A system of linear equations consists of multiple linear equations. The solution to a linear system, if one exists, is usually the point that all of the equations have in common. Occasionally, the solution will be a set of points.
There are four commonly used tools for solving linear systems: graphing, substitution, linear combination, and matrices. Each has its own advantages and disadvantages in various situations, however I often wondered about why the linear combination approach works. My earlier post explains why it works from an algebraic perspective. This post will try to explain why it works from a graphical perspective.
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:
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
A few more interactive GeoGebra applets have been added to my collection.
Each of these graphs the function indicated in the name, with parameters that you can adjust using sliders. As you move a slider, you can watch how that parameter affects the graph of the function, and see what the resulting function definition looks like.
Each of these pages also has a set of questions following the grapher that are intended to lead students through the process of playing with and considering the effect of each constant on the graph (without giving away too much, hopefully…).
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:
Towards the end of the unit(s) on Linear Equations and their graphs, students can feel a bit overwhelmed. The following is an attempt to summarize and link the key concepts you need to be comfortable with.