Using your Graph to interpolate, extrapolate and calculate slope.

- D. Gilliland, Honors Physical Science @ Sarasota High -

Interpolate
Extrapolate
Slope

OK - so you made a graph. Now let's see how you can interpolate, extrapolate and calculate the slope of the linear curve to obtain information not available on our data table.

But before we do let's make sure we understand a few things.

A scientific graph shows the relationship between the independent variable (the variable you changed in the experiment) and the dependent variable (the change that that was produced in the experiment). When graphing your data remember to place your independent variable on the x-axis and the dependent variable on the y-axis.

 

Once you have graphed your data you can use your graph to find the values of points on the graph that you did not test in your experiment.

Suppose a scientist dropped an object and measured its speed at various times while it fell. After collecting her data she graphed it.

You can see that time is the independent variable (controlled by the experimenter), speed is the dependent variable (speed depends on the time the object has fallen) and the curve is linear. This graph shows that speed and time in free fall are directly proportional - if you increase one the other also increases by a set amount.

But the graph now allows you to find the speed of a free falling objects at times other than what the experimenter collected. For instance, the free falling speed at 0.250 sec. was not part of the data collected. But you can easily determine that speed of the falling object at 0,250 sec. with your graph.

To find the speed at 0.250 sec. you would:

  1. Go to 0.250 sec. on the x-axis.
  2. Come straight up on the graph until you reach the curve.
  3. Come straight across to find the speed.

You would find that the free falling object in 0.250 sec would attain a speed of 3.00 m/sec.

Top of Page

Suppose you were asked to tell how long it would take a free falling object to obtain a speed of 2.50 m/sec. How would you do it?

You would solve this problem by:

  1. Finding the speed 2.50 m/sec. on your y-axis.
  2. Go across the graph until you hit your curve.
  3. Go straight down from the curve to find your time.

You would find that the free falling object would obtain a speed of 2.50 m/sec. in 0.193 sec.

In both of these above cases you have used interpolation (inter-po-lay-shun) - finding the value of a point that lies within the range of your data points - to answer the questions. You can see that your lowest speed collected was 0.900 m/sec while the highest speed collected was 3.80 m/sec. To find any speed between those two values requires interpolation.

But to find the time it would take for the falling body to attain a speed of 4.00 m/sec would require extrapolation (ex-strap-o-lay-shun) - finding the value of a point that lies outside the range of your data points. In this case it would take 0.350 seconds to attain a speed of 4.00 m/sec.

With a linear graph we can calculate how much our dependent variable changes when we change our independent variable by one. This is called slope and is obviously very important to a scientist. (You cannot calculate the slope of a parabola or hyperbola).

Top of Page


The slope of a curve has the following formula:

Let's use the slope of our curve in the speed vs. time graph for our falling object.

While it's nice being able to find the speed of a falling body at several times, it would be better to be able to calculate how much the speed changes every second it falls. Once we know this we can calculate the speed at any time - whether it is on our graph or not. The slope of our curve will give us this.

To determine the slope of your curve follow these steps:

  1. Find two values on your x-axis and subtract them to find the difference between them (including your units). Select two values that are (a) both on a line since there is less error than if you were to estimate a value between the lines and (b) are fairly far apart. If you are off a little in your estimation when the values have a great difference it won't have as large an error on your slope as if they were close to each other.(A-B in the diagram below)
  2. Go to each of the two values you chose on your x-axis and find the corresponding value on the y-axis. To do this go to your value on the x-axis, go straight up to the curve then straight across to the y-axis scale - just like you did at the top of this program.
  3. Subtract the two y-axis values to find the difference between them (include your units in your work).(C-D in the diagram below)
  4. Divide the y-axis difference by the x-axis difference (include your units).
  5. Be sure to include your units in your slope to give it meaning. What does 3.5/1 mean? Not much. Now what does 3.5 pizzas/ 1 person mean? A big stomach ache! Units give numbers a meaning. Remember we are dealing with measurements in science - not numbers.

You now have the slope of your curve. The slope of the graph of a falling object is shown below.

 

The slope of this curve tells you that according to the data collected, falling objects increase their speed by 9.60 m/s every second they are falling. With the slope you can easily calculate the speed of a falling object at any time. This is the importance of slope - it shows how the dependent variable changes when you change the independent variable by one.

Should you have any further questions on this please post them on the message board or ask your teacher.

Top of Page

NetTutor------------------------main menu

2000, Doug Gilliland - The Physical Science Series