Mathematics
Stopping Distances

How to Interpret a Graph



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Stopping Distances
Ludovic Harold Tesla's image for:
"How to Interpret a Graph"
Caption: Stopping Distances
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Image by: sundayeveningmath
©  http://2.bp.blogspot.com/-k2cQBTwKY4Y/TZiIBVWDVCI/AAAAAAAAABc/rjzTiFgjqcU/s1600/winter-stopping-distances.gif

In mathematics there are many forms of graphs; but every graph has many things in common. A graph is basically an x-axis a y-axis and numerous coordinate points (x,y) somewhere on the graph. Most of mathematics works with functions and not just points; but every function can be broken down into separate lines comprised of an infinite amount of infinitesimally small points. Graphs are quite simple to read but when one really understands what a graph represents they can drastically improve their understanding of any function that is graphed. To start we will create a graph and describe all of its parts. If you notice the image preceding this article you will see a graph of speed vs. stopping distance.

X-Axis: This is the horizontal line you notice along the bottom of the graph. It is labeled “Speed” and has a unit of mph or miles/hours. In general the x-axis is the independent variable; this means that this is the data we collect and is what causes/affects the y-axis variable. This means that our travelling speed is what determines how long it will take us how to stop and not the other way around.

Y-Axis: This is the vertical line you notice along the left of the graph. It is labeled “Stopping Distance” and has a unit of meters. This data is generally known as the dependant variable because it’s value depends on what x is. This data can also be collected but is usually the process of finding it is using the x-value to calculate the y-value.

Function:

The function is the line you see in the box created by the adjoining x-axis and y-axis. Looking at this we can find how our data actually represents the real world application. To create a function one would collect data and graph its points with coordinates (x,y) for example (1,5),(2,7),(3,9). Then from our points we would create a function that would accurately represent our points. From our 3 points earlier we would create a line y=2x+3. This is because (5)=2(1)+3, (7)=2(2)+3, and (9)=2(3)+3. From these points we than can extrapolate our function and find the y that corresponds to an x-value that we did not test for example (5,?) y=2(5)+3=13 so our point would be (5,13).

We can actually find the values of a graph by just interpreting it even without a function. To do this we would draw a vertical line from the x-value we were curious about until it hits the line/curve of our function. Then from that point we would draw a horizontal line to the left until it hits the y-axis and that value on the y-axis would be the corresponding y-value to our x-value.

Key:

Most graphs do not have a key but some do just so they can describe multiple situations on one graph. The key in this graph is the box that says “dry asphalt” and “black ice”. To read this we would notice that blue indicates the dry asphalt and red indicates black ice. This means that the red line is the speed and stopping distance for cars on black ice; while the blue line is for cars on dry asphalt.

From all of these parts of a graph we can read any graph and determine everything about it. Even with no background information about the graph we can make observations about the function and know what it means. Just from reading the line on our example graph we can notice that the faster we travel in a vehicle the longer distance we travel when we try to stop; and it takes longer to stop on black ice than dry asphalt even when travelling the same speed.


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