Math Homework. Do It Faster, Learn It Better.

Comparing Linear, Polynomial, and Exponential Growth

Consider the three functions y = x , y = x 4 and y = 2 x .

The first is a linear function ; its graph is a straight line.

The second is a polynomial function .

The third is an exponential function .

Suppose you graph all three on the same axes, and ask the question, “Which function grows the fastest for large values of x ?”

In this graph, it appears that the answer is y = x 4 . It is greater than y = 2 x for all values shown.

However, if we zoom out the y -axis quite a lot, we see that at x = 16 , y = 2 x overtakes y = x 4 , and after this point, y = 2 x grows faster. (Note that with the axes scaled this way, y = x grows so slowly that it is indistinguishable from the x -axis.)

In fact, it can be shown that this is true for ANY exponential function and ANY polynomial function with positive growth. The exponential function will eventually outstrip the polynomial function.