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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:

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.

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.

Graphing Exponential and Logarithmic Functions

College Algebra Diagnostic Tests

If your student needs help comparing polynomial, exponential, and linear growth, 1-on-1 tutoring is a clear option. During tutoring sessions, your student's tutor can choose the appropriate pace -- whether they want to steam ahead or slow down to hammer down key concepts. Tutors can also personalize the lesson plan by catering to your student's learning style, hobbies, and much more. Speak with our Educational Consultants today and rest assured: Varsity Tutors will pair your student with a suitable tutor.

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