Where $a_n,a_{n-1},....,a_1,a_0$ are constant and real numbers and n is a non-negative integer

- a
_{n},a_{n-1},...,a,a_{0}are called the coefficients for x^{n},x^{n-1},..,x^{},x^{0} - n is called the degree of the polynomial function
- A Constant Function is the polynomial function with zero degree
- n is a non-negative integer.It can not be fraction also

Example of Polynomial functions

$3x^2 + 4x+6$

$9x^3 -4x^2 +1$

$4x +1$

$4x^2 + \sqrt {2} $

The below are not polynomial functions

$x^{-4} + 1$

$ \sqrt {x} + 2x +1$

$ x^{2/3} +1$

$3x^2 + 4x+6$

$9x^3 -4x^2 +1$

$4x +1$

$4x^2 + \sqrt {2} $

The below are not polynomial functions

$x^{-4} + 1$

$ \sqrt {x} + 2x +1$

$ x^{2/3} +1$

Type of Polynomial function |
Example |
Degree |

Constant Function |
$1$ |
n=0 |

Linear Function |
$4x +1$ |
n=1 |

Quadratic function |
$x^2 +4x +1$ |
n=2 |

Cubic Function |
$3x^3 -4x^2+x +11$ |
n=3 |

Quartic Function |
$x^4 +2x^3 +4x^2+x +1$ |
n=4 |

Function f(x) =0 is also an polynomial function with undefined degree

Domain = R

Range is dependent on the type of polynomial function.

For Linear function ,Range is R

For constant function Range is {c}

For Quadratic function like ${x^2 +1}$ , Range is $[1,\infty)$

We can apply the below steps to find the Range of Quadratic Functions easily

Step 1: Quadratic Function will be of type

$y=ax^2 + bx + c$

Step 2: Quadratic function will have minimum value or maximum value depending on the coefficent a

if a > 0 , It will have minimum value

if a < 0, it will have maximum value

Step 3: The minimum or maximum values can be found using

a. First find the value of x where it will be minimum or maximum

$x_0 = - \frac {b}{2a}$

b. Now minimum or maximum value can be found using

$y_0 = f(x_0)$

Step 4: Range will be given as

if a > 0 , Range will be $[y_0,\infty)$

if a < 0 , Range will be $(-\infty,y_0] $

For Cubic functions ,Range is R
Step 1: Quadratic Function will be of type

$y=ax^2 + bx + c$

Step 2: Quadratic function will have minimum value or maximum value depending on the coefficent a

if a > 0 , It will have minimum value

if a < 0, it will have maximum value

Step 3: The minimum or maximum values can be found using

a. First find the value of x where it will be minimum or maximum

$x_0 = - \frac {b}{2a}$

b. Now minimum or maximum value can be found using

$y_0 = f(x_0)$

Step 4: Range will be given as

if a > 0 , Range will be $[y_0,\infty)$

if a < 0 , Range will be $(-\infty,y_0] $

Here are the graph for function

$f(x)=3$ ( Constant function)

$f(x)=x+2$(Linear Function)

$f(x)=3x+1$(Linear Function)

For Quadratic function, the graph is a parabolic graph. Lets see few example

1. $f(x) =2x^2$ , $h(x) =5x^2$ , $g(x) =10x^2$

We can see the graph is a upward parabola and as we increase the coefficient of $x^2$,the graph is stretched vertically

2. $f(x) =-2x^2$ , $h(x) =-5x^2$ , $g(x) =-10x^2$

We can see the graph is a downward parabola and as we increase the coefficient of $x^2$,the graph is stretched vertically

3. $f(x) = x^2 + 2x +1$ , $g(x) = x^2 + 4x +1$ ,$h(x) =x^2 +6x +1$

We can see the graph is a upward parabola and with increase in coefficient of x ,it drifts downwards on the left

4. $f(x) =x^2 +1$ and $h(x) =-x^2 -1$

Here since their are no real roots of the given quadratic function , the graph is not touching the x-axis. First graph is upward above x-axis and second graph is downward below x-axis

So , the intersection of parabola graph depends on the real roots of the function

5.$f(x) =x^2 -5x +6$ and $h(x) =-x^2 +7x -12$

Here since their are real roots of the given quadratic function ,the graph is intersecting the the x-axis.

Lets see few example of the cubic function

1. $f(x) =x^3$

We can see it extends in both upward and downward direction

2. $f(x) = x^3 + 1$

Here is the cubic function with one real root

3.$f(x) = x^3 -3x^2 +3x -1= (x-1)^3$

Here is the cubic function with three equal real root

4.$f(x) = x^3+12x^2+39x+28=(x+1)(x+4)(x+7)$

Here is the cubic function with three different real root.

1. $f(x) =x^4 -5x^2 +4 =(x-1)(x-2)(x+1)(x+2)$

This is a quartic function and it has four real roots .

1. The turning point in a graph is defined as the points from where graph from upward to downward or downward to upward. The turning points in the graph is always less or equal to (n-1) of the polynomial function.So a quartic function has maximum 3 turning points in the graph.A quadratic equation has maximum one turning point. A Cubic equation has maximum 2 turns. You can validate these in above given example graph also

2. The zero's or root of the polynomial function are point at which graph intersect x-axis , i,e the point where the value of y=0. The real roots of the polynomial function is always less or equal to the degree n of the polynomial.

How to Factor Polynomials

How to Solve Quadratic equations

We have already discussed above on how to find domain and range also

a. $y =x^2 +5$

b. $y = x^3 $

c. $y=x^2 -7x +5$

a. Clearly it is defined for all $x \in R$

b. Clearly it is defined for all $x \in R$

c. Clearly it is defined for all $x \in R$

2. Find the domain and range of function

$y=3x^2+12x−12$

We have ,

$y=3x^2+12x−12$

Clearly it is defined for all $x \in R$

Comparing it to $y=ax^2 + bx +c$

Then a=3, b=12 c=-12

Now since a> 0,then it will have minimum value

Now Minimum value will occur at

$x_0 = - \frac {b}{2a} = -2$

So minimum value is

$y_0= f(x_0) = 3 \times (-2)^2 + 12 (-2) - 12= -24$

So range is $[-24, \infty)$

3. Let $f : R \rightarrow R$ , $f(x) = x^2 + 2[x] -1$ for each $x \in R$

Find the values of f(x) at x= 1.2 ,-.5 ,-2.1

$f(x) = x^2 + 2[x] -1$

$f(1.2) = (1.2)^2 + 2[1.2] -1 =1.44 +2-1=2.44$

$f(-.5) = (-.5)^2 + 2[-.5] -1 =.25 -2-1=-2.75$

$f(-2.1) = (-2.1)^2 + 2[-2.1] -1 =4.41 -6-1=-2.59$

**Notes****Assignments**