A linear function is a function of the form f(x) = mx + b whose graph is a straight line. When written in this form, the slope of the graph is m and the y-intercept is b. Except for constant functions, linear functions are the simplest of all mathematical functions. This is because the largest exponent of the variable x is 1 in a linear function. So a linear function is sometimes said to be a polynomial function of degree 1. In general, polynomial functions that have degree higher than 1 have more complicated graphs and require a higher level of mathematical analysis than do linear functions. The simplest linear functions are those which pass through the origin. They have y-intercept equal to 0, so their equations take the form f(x) = mx, or, equivalently, y = mx. The slope m in this equation is calculated as follows: Pick two points on the line and calculate the difference in values of the y-coordinates of those points and divide it by the difference in the x-coordinates of the points. This calculation is sometimes called "rise over run" or "rise divided by run." In any case, the slope measures the rate of change of the y-coordinates as one moves from left to right along the graph. Essentially, the absolute value of the slope measures the "steepness" of the line. The bigger the absolute value of the slope, the steeper the line. Note also that if the slope is positive, then the line "rises" from left to right; if the slope is negative, then the line "falls" from left to right; and if the slope is 0, then the line is horizontal.
Because of their relative simplicity, linear functions are often the first choice of a mathematician or scientist for modeling data collected from real world situations. If a linear model is a good fit for the data, then the analysis of the real world situation is likely to be much simpler than if the model has higher degree. Thus statisticians have developed a carefully worked out theory for deciding when a linear function is a "good enough" fit for a data set. The theory is called linear regression and allows the statistician to determine the slope and the y-intercept of the so-called "line of best fit." It also provides for the calculation of a number called the "correlation coefficient" that tells how close this best fit line actually comes to "capturing" the trend of the data. Therefore, this line of best fit is also sometimes called a "trend line." In some cases, even when a scatter plot of a data set does not look linear, it is possible to use mathematical transformations to "linearize" the data. Then the techniques of linear regression can be used with the linearized model to determine the form of the best fit function for the original data. This, for example, is the theory behind semilogarithmic graph paper or the "exponential regression" feature found on modern graphing calculators. If the raw data appears to be more exponential than linear, a plot of first coordinates against the logarithms of the second coordinates will appear more linear. We say that a logarithmic transformation has been done on the second coordinates of the original data set. Then linear regression is carried out on the transformed data set, which allows the statistician to arrive at an appropriate exponential model for the original data. A graphing calculator with an exponential regression feature does all of this automatically, returning the best fit exponential function model within a few seconds. Similar techniques may be carried out on non-linear data which appear to be logarithmic, power, higher degree polynomial, or trigonometric. The point is that the basis of all these different types of regression analysis is linear regression.
There are many other examples of the use of linear functions to approximate non-linear functions in mathematics. For instance, to approximate the value of a non-linear function at a given point on its curve, one can use the linear function which is tangent to the curve at a nearby known point. The techniques of calculus can be used to determine this tangent line function and so long as the point of tangency is "close" to the point whose second coordinate is being approximated, this approximation should be quite good. In fact, it can be shown that the tangent line to a curve at a point is the best linear approximator to the curve at that point. The repeated use of this technique is the basis for Euler's method for numerically solving differential equations whose solution curves are non-linear.
In summary, although linear functions are themselves relatively simple to analyze, that simplicity has made them the basis for analysis of much more complicated non-linear phenomena. Thus linear functions are arguably among the most important functions in mathematics.
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