3.2 Linear Functions

The basic fundamental function, the one that calculus is based upon, is the linear function. A linear function is a function whose graph consists of segments of one straight line throughout its domain.

Such a line is, you may remember, determined by any two points on it, say \((a, f(a)), (b, f(b))\). Thus, you can pick any \(a\) and any \(b\) in its domain and determine the line from the two values, \(f(a)\) and \(f(b)\).

What is a formula for such a function?

We can determine the linear function which takes value \(f(a)\) at \(a\) and \(f(b)\) at \(b\) by the following formula:

\[f(x) = f(a)\frac{x-b}{a-b} + f(b)\frac{x-a}{b-a}\]

The first term is \(0\) when \(x\) is \(b\) and is \(f(a)\) when \(x\) is \(a\), while the second term is \(0\) when \(x\) is \(a\) and is \(f(b)\) when \(x\) is \(b\). The sum of the two is therefore \(f(a)\) when \(x\) is \(a\) and \(f(b)\) when \(x\) is \(b\). And it is a linear function. Linear functions have a term that is \(x\) multiplied by some constant, and may also have a constant term as well.

A more convenient and suggestive form for this function can be gotten by putting the x terms together:

\[f(x) = mx + c = \frac{f(b) - f(a)}{b-a}x + \frac{f(a)b - f(b)a}{b-a}\]

The number \(m\) which occurs here is called the slope of this line. Notice that \(m\) is given by the ratio of the change of \(f\) between \(x = b\) and \(x = a\) to the change in \(x\) between these two arguments:

\[m = \frac{f(b) - f(a)}{b-a}\]

If \(f\) is plotted, where \(f(x)\) meets the \(y\) axis is what we call \(c\) here. It is called the y-intercept of this line, which is the value of \(y\) when \(x\) is \(0\).

There is a mathlet here which allows you to vary the slope \(m\) and y-intercept c and see what that does to a line. You should fiddle with this mathlet and from it get an idea what the slope \(m\) tells you about the line. Using it you can construct your own examples.

You can actually construct a spreadsheet that can do the same thing as this applet. You would be wise to do so. Directions on exactly how can be reached by clicking here.

I know all this stuff. Why do you waste my time with it ?

All this may sound simple to you, but if you understand it, you are well on your way to understanding calculus. Realize that calculus consists of studying functions through studying the slopes of the straight lines they resemble near any given argument. Here are some exercises to help you get used to these things.


3.3 Play with the applet until you get a feel for the geometric meaning of the slope of a line. Then take a piece of paper, draw x and y axes on it and put scales on them, and have a friend draw some straight lines on the paper. Without measuring, guess the slopes of the lines. Now measure the lines (change in y over change in \(x\)) and see how good your guesses were.

3.4 When is the slope of a line negative? When is it \(0\)? When is it \(1\)? When \(-1\)? If you use the same scale for \(x\) and \(y\), what does slope 10 look like? How about slope \(-\frac{1}{10}\)?

3.5 Follow the directions that you can get to above to construct a spreadsheet that can work as the applet here. Try it with the various slopes of the last question.

3.6 Construct the linear function, \(g\), with slope 2 satisfying \(g(1) = 1\); graph it. What is \(g(4)\)? Do the same for the linear function, \(h\), which satisfies \(h(1) = 4\), \(h(4) = 12\). What is the slope of \(h\)?