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Below you will find information about three types of relationships commonly found in the study of science including physics. Before you read and study this material, you may want to review the information about plotting points on graphs and displaying and interpreting data on graphs. The link below will take you to this information.

The Basics of Graphing and Displaying Data

Direct Variation: Direct Proportionality / Linear Relationship / y a x

Inverse Variation: Inverse Proportionality / y a 1/x

Parabolic Relationship: y a x^2 .

Direct Variation: Also known as a linear relationship, it involves a relationship between two variables defined or described by a straight line. As the independent variable is increased the dependent variable increases proportionally (and vis-a-vis). The variables' values are said to be proportional to one another. The "generic" Equation defining this type of relationship or function is y = mx + b. The variable value m is equal to the slope of the line. The variable value b is the value of the y intercept, where the line graph crosses the y axis.

Below is an example of a position-time data for a situation where an object is moving along with a constant velocity. Examine the data table and then the graph of the data. A scatter plot is being used to visualize the data in graphical form. You may connect the dots plotted to see the continuous function nature of the relationship and to do interpolation. (Look up scatter plot and interpolation, if you are unfamiliar with the concepts.)

Table of Position vs Time data
Time (s)	Position (m)
0.0	0.0
1.0	10.0
2.0	20.0
3.0	30.0
4.0	40.0
5.0	50.0

Position vs Time Graph - Direct Variation; Linear

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Inverse Variation: Also known as a hyperbolic relationship, it involves a relationship between two variables defined or described by a hyperbola. As the independent variable is increased the dependent variable does the opposite. The variables' values are said to be inversely proportional to one another. The generic equation defining this type of relationship or function is k = x y, where k is a numerical constant that remains the same as x and y change respectively. In a single graph of a a hyperbola the mathematical product of x and y remains a numerical constant represented by the symbol k.

Below is an example of a volume vs pressure data set for a situation where a specified volume of gas is being compressed by a defined amount of increase in the pressure being applied to a moveable piston pressing downwards on the gas. Examine the data table and then the graph of the data. A scatter plot is being used to visualize the data in graphical form. You may connect the dots plotted to see the continuous function nature of the relationship and to do interpolation. (Look up scatter plot and interpolation, if you are unfamiliar with the concepts.)

Table of Volume vs Pressure data
Volume (ml)	Pressure (pascals)
22,400	1
11,200	2
5,600	4
2,800	8
1,400	16

Volume vs Pressure - Inverse Variation; Hyperbolic curve

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Parabolic Relationship: This type of relationship involves two variables that have a more complex relationship than either the linear or hyperbolic functions described above. The value of the dependent variable is proportional to the square of the independent variable. The generic equation defining this type of relationship or function is y = kx^2, where k is a numerical constant that remains the same as x and y change respectively.

Below is an example of a time vs position data set for a situation where an object is being accelerated over a short time interval. Examine the data table and then the graph of the data. A scatter plot is being used to visualize the data in graphical form. You may connect the dots plotted to see the continuous function nature of the relationship and to do interpolation. (Look up scatter plot and interpolation, if you are unfamiliar with the concepts.)

Note: Look particularly at the second graph below that represents the data table. It shows that there is indeed a direct variation between the position of an object being accelerated and the time over which it has been accelerated squared.

Table of Position vs Time
Time (s)	Position (m)
0.0	0.0
1.0	4.9
2.0	19.6
3.0	44.1
4.0	78.4
5.0	122.5

Position vs Time - Parabolic Relationship; Parabolic curve

Position vs Time^2 - Parabolic Relationship; Parabolic curve

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