Graphing – MathMaine

Linear Systems: Why Does Linear Combination Work (Graphically)?

A system of linear equations consists of multiple linear equations. You can think of this as multiple lines graphed on one coordinate plane. Three situations can arise when looking at such a graph. Either:
1) No point(s) are shared by all lines shown
2) There is one point that all lines cross through
3) The lines lie on top of one another, so there are many points that the lines have in common.

There are four commonly used tools for solving linear systems:
– graphing
– substitution
– linear combination
– matrices
Each has its own advantages and disadvantages in various situations, however I often used to wonder 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.

Consider the linear system:

$\begin{cases}y=-3x+2\\y=x-6\end{cases}$

which, when graphed, looks like:

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.

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).

Interactive Graphs for Exponential and Logarithmic Functions

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…).

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

Quadratic functions:

GeoGebraBook of Quadratic Applets, which contains

Exponential and Logarithmic functions:

GeoGebraBook of Exponential and Logarithmic Applets, which contains

Rational functions:

Interactive Rational Function Graph

Trigonometric functions:

GeoGebraBook for Exploring Trig Functions, which contains

Unit Circle Symmetries:

GeoGebraBook of Unit Circle Symmetry applets, which contains:

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

Analyzing Linear Equations: a summary

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.

Lines

What is the least amount of information you need to know in order to be able to identify a line exactly? Two pieces of information: a point that the line passes through, as well as either a second point on the line or the line’s slope.