:

  • The solid line has been “dilated vertically by a factor of 2″ to produce the dashed line, or
  • The dashed line has been “dilated vertically by a factor of 0.5″ to produce the solid line.

Dilating by a negative factor will cause both a dilation and a reflection about the axis to occur (points that were on one side of the axis are now 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 is Great!

I finally figured out how to post GeoGebra applets on Google Sites so that I can link to them from WordPress.com (which does not support them).

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 ones for Linear functions:

Quadratic functions:

Exponential and Logarithmic functions:

Rational functions:

Trigonometric functions:

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

Why Quadratics?

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 Continue reading Why Quadratics?