Why Use Graphs ?
It has once been said that a picture is worth a thousand words. This is very true in science. In science we deal with numbers, some times a great many numbers. These numbers, or measurements, are usually data taken during a scientific experiment. A graph helps scientists to look at a large set of data and determine how the measurements are related to each other.
Lets look at the following set of data taken from an experiment to determine the effect of sleeping on test grades. This data was taken from a group of 20 students who recorded the amount of time they slept each night during one week and their test score in the class at the end of that week. The amount of time sleeping that is recorded in our table is the average for the week.
Time Sleeping
(Hrs)
Test
Grade
Time Sleeping
(Hrs)
Test
Grade
6 75 4 62
7 76 8 88
3 61 3 71
7 92 5 82
8 91 6 85
3 65 6 84
9 95 5 98
5 78 9 94
1
12
45
75
13
1
86
22
It would be difficult, but not impossible, to state any good conclusions from this set of data.
Imagine if there were a thousand sets of data to look at. You could get a very bad headache and a very poor conclusion by just looking at a table of data. A graph allows us to draw a picture of this data so we can see the trends better. For instance does our data show that more sleep means better test grades or worse test grades ?
So how do you draw a good scientific graph ? By drawing a complete graph of course.

Drawing a Graph
A complete graph has many parts to it, all of which are important. We will draw a complete graph using our data as we discuss the steps for drawing a complete graph.
Step 1 : Setting up the axes.
All graphs that we do in this course will have two axes (horizontal and vertical). This will not always be the case. a) Which variable goes on which axis. In science the independent variable , that is b) c) the variable which is changed by the person doing the experimenting, goes on the horizontal (x-axis). The
An appropriate scale needs to be shown on each axis. The spacing needs to be like a ruler, but you don’t always need to start at zero and the scales for each axis can be different. dependent variable , that is the variable which changes in the experiment when the independent variable is changed, goes on the vertical axis (y-axis).
1
For our experiment the time sleeping would be the independent variable and the test score would be the dependent variable.
Each axis will need to be labeled with its name and the units of measurement.
For our experiment the Time Sleeping axis will go from 0 to 15 hours and I will count by 1. The test score axis will go from 0 to 100 and I will count by 10.
Our graph will look like this at the end of step 1 :
1 In a table, the first column is usually the independent variable.

Step 2 : Plotting the points
We now make a dot for each data point. Sometimes you just have to estimate where to put the dot. Notice, we do not connect the dots.
At this point the graph is really beginning to show us a trend. Do you see it ? An important note to make here : You may or may not like the conclusion this data gives. You have to accept the conclusion based on this data though, because your data is your data.
After this step our graph looks like this, but it is not complete.
Step 3 : A Curve of Best Fit
Remember we are looking for a trend in this data, and if possible we would like to write a mathematical equation to model this data and to make predictions from our experiment. To do this we fit a curve of best fit to this data. It won’t necessarily go through all of the data points, but it will show you a trend to the data. We will consider three types of relationships (although there are many more)
Type 1: A Linear Relationship
This type of relationship shows that the two variables change together (this is sometimes called a direct variation). More precisely though, these two variables change at a constant rate called its constant of variation also known as its slope.
15

The General Equation :
The Graph : y = kx or y = mx +b (k and m are the slope)
Type 2: A Quadratic Relationship
Line
This type of relationship also shows that the two variables change together, but that the dependent variable changes faster than the independent variable. More precisely it changes with the square of the independent variable.
The General Equation : y = kx
2
or y = ax
2
+bx +c
Type 3:
The Graph :
An Inverse Relationship
Parabola
This type of relationship shows that the variables change in opposite directions.
For example, if one variable increases the other decreases. This is sometimes called an indirect variation.
The General Equation : y = k/x
The Graph :
Hyperbola
There are many other types of relationships. We can have cubic relationships, inverse square relationships, sine and cosine relationships, …, but for our data I think a linear relationship fits it best so I will draw in a line of best fit. To do this there is no real good way, just use a ruler and try and get as many points above the line as below it. No two people will get exactly the same line this way.

Our graph now looks like this :
It is still not finished. We need a title.
Step 4 : The Title
No graph is complete without a title. Now when we talk about a title in science we mean a descriptive title not a literary title, save that for English class. Titling this graph “A Glorious Relationship Between Most Excellent Test Scores and the
Amount of Blissful Sleep One Has Received.” would not be an appropriate title.
A quick way to title a graph is the write it as Dependent Variable vs Independent
Variable. Yes it is boring but it does the job.
Our title will be “ Test Score vs Sleep Time”
Our complete graph now looks like this :

