Functions
It is helpful in visualizing problems to know what certain functions look like. The linear algebraic function (see Fig. I 8 ) is y = mx + b, where m is the slope of the straight line and b is the intercept, the point where the line crosses the x axis.
Fig. I 8
The next most complicated function is the quadratic (see Fig. I 9), and the simplest quadratic is y = x2, a curve of increasing slope symmetric about the y axis. Quadratics are also called parabolas. Adding a constant a in front of the x2 either sharpens (a > 1) or flattens (a < 1) the graph. Adding a constant to obtain y = ax2 + c serves to move the curve up or down the y axis. Adding a linear term, producing the most complicated quadratic, moves the curve up and down and sideways. If a quadratic is factorable then the places where it crosses the xaxis are obtained directly in factorable form. This discussion of parabolas is continued in the chapter on projectile motion.
With a little experience you should be able to look at a function y = x2 + 2x 8 (see Fig. I 9) and say that the x2 means it is a parabola, the coefficient of 1 means it has standard shape, and the other two terms mean that it is moved up and down and sideways. Factor to y = (x + 4)(x 2), and the curve crosses the x axis at x = 2 and x = 4. Because it is a parabola the curve is symmetric about x = 1.
Cubic curves have the general shape shown in Fig. I 10. Adding a constant term moves the curve up or down the y axis. A negative in front of the x3 term produces a mirror image about the x axis. Quadratic and linear terms in a cubic produce peaks and troughs in the curve.
Logarithms and Exponents
Logarithms and exponents are used to describe several physical phenomena. The exponential function y = ax is a unique one with the general shape shown in Fig. I 11.
This exponential equation y = ax cannot be solved for x using normal algebraic techniques. The solution to y = ax is one of the definitions of the logarithmic function: x = logay.
The language of exponents and logarithms is much the same. In exponential functions we say “a is the base raised to the power x.” In logarithm functions we say “x is the logarithm to the base a of y.” The laws for the manipulation of exponents and logarithms are similar. The manipulative rules for exponents and logarithms are summarized in the box.
The term “log” is usually used to mean logarithms to the base 10, while “In” is used to mean logarithms to the base e. The terms “natural” (for base e) and “common” (for base 10) are frequently used.
