SECTION 1.9: Proportionality, Power Functions, and Polynomials

When a quantity y is proportional to a quantity x, then it is implied that y = kx where k is described as the constant of proportionality (the number of times by which quantity y changes with change in quantity x).

The quantities are said to be directly proportional when the increase (decrease) in value of one results in the increase (decrease) in value of the other.

The quantities are said to be inversely proportional if one quantity is proportional to the reciprocal of the other. This implies that the increase in one quantity results in a decrease in the other quantity and vice versa.

For example, to say velocity = displacement/ time indicates that velocity is directly proportional to displacement and inversely proportional to the quantity time.

It is said that y is a power function of x if y is proportional to a constant power of x, i.e., y = k(x^p), where k is still the constant of proportionality and p is the power.

Some graphs of basic power functions can be explored: It is seen the (x^2) is consistently concave up. Values of y decrease for -x and increase for +x. The graph of (x^3) is concave down for all values of -x and concave up for all values of +x. The graph of (x^1/2) is increasing and concave down.

Sums of power functions with non-negative integer expoenets are called polynomials. They can be written in the following form:

y = an*(x^n) + an-1*(x^n-1) + ... + a1*x + a0

where,
n is a non-negative integer called the degree of the polynomial
an is a non-zero number called the leading co-efficient, and
an*(x^n) is called the leading term.

If n=2, the polynomial is quadratic.
If n=3 the polynomial is cubic.
If n=4, the polynomial is quartic
If n=5, the polynomial is quintic and so on.

Personally, I find it challenging to graph polynomials especially when they get more complicated (as in fractions) and their asymptotes are not as obvious.

The relevance of such polynomials and the concept of proportionality is extremely important in Economics to understand the fluctions in the demand and supply with changes in price of goods. The price at which maximum revenue can be attained also depends on the kind of relationship between the good and the people's demand for it. Also, demand for complementary goods is dependent on the concept of direct proportionality as is the demand for substitute goods dependent on the concept of inverse proportionality.