# How Polynomials Behave

A polynomial looks like this:
 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 will use this simplified definition:

you can draw it without lifting your pen from the paper

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

## How the Curves Behave

Let us graph some polynomials to see what happens ...

f(x) = xn

Which actually does interesting things:

 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

And:

 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

## Power Function of Degree n

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

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

The "a" changes it this way:

• Larger values of a squash the curve (inwards to y-axis)
• Smaller values of a expand it (away from y-axis)
• And negative values of a flip it upside down
 Example: f(x) = ax2 a = 2, 1, ½, -1 Example: f(x) = ax3 a = 2, 1, ½, -1

We can use that knowledge when sketching some polynomials:

### Example: Make a Sketch of y=1-2x7

• You know how x3 looks,
• x7 will be 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:

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:

### 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.

### 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.

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

## What Happens at the Ends

And when you 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 to the 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 etc

This is officially called the "End Behavior Model".

And yes, we have come to the end!

## Summary

• Graphs will be 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:
• Larger values squash the curve (inwards to y-axis)
• Smaller values expand it (away from y-axis)
• And negative values flip it upside down
• Turning points: there will be "Degree-1" or less.
• End Behavior: use the term with the largest exponent