Analyzing this Graph
What does this graph tell you. If you are honest it tells you the trend is that people who get more sleep seem to get higher test scores. Once again if you are honest, this seems logical. Also notice that not all the data points exhibit this trend, it is only all the points taken together.
We can also look at our line of best fit and predict from it. For instance it seems that if I sleep 8 hours I should be able to score 100 on the test. I do need to caution you at this point, because the amount of sleep is not the only thing which affects test scores, and we need to be aware of this as we analyze our graph.
Finally we can write an equation for our line of best fit, to do this we need to find the slope and the y - intercept.
The y-intercept : Reading off the graph I would say our line of best fit crosses the yaxis at a test score of 33. This will be my y-intercept. What does this physically mean? It is the score you could expect to receive if you didn’t sleep at all the night before. Remember, this is only a prediction based on the trend of our data, it should not be treated as anything more than this.
The Slope : I need to find two points on my line of best fit (not just two data points). I choose (2,49) & (8, 100). See the graph below. I then apply the slope formula I learned in Algebra class. m  y
2 x
2
 y
1
 x
1

100  49
8  2
 8.5
.
What does this mean? It means you can expect to raise your test
 score by 8.5 points forevery additional hour of sleep you get the night before. Again remember, this is only a prediction based on the trend of our data, it should not be treated as anything more than this.

So my equation is y = 8.5x + 33. However, using y & x for my variables isn’t very descriptive.
It is ususally a good idea to use variables that describe your data. As an example, since my dependent variable is “Test Score” I might want to us an S for my dependent variable (for Score) rather than y. I might use t for my independent variable (for time) rather than x. My equation could then be written as S = 8.5t + 33. Now I can use this equation to predict the test scores you can get if you know the number of hours you sleep. Try it.
Graphing Calculators
It is acceptable to use a graphing calculator to determine the slope, y-intercept and the equation of the line of best fit for your data. However if you do this it is a good idea to mention this when you are expected to show your work. “The equation for the line of best fit was found using a TI-84+ graphing calculator.” would be an appropriate way to do this.
There is a catch though. If you are required to graph your data and draw your line of best fit, then the line of best fit you draw should match up with the equation you give. For example, if your equation is F = 3.5m + 12, then the line you draw for this should have a y-intercept of 12 and a slope of 3.5. If the line you draw has a y-intercept of 20 and a slope of 1, then this doesn’t match and you will lose points.
Linearizing Data (New Stuff)
Strange as it may seem, even when we have data that is not linear, we can make it linear by choosing to graph different variables. This is usually done when you know what the theoretical equation for the relationship is and you want to use the slope of your line of best fit to find the value of some constant or other variable.
Kinetic Energy Example
Normal Graph Linearized Graph The equation & data are:
𝐊 =
𝟏
𝟐 𝐦𝐯 𝟐 v
(m/s)
0
10
20
30 v 2
(m 2 /s 2 )
0
K (J)
0
100 1250
400 5000
900 11250
40 1600 20000
50 2500 31250 Graphing the in our normal way we get a quadratic relationship that any graphing calculator can give you the equation. This equation then can be used to find m – which turns out to be the mass.
If we instead graph K vs. v 2 , we get a nice linear relationship. Any linear relationship has the form y =
(slope)x, and since K plays the part of y and v 2 plays the part of x, whats left is the slope which is
1
2 𝑚 . This can be used to find m.

The equation & data are: 𝐚 =
𝐅 𝒎 m
(kg)
5
1/m
(kg -1 ) a
(m/s 2 )
0.200 400.00
10 0.100 200.00
20 0.050 100.00
30 0.033 66.67
40 0.025 50.00
50 0.020 40.00
Acceleration Example
Normal Graph
Graphing the in our normal way we get an inverse relationship that most graphing calculators can give you the equation. This equation then can be used to find F – which turns out to be the force.
Linearized Graph
If we instead graph a vs. 1/m, we get a nice linear relationship. Any linear relationship has the form y =
(slope)x, and since a plays the part of y and 1/m plays the part of x, whats left is the slope which is F.
Area Under the Curve (New Stuff)
When you are analyzing a graph the slope tells you the rate of change of the dependent variable with respect to the independent variable. This can be very useful. Another way to analyze a graph though is to look at the area under the curve (actually the name is deceiving. We mean the area between the graph and the horizontal axis.). This area sometimes gives you another physical quantity, gives you the continuous sum of all of the values of the dependent variable over a certain interval of the independent variable, or can be used to find the average value of the dependent variable over some interval of the independent variable. Understanding why this is requires some knowledge of calculus, so you are just going to have to trust me, but if you really want to know, I’ll be glad to enlighten you.
In AP Physics B, we will calculate the area under a curve by breaking the area up into smaller areas consisting of known shapes and then adding these areas together. The shapes we are going to use are the rectangle (A=lw), and the triangle (A=0.5bh). The only catch is that areas below thw horizontal axis are negative. {Negative area – very interesting.}

Impulse Example
Below is a graph of the Force vs. Time on an object over a 10-s time interval.
Find the total area under the curve:
A = A
1
+ A
2
+ A
3
- A
4
= 0.5(2)(5) + (4)(5) + 0.5(1)(5) – 0.5(3)(2) = 24.5-N•s
It turns out that this area is the impulse which is also the change in momentum of the object.
Find the average value of the force over the entire 10-s time interval:
F avg
= A/10 = 24.5/10 = 2.45-N