Power Functions

Hello friends!

This is the first post on this blog. We're going to learn about Power Functions. It's the first subchapter in Chapter 1 (Polynomial Functions). I know that most of you have known about this. But, I hope this post can help you who need more explanation.

Before we start, there is prerequisite skills you'd better know ^{_{for Chapter 1}}. It's as follows :
-Function Notation
-Slope and y-intercept of a Line
-Equation of a Line
-Finite Differences
-Domain and Range
-Quadratic Functions
-Transformations 

Ok, let's start!  
 

What is Power Functions?

"Power Functions has the form f(x) = axⁿ, where x is a variable, a is a real number, and n is a whole number.The simplest type of polynomial function is power functions."

source : advanced function textbook (McGraw-Hill Ryerson)

A function is called a polynomial function if it satisfies :

 
^{_{source : wikipedia}} 
 
 where

a_n is the leading coefficient which has the greatest power of x
x is a variable
n is non-negative whole number
a₀, a₁,a₂, ..., a_n are coefficients and real numbers
a<sub>0 is the constant term (Because x^o= 1)

  ^{For example : 
 f(x) =}</sub> 3x⁵ + 2x³ - 4x² - 2
^{_{It can be written like f(x) =}} 3x⁵ + 0x⁴ + 2x³ - 4x² + 0x¹ – 2x<sup>0


<sub>Power Functions have special names which depends on their degree.
y = a                   Degree = 0
                           Name = Constant

y = ax                 Degree = 1
                           Name = Linear 

y = ax2               Degree = 2
                           Name = Quadratic

y = ax3               Degree = 3
                           Name = Cubic

y = ax4                Degree = 4
                            Name = Quartic

y = ax5                Degree = 5
                            Name = Quintic</sub></sup>     

^{_{Even-degree power functions is a function which has an even non negative whole number as the greatest degree. It usually has line symmetry in the y-axis, x = 0.}}

^{_{Odd-degree power functions is a function which has an old non negative whole number as the greatest degree. It usually has point symmetry about the origin, (0,0).}}  

^{_{What is End Behaviour?}}

^{_{End Behaviour of polynomial function desribes the graph of f(x) as x approaches positive infinity or negative infinity.}} 

 

^{_{source : http://mathworld.wolfram.com/Quadrant.html}}

^{_{Even-degree polynomial function end behaviour :}}

^{_{The leading coefficient is positive : The graph extends from quadrant 2 to quadrant 1.}}

^{_{The leading coefficient is negative : The graph extends from quadrant 3 to quadrant 4.}}

^{_{Odd-degree polynomial function end behaviour :}}

^{_{The leading coefficient is positive : The graph extends from quadrant 3 to quadrant 1.}}

^{_{The leading coefficient is negative : The graph extends from quadrant 2 to quadrant 4.}}

  <sup><sub>Examples :

  
Even degree.
Positive leading coefficient.
Domain = {x ϵ R }
Range = { y ϵ R | y ≥ 0 }
It has line symmetry in y-axis.
It extends from quadrant 2 to quadrant 1.</sub></sup>

<sup><sub>Even degree.
Negative leading coefficient.
Domain = {x ϵ R }
Range = { y ϵ R | y ≤ 4 }
It doens't have line symmetry.
It extends from quadrant 3 to quadrant 4.



                           
Odd degree.
Positive leading coefficient.
Domain = {x ϵ R }
Range = { y ϵ R }
It has</sub></sup> point symmetry about the origin, (0,0).
^{_{It extends from quadrant 3 to quadrant 1.}}

^{_{Odd degree. 
Negative leading coefficient.
Domain = {x ϵ R }
Range = { y ϵ R }
It has}} point symmetry about the origin, (0,0). 
^{_{It extends from quadrant 2 to quadrant 4.}} 

source pictures : 
http://algebra.freehomeworkmathhelp.com/
www.uiowa.edu
hotmath.com

<sup><sub>These are the videos which may help you to get 
to know about this topic simpler :


Power Functions : CLICK HERE
Polynomial Function : CLICK HERE
End Behaviour : CLICK HERE
Domain : CLICK HERE
Range : CLICK HERE</sub></sup> 
 

^{_{Well, that's all about Power Functions in Chapter 1. I hope everyone who reads this post will understand more about it. Anyone who has questions or suggestions, feel free to comment here. We can discuss about other topics related to Advanced Functions too. Have a nice day!}}

-Y U L I T A-