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Linear Extrapolation Method

Science advances through the carrying out of experiments and collection of data. Often it is only possible to collect data within certain limits of the independent variable. If there is a linear dependence of the dependent variable (measured quantity) with the independent variable (varied quantity), then it is possible to do a linear extrapolation to find values of the dependent variable outside of the measured range.

Limits of Experiments
- Linear relationships are common in science, and are the simplest type of graph that can be obtained. Often an experiment will be carried out which is limited by the equipment setup. For example the measurement of temperature with changing pressure is limited by the range of pressures that can be controlled and also by the range of temperature that can be measured. This can result in a set of data points over a limited range of the parameter space. When this occurs, a linear extrapolation can find the value of the dependent variable at a point on the graph that could not be directly measured.
Gradient
- The first process in carrying out a linear extrapolation is the determination of a straight line equation that corresponds to the data. To determine the straight-line equation, two points on the graph are needed. Its usually best to go for the lowest point and highest point to get an average gradient. The gradient of the straight-line is calculated from the equation: Gradient = Difference in y / Difference in x
  
  For example, if the two points on the graph are (1,1) and (5,5) then the gradient is:
  
  gradient = 5 - 1 / 5 - 1 = 1
y-Intercept
- Once you have the gradient, the equation of the straight line can be obtained through substitution.The equation of a straight line is: y=mx + c. The gradient is m and c is the y-intercept. Following the example, m=1 so the equation so far is: y=x + c. The value of c can be obtained by substituting one of the points into the equation: Using the point (5,5): 5=5+c therefore c=0. The equation of the straight line in this case is y=x
Linear Extrapolation
- Once the equation of the straight line has been obtained, the linear extrapolation can be carried out. Simply determine the point on the x-axis that the value of y is needed, and plug this value into the equation of the straight line to obtain the answer. Following the example, if the value of y is needed for x= 1000:
  
  y=x=1000

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