Comparing Linear, Polynomial, and Exponential Growth

One of the most interesting things about algebra is that different kinds of equations take different shapes when we graph them on a plane. But what exactly is the difference between linear, polynomial, and exponential growth when we graph these functions? Let's find out:

Putting all three functions on a plane

Consider these three functions:

$y=x$

$y=x^4$

$y=2^x$

What do these functions look like when we graph them? Let's take a look:

The interesting thing about graphing functions is that their shape represents their type. For example, the blue function is linear, so it takes the shape of a straight line. The red function is polynomial, so it takes the form of a parabola. The green function is exponential, so it takes the shape of an upward curve.

Making our comparisons

Once we understand how the functions work, it's easier to compare them. For example, what if we wanted to find the function that grows the fastest if we increase the value of x? Take a look:

At first glance, it might seem like the parabola is the clear winner. But what happens when we zoom out? Take a look:

As we can see, the equation for exponential growth overtakes the parabola at $x=16$ . This teaches us a useful lesson: At some point, the exponential function with base greater than 1 will always outstrip the polynomial function.

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