Intro to Power Functions

With practice the concept of slope for linear functions becomes intuitive. It makes sense that the line that fits the equation $y=2x$ has a steeper ascent then the line that fits the equation $y=1/2x$. You only have to move horizontally one unit to change your vertical direction two for the former when you graph $y=2x$. How many blocks do you need to move horizontally to change your vertical direction by one for the line $y=1/2x$?

 When we express concepts like $y=x^{2}$ the abstract behavior of what is being represented becomes a little harder to see.  

A monomial of one variable, let's say x, is an algabraic expression of the form

$cx^{m}\$    

where

• $c$ is a constant, and
• $m$ is a non-negative integer (e.g., 0, 1, 2, 3, ...).

The integer $m$ is called the degree of the monomial.

The idea of a monomial of degree zero appears a bit mystical since it always represents one, except when the value of the variable is set equal to zero when the result is undefined. This idea allows us preserve the value of the constant in the monomial. We know that $cx^{0}$ is always equal to $c$ because even though we have 0 x's (somethings) we still have a c. When x = 0 things are difficult because the value we started with, 0, represents nothing.

For a monomial of power 1 we are multiplying C by one instance of our variable. When $x=0$ we get $c*0=0$. When $x\neq 0$ we are multiplying c by 1 x. If x is less than 1 then c gets smaller, if x is more than 1 c gets bigger. When x is between 0 and -1 c gets smaller slower, when x is less than -1 c gets smaller faster.

A monomial with power two is one that "squares" the value of x. The reference to square is because using the multiplication operation once allows us to measure area. If you have something that is one unit on each side this is called a square unit. If you divide both sides of your square unit in half, you get 4 quarter units. We represent this with math by doing the multiplication $1/2*1/2=1/4$ Squaring something is a non-intuitive operation until you become comfortable with the graph of the function. We can see this with the story of the mathematician who was offered a reward by his king. The mathematician said he wanted a single grain of wheat, squared every day for 30 days. For the first seven days the king's servants delivered 1, 2, 4, 16, 256, 65,536 grains of wheat to the mathematician. On the seventh day the value was 4,294,967,296 (4 gig in computer terms)... Sometimes the story ends with the king re-negotiating, sometimes the story ends with the king executing the mathematician to preserve his kingdom, and sometimes the king is astute enough not to take the deal.

A monomial with power three is one that "cubes" the value of x. This is because we use the operation x*x*x to measure the volume that a given area of x*x takes up. If you have a cube that is 1 unit on each side and cut each side in half you will find that you have created 8 cubes. If the mathematician had asked to have the single grain of wheat cubed than the servants would have delivered 1, 8, 512, $134\times 10^{6}$, $242\times 10^{22}$ grains of wheat and the kings deal would have needed to be re-negotiated two days earlier.

Power Function

A power function is a function that can be represented in the form

$f(x)=kx^{p}$

where $k$ and $p$ are real numbers, and $k$ is known as the coefficient. We can also think of a power function as a monomial function; that is, a power function takes the form $y=f(x)$, where $f(x)$ is a single-variable monomial.

Resources

• Intro to Power Functions and Polynomial Functions, Book Chapter
• Guided Notes

Licensing

Content obtained and/or adapted from:

• Polynomials, Wikibooks: Algebra under a CC BY-SA license