Power Functions



POWER functions



Power functions are basically the simplest of all polynomial functions. Very basic, but can be tricky at times. Just read the functions carefully, and be sure not to overlook any detail because every part of a power function is significant.

It's a lot easier than you think to imagine the graph of a power function just by STARING at a power function. Of courseeeeeeeeeeeeee, you need some basic knowledge to be able to do magic! No worries, the table below explains everything.

 

 

Key Features	y = xn ; where n is an odd number	y = xn ; where n is an even number
Domain	{ x ϵ R }	{ x ϵ R }
Range	{ y ϵ R }	{ y ϵ R | y ≥ 0 }
Symmetry	Point symmetry about (0,0)	Line symmetry at x = 0
End Behavior	As x → ∞ , y → ∞ As x → -∞ , y → -∞	As x → ∞ , y → ∞ As x → -∞ , y → ∞

Before further explanation, there are new terms that need to be remembered as it will help in understanding future polynomial questions.

          Degree = The highest power in a power / polynomial function

               e.g. y = 3x² + 2x + 1    this means, the degree of the given power function is 2 (since 2 > 1)

          Leading coefficient = The coefficient of the term with the highest power in a power/polynomial function

               e.g. y = 3x² + 2x + 1    this means, the leading coefficient is 3, since it is the coefficient of x²

Tip :

1) The range of a power function remains +∞ at all times ONLY for a power function with an even degree.

2) Only the range of a power function with an even degree has restrictions.

3) For a point symmetry, the coordinate of that point must be stated to show a clearer understanding of the question. Whereas for a line symmetry, the value of x at the symmetry must be stated. (note: x need not necessarily be 0. It can be a whole number depending on the power function)

To practise, try imagining the shapes of the graphs just by analyzing the power function given. And when you've become an expert, try vice versa. When you're an expert in both the ways, you deserve a pat on the shoulder! =)

Wait, wait, wait. It's not the end yet! Just a liiiiittle more...

End Behavior	Function
Extends from quadrant 3 to quadrant 1	y = xn where n is an odd number (if the coefficient of x is a positive)
Extends from quadrant 2 to quadrant 4	y = -xn where the coefficient of x is a negative number (if n is an odd number)
Extends from quadrant 2 to quadrant 1	y = xn where n is an even number (if the coefficient of x is a positive)
Extends from quadrant 3 to quadrant 4	y = -xn where the coefficient of x is a negative number (if n is an even number)

Note that polynomial graphs vary by THEIR DEGREE (odd/even) & THE SIGN OF THEIR LEADING COEFFICIENT (positive/negative)

If you intend to use a graphing calculator, refer to this link Click Here for a step-by-step guideline on how you can plot a graph by just keying in a function in your graphing calculator (I am referring to Texas Instrument TI-84 which looks like below)

 

Happy Math! Until next time...