4.1 More Complicated Functions

Differential calculus is about approximating more complicated functions by linear functions. We now address the question, what more complicated functions do we want to deal with?

Most all of the functions we will talk about can be formed by starting with three basic functions, and applying the operations of addition, subtraction, multiplication, division, inversion (like in going from the square to the square root) and substitution to copies of them.

We can define even more functions by using calculus, but these need not be investigated now.

The three basic functions are the identity function, the sine function and the exponential function. For the moment we will start with only the first, the identity function.

If we multiply copies of the identity function together, we get powers of it, like \(x * x\) (which is \(x\) squared), or \(x * x * x\), which is \(x\) cubed, and so on. Any function consisting of a positive power multiplied by a constant is called a monomial. If we add or subtract a finite number of these, we get what are called polynomials.

The simplest polynomials are the linear functions we have already mentioned. The next more complicated ones are quadratic functions; these have the form, \(ax^2 + bx + c\), where \(a, b\) and \(c\) are numbers. Cubic functions have a cube term in the, quartic functions a term like \(dx^4\), and so on.

We can evaluate and plot quadratic functions with very little more effort than we expended on linear functions. The only difference is that we should add a quadratic coefficient say in B6, and enter =B$6*A10*A10+B$2*A10+B$3 into B10 (and then copy this down column B.)

For example, try this, putting \(1\) in B6. After entering the instruction above in A10, you have to copy it into B11 through B500, and you can now plot any quadratic by changing your parameters.

When you do this you will find something that is sort of nice, all quadratics look more or less alike except that some are upside down.

That is, if you plot a quadratic and don't pay attention to the scales of your graph or which end is up, and where its peak or valley is, you cannot tell them apart. Quadratics with a given sign for the quadratic coefficient, are all alike except for scale and location of their high and low points.
A second nice fact about quadratics is that we know how to solve some equations of the form \(f(x) = 0\), when \(f\) is quadratic.

What equations are those?

Well, we know how to solve equation

\[x^2 = A\] which means the same thing as: \[x^2 - A = 0\]

when A is a positive number. We can solve them because a solution is, by definition, the square root of A.

Actually we define \(\sqrt{A}\) (also written as \(A^{\frac{1}{2}}\)) to be the positive number whose square is \(A\), when \(A\) is positive, and the two solutions to this equation are \(\sqrt{A}\) and \(-\sqrt{A}\).

By arithmetic manipulations you can reduce any quadratic to this solvable form, and solve it, and you will get the famous quadratic formula for solutions.

How is that and what is that?

The equation \(ax^2 + bx + c = 0\) can be rewritten (when \(a\) is not \(0\), after dividing by \(a\)) as

\[x^2 + \frac{bx}{a} + \frac{c}{a} = 0\]

which is the same as

\[(x + \frac{b}{2a})^2 = \frac{b^2-4ac}{4a^2}\]

Thus, the square root of the left hand side is plus or minus the square root of the right hand side here.

\[x + \frac{b}{2a} = \frac{\sqrt{(b^2 - 4ac)}}{2a}\]


\[x + \frac{b}{2a} = -\frac{\sqrt{b^2 - 4ac}}{2a}\]

This is a peculiar way to write the standard quadratic formula.

Exercise 4.1 Find two solutions to each of the following equations:

\[x^2 - 3x - 4 = 0\]

\[4x^2 - 3x - 1 = 0\]