## SAT Physics Conventions and Graphing - Slope and Area

GRAPHING VARIABLES
The graphing techniques of mathematics are used in science to compare dependent and independent variables. In mathematics, you are familiar with the traditional x- and y-coordinate axes. In science, the x-axis represents the independent variable and the y-axis represents the dependent variable. The value of the dependent variable depends upon the independent variable.
Graphs are always tided so that the dependent variable is listed first, and the independent variable is listed second. As an example, a position versus time graph would have position (dependent variable) plotted on the y-axis and time (independent variable) plotted on the x-axis.

Slope
Slopes are very important and are often the key to answering many of the graphing questions on the SAT Subject Test in Physics. Slope is determined by dividing the rise (y-axis value) by the run (x-axis value). The trick is to look at the units written on the axes of the graph. If you divide these units, you can easily identify the significance of the slope. (A) What is the value and significance of the slope in the time interval from 0 to 3 seconds?

WHAT'S THE TRICK?

Determining the slope is simply a matter of dividing the rise (y-axis values) by the run (x-axis values). The significance of the slope is determined by examining the resulting units. The resulting units, meters per second (m/s), are the units of velocity. Therefore, the slope of the position versus time graph is equal to velocity. During the first 3 seconds, the object has a velocity of 5 m/s.

Slope of a Graphed Function (continued)
(B) What is the value and significance of the slope in the time interval from 3 to 5 seconds?
WHAT'S THE TRICK?
The slope in the interval between 3 and 5 seconds is zero. During this time interval, the object has a velocity of zero and the y-axis value (position) is not changing. The object’s position remains constant at a location 15 m from the origin.

Area
The area formed by the boundary between the x-axis and the line of a graph is also very useful. Areas are calculated by multiplying the height (y-axis value) by the base (x-axis value).
In problems where the area forms a triangle, the area is found with ½height x base. In cases where the line of the graph is below the x-axis, the area is negative. See Figure 1.1. As with slope, you can easily determine the significance of the area. By multiplying the units written on the axes of the graph and then looking at the resulting units, you can quickly determine the significance of the area. What is the value and significance of the area of the graph during the time interval between 0 and 10 seconds?
WHAT'S THE TRICK?
Determine the area, and examine the resulting units.
area = height x base = (10 m/s)(10 s) = 100 m
Meters (m) are the units of displacement. The area under a speed versus time graph is therefore the displacement of the object during that time interval. The object graphed above traveled 100 m in 10 seconds.
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