Analyzing+Graphs


 * Analyzing graphs **


 * Introduction **

A graph is a way to represent the relationship between two variables by use of data points. In physics, it is important to be able to read and interpret the data that graphs display.

To start with, a graph is composed of two axes (pronounced 'ack-sees'), labeled x and y. The x axis normally represents the independent variable (that is, what you are controlling) while the y axis represents the dependent variable (the result of the change in the independent variable). This type of graph is aptly referred to as an “Independent versus Dependent” graph.

The standard form of a graph, as taught in math class is This generally applies to physics, but with a few variations.

There are several types of graphs, and it is important to know how to analyze all of them.


 * Types of Graphs **

These graphs generally fall into one of seven categories: linear, radical, inverse, quadratic, exponential, sinusoidal, and absolute value.

**Linear** graphs have a direct relationship, meaning that for every change made to the Independent Variable (from here on referred to as “x”) a proportional change is made to the Dependent Variable (hereafter referred to as “y”). These graphs have a straight line, and are very simple to graph.

The general form for a linear graph is

The graph will usually look like this, with a few slight variations with regards to extra variables.



**Quadratic** graphs occur when the “y” is the result of squaring the “x”. These graphs have a changing slope, slowly becoming steeper as they go, making them harder to graph.

These graphs have the general equation

The graphs will look like this



An **inverse** graph occurs when the “y” decreases as the “x” increases. These graphs can also be linear—in which case they are referred to as “inverse linear” graphs, and have a negative, but straight, slope. They can also be quadratic, or “inverse quadratic”. This is a very loose term for a graph, and really just refers to the graph being negative.

The standard form for this graph is Inverse Linear graphs will look like this, but the general trend of reversing the graph will remain true for all variations of Inverse graphs.



**Radical** graphs (also referred to as "Inverse Quadratic" graphs) are the result of taking the root of the “x” to get the “y”. These graphs are somewhat similar to quadratic graphs, but turned onto their sides. Sometimes, the “x” and “y” axes are switched to make the graph. They begin steeply, but soon even out.

These graphs are based on the equation



They generally look like this



**Sinusoidal** graphs are the representation of either the sine or the cosine of the "x". These graphs are drawn as waves. The sine and cosine graphs begin in different places, but otherwise look very much the same.

Sinusoidal refers to both Sine and Cosine, which will look like this-- notice that the cosine graph is just shifted back one half wavelength from the sine graph.








 * Absolute Value** graphs do not ever have negative "y" values. They have a "V" shape to them, and reflect the positive line over the "y" axis.

These graphs are seen with two different notations or

And they look like this




 * Other Useful Information **

It is important to remember to take into account the scale of the graph. Both when trying to interpret one and when trying to make one. When reading a graph, remember that the scale will not always be one, be sure to check the scale before doing anything else. When making a graph, remember to pick a scale and stick to it. If you use numbers like 14, 78, 100... and so on, you will only confuse those who are attempting to read your graph and they will not be able to interpret your data. Keep in mind that the purpose of graphs is to communicate data.

Another important thing in physics in being able to interpret graphs is knowing how to use slope and area. The slope of a line is really a ratio, the result of "y" divided by "x". Less known is the information that the area under a graph is the product of "x" multiplied by "y". This knowledge will be very helpful as you try to analyze graphs.


 * References **

http://homepage.mac.com/astronomyteacher/dvhs/grapheq.html