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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.