Power Functions

Functions with a form like $y=x^2$y=x2, $y=2x^2$y=2x2 and $y=3x^3$y=3x3, etc. are known as power functions.

The following applet lets you see the effect of increasing the powers of $x$x  for the function $y=x^n$y=xn. 

• Note how for odd powers the graph moves in opposite directions at the extremities, and for even powers the graph moves off in the same direction at the extremities. 
• Also look carefully at the shape of the curves as they come close to the origin.
• What else do you notice about the curves?

For odd powers greater than $1$1, the curves rise toward the origin with a decreasing positive gradient until, when the origin is reached, they become momentarily horizontal and then continue upward with an ever increasing gradient. That "terracing" shape around the origin is known as a horizontal inflection.

For even powers the curve changes direction at the origin. This is called a minimum turning point.

 

Concepts of power functions

Power functions have the general form $y=ax^n$y=axn where $n$n is any number. When $n$n is a positive integer, we can learn to sketch these functions by considering a few simple principles of powers.  

Whenever a non-zero number is raised to an even integer power, the result is always positive. So for example $\left(3\right)^4=+81$(3)4=+81 and also $\left(-3\right)^4=+81$(−3)4=+81.  This results in the graphs of all power functions that have even integral powers to have some similar properties and general shape.

Whenever a non-zero number is raised to an odd integer power, the result is positive when that number is also positive, and negative when that number is also negative. So $\left(3\right)^3=+27$(3)3=+27 and $\left(-3\right)^3=-27$(−3)3=−27.  This results in the graphs of all power functions that have odd integer powers to have some similar properties and general shape.

Whenever a number lies in the interval $-1−1<x$<$<$1$1, squaring it or cubing it (or indeed raising it to any integer power) reduces the absolute value of its size. So $\left(-0.5\right)^3=-0.125$(−0.5)3=−0.125 and $\left(0.9\right)^2=0.81$(0.9)2=0.81 . 

To demonstrate the effect of these principles on power graphs, we have graphed the functions given by $y=x$y=x, $y=x^2$y=x2, $y=x^3$y=x3 and $y=x^4$y=x4 between $x=-2$x=−2 and $x=2$x=2. 

Note the following four properties:

• Each graph passes through the origin.
• The function values for $y=x^2$y=x2, $y=x^3$y=x3 and $y=x^4$y=x4 within the interval $\left[-1,1\right]$[−1,1] lie closer to the $x$x-axis than the function values for $y=x$y=x. As $x$x extends beyond the interval $\left[-1,1\right]$[−1,1], the functions values grow further away  from the $x$x-axis than those of $y=x$y=x.    
• The odd powered graphs of $y=x$y=x and $y=x^3$y=x3 pass through $\left(-1,-1\right)$(−1,−1) and $\left(1,1\right)$(1,1) and have rotational symmetry about the origin. Specifically the function value $f\left(-x\right)$f(−x) is the negative of the function value $f\left(x\right)$f(x). That is $f\left(-x\right)=-f\left(x\right)$f(−x)=−f(x).
• The even powered graphs of $y=x^2$y=x2 and $y=x^4$y=x4 pass through $\left(-1,1\right)$(−1,1) and $\left(1,1\right)$(1,1) and possess reflective symmetry about the $y$y - axis. specifically the function value $f\left(-x\right)$f(−x) is the same as the function value $f\left(x\right)$f(x). That is $f\left(-x\right)=f\left(x\right)$f(−x)=f(x).

These properties hold for all integral powers. 

Effect of the coefficient $a$a

The coefficient $a$a in the power function form $y=ax^n$y=axn scales the function values of $y=x^n$y=xn by a factor of $a$a. So for example, if $a=\frac{1}{2}$a=12​, then each value of the function $y=x^n$y=xn is halved, so the graph looks compressed. If $a=2$a=2, the function values are doubled and the graph looks vertically stretched instead. 

If $a$a happens to be negative, the effect on the graph is a reflection across the $x$x-axis. That is, function values that are negative become positive and function values that are positive become negative.

As an example compare the graphs of $y=x^2,y=2x^2,y=-\frac{1}{2}x^2$y=x2,y=2x2,y=−12​x2 and $y=-3x^2$y=−3x2 as shown here:

 

Worked Examples

Question 1

Question 2

Question 3