How Polynomials Behave

When we talk about solving a polynomial, it means we have to find the roots of the polynomial. The root is also called the zero of the polynomial function. These are the points where the value of the function is zero.

 

If we observe the polynomial function above as shown in the figure above, the value of function is varying either above x-axis or below x-axis in between the roots (zeroes) of the function.

Please note that the value of function is negative where the curve is below x-axis and is positive where the curve is above x-axis.

Degree

Name

Example

Graph

0

Constant

3

1

Linear

3x+2

2

Quadratic

x2-2x+3

3

Cubic

2x3-4x2

4

Quartic

x4-2x+2

….

 

How Polynomials Behave 

The graphs of a polynomial function are always continuous and smooth.

Let us draw some of the curves for a polynomial function .

Let, f(x)=xn

When n=0, 2, 4…. (even), then function behaves in a similar fashion.

·         Always stay equal or above 0.

·         Always pass through (0,0), (1,1) and (-1,1)

·         When the value of n is high, the curve flattens out near 0 and rises more sharply.

And, when n takes the odd values, the function behaves in a certain similar pattern.

·         The curve moves from negative x and y to positive x and y.

·         The curve passes through (0,0), (1,1) and (-1, -1) always.

·         When the value of n is high, the curve flattens out near 0 and rises/fall more sharply.

Let us consider, f(x) = axn

·         When the value of a is high, the curve moves inwards toward y-axis.

·         When the value of a is low, the curve expands away from y-axis.

·         Negative value of a flips it upside down.

Example: f(x) = ax2

a = 2, 1, ½, −1

 

Example: f(x) = ax3

a = 2, 1, ½, −1

 

Local Maximum and Local Minimum

The local maximum and local minimum are called the Turning Point on x. 

How may Turning points does a polynomial have?

Number of turning points ≤ Degree – 1  

 Degree of a polynomial is the largest exponent of the variable.

4x3+2x2+7 has the degree 3.

Number of turning points ≤ 3-1 ≤ 2

Hence, the above polynomial has 2 or less than 2 turning points.

 When we move far from zero either to the right or the left of zero on the x-axis, the graph resembles, y = axn, where n is the highest degree of the polynomial. 

Summary

1.       Graphs of a polynomial are continuous and smooth.

2.      Even exponents of a polynomial behave in the same way: 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.

3.      Odd exponents behave in the same way: 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.

4.      Factors:

a.      Larger values squeeze the curve (inwards to y-axis)

b.      Smaller values expand it (away from y-axis)

c.       And negative values flip it upside down

5.      Turning points= "Degree − 1" or less.

6.      End Behavior: use the term with the largest exponent.

Our Categories