How Polynomials Behave

A polynomial looks like this:

Polynomial expression 2x^4 plus 6x minus 5

example of a polynomial

Continuous and Smooth

There are two main things about the graphs of Polynomials:

The graphs of polynomials are continuous, which is a special term with an exact definition in calculus, but here we'll use this simplified definition:

Pencil drawing an uninterrupted smooth curve on paper we can draw it without lifting our pen from the paper

The graphs of polynomials are also smooth. No sharp "corners" or "cusps"

Smooth polynomial curve contrasted with a curve having a sharp corner

Rollercoaster track curving smoothly through hills and loops

Example: Rollercoaster

Imagine you're riding a rollercoaster: the ups and downs of the track can be modeled by a polynomial equation.

And (you hope) there are no sharp changes.

How the Curves Behave

Let's graph some polynomials to see what happens ...

... and let's start with the simplest form:

f(x) = xⁿ

Which actually does interesting things:

Even power curves y=x^2, x^4, and x^6 flattening near zero and steepening above 1

Even values of "n" behave the same:

Always above (or equal to) 0
Always go through (0,0), (1,1) and (-1,1)
Larger values of n flatten out near 0, and rise more sharply above the x-axis

Odd power curves y=x^3, x^5, and x^7 passing through quadrant 3 to quadrant 1

Odd values of "n" behave the same:

Always go from negative x and y to positive x and y
Always go through (0,0), (1,1) and (−1,−1)
Larger values of n flatten out near 0, and fall/rise more sharply from the x-axis

Power Function of Degree n

Next, by including a multiplier of a we get what's called a "Power Function":

f(x) = axⁿf(x) equals a times x to the "power" (ie exponent) n

The a changes it this way:

Values greater than 1 squash the curve (inwards to y-axis)
Values between 0 and 1 expand it (away from y-axis)
And negative values of a flip it upside down

Example: f(x) = ax²
a = 2, 1, ½, −1

Parabolas of y=ax^2 scaled by a equals 2, 1, 1/2, and -1

Example: f(x) = ax³
a = 2, 1, ½, −1

Cubic curves of y=ax^3 scaled by a equals 2, 1, 1/2, and -1

See it here (try the sliders):

images/function-graph.js?fn0=a*x^2&fn1=b*x^3&xmin=-5.294&xmax=5.259&ymin=-1.447&ymax=4.553&vara=1|-2|2&varb=1|-2|2

We can use that knowledge when sketching some polynomials:

Example: Make a Sketch of y=1−2x⁷

Start with the simplest "odd power" graph of x³, and gradually turn it into 1−2x⁷

We know how x³ looks,
x⁷ is similar, but flatter near zero, and steeper elsewhere,
Squash it to get 2x⁷,
Flip it to get −2x⁷, and
Raise it by 1 to get 1−2x⁷

Like this:

Five-step transformation from y=x^3 to y=1-2x^7 by flattening, stretching, flipping, and shifting

So by doing this step-by-step we can get a good result.

Turning Points

A Turning Point is an x-value where a local maximum or local minimum happens:

Cubic curve with turning points labeled at the local maximum crest and local minimum trough

How many turning points does a polynomial have?

Never more than the Degree minus 1

The Degree of a Polynomial with one variable is the largest exponent of that variable.

Polynomial 4x^3 minus 5x^2 plus 3 with the exponent 3 identified as the degree

Example: a polynomial of Degree 4 will have 3 turning points or less


x⁴−2x²+x has 3 turning points		x⁴−2x has only 1 turning point

The most is 3, but there can be less.

We may not know where they are, but at least we know the most there can be!

Also: turning points change in pairs, because if the curve turns away, it has to turn back. So a degree 4 polynomial can have 1 or 3 turning points, but never 2.

What Happens at the Ends

And when we move far from zero:

far to the right (large values of x), or
far to the left (large negative values of x)

then the graph starts to resemble the graph of y = axⁿ where axⁿ is the term with the highest degree.

Example: f(x) = 3x³−4x²+x

Far left or right, the graph will look like 3x³


Near Zero, they are different		Far From Zero, they become similar

This makes sense, because when x is large, then x³ is much greater than x² and so on

This is officially called the "End Behavior Model".

Summary

Graphs are continuous and smooth
Even exponents behave the same: above (or equal to) 0; go through (0,0), (1,1) and (−1,1); larger values of n flatten out near 0, and rise more sharply
Odd exponents behave the same: go from negative x and y to positive x and y; go through (0,0), (1,1) and (−1,−1); larger values of n flatten out near 0, and fall/rise more sharply
Factors:
- Values greater than 1 squash the curve (inwards to y-axis)
- Values between 0 and 1 expand it (away from y-axis)
- And negative values flip it upside down
Turning points: there are "Degree − 1" or less
End Behavior: use the term with the largest exponent

486, 487, 488, 4020, 1128, 4022, 1129, 2404, 2405, 4021