How to Straighten a Curve

It is the nature of physicists to be able to see relationships between natural phenomena.  This is most clearly and simply done through a graphical analysis, where one quantity that changes will be plotted on the x-axis of a graph, and the y-axis will display the resulting change (or lack thereof) in another quantity.  However, these relationships can often be difficult to determine, and even harder to understand.  The most simple for of a graph is a linear equation, that is a straight line that passes through the y axis at some point.  These equations are extremely useful to physicists because the equation of the line can be determined very easily.  Simply take the slope of the line, multiply it by the changing value that is plotted along the x-axis, and add a constant term equal to the y-intercept.  Therefore, it is easy to see why physicists would like to transform more complex relations into much similar ones.  

The process of actual curve straightening is quite simple.  The theory is that we can modify the x-axis values in order to change the shape of the graph.  By doing enough changes (usually no more than two), we can successfully create a linear graph from which we can come to better understand the phenomena we are analyzing.  

There are two types of graphs that many physicists are often charged with straightening.  The first is a simple parabola, that is a graph that curves up exponentially.  The most simple parabola can be found by graphing the equation y = x squared.  In fact, every parabola involves something to an even power, usually the second power.  Because of this knowledge, a parabolic graph can be straightened by square rooting each of the x data values, and graphing them against the original corresponding y values.  This will result in a linear relationship that we as physicists can interpret, analyze, and apply.  

Sometimes a physicist might be presented with a hyperbolic-shaped graph.  This graph is equally easy to straighten.  The simple form of a hyperbola can be found by graphing the equation y = 1/x.  Because all hyperbolas are extensions of this function, we can modify our x-axis values by inverting them (to take an inverse, simply divide 1 by the value).  This will have the effect of straightening the curve of the graph.   

However, in more complex situations, a graph may be a combination of these two types of curves: an inverse square function.  This actually looks like a reflected parabola if one only examines the values for x and y that are greater than zero, and the basic equation for an inverse square is the equation y = 1/x squared.  In order to straighten a curve such as this, we must apply both techniques, although the identities of square roots permit us to perform them in either order.  Simply modify the x-axis values by taking the inverse and square root of each value.

Curve straightening is one of the most valuable techniques a physicist possesses.  In fact, this skill is so useful that it is often applies in other scientific disciplines such as chemistry an biology.  There are profound implications of curve straightening, as it allows us to understand the most mysterious and unnerving phenomena that are present in our physical universe.