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Standard level

3.02 Power functions

Lesson

Power functions

Functions with a form like $y=x^2$y=x2$y=5x^3$y=5x3 and $y=ax^n$y=axn are known as power functions.

The following applet lets you see the general shape of power functions where $a=1$a=1 and the powers are positive integers.

  • What happens at the extremities of the graph? Is it different for odd and even powers? 
  • Also look carefully at the shape of the curves as they come close to the origin.
  • Do you notice anything else about the curves?

 

 

Odd degree power functions all have the following properties:

  • The graph has $180$180$^\circ$° rotational symmetry about the origin, that is $f(-x)=-f(x)$f(x)=f(x) for all $x$x
  • The graph moves in opposite directions at the extremities
  • They all pass through $\left(0,0\right)$(0,0), $\left(-1,-1\right)$(1,1) and $\left(1,1\right)$(1,1)
  • Domain and range are both $\left(-\infty,\infty\right)$(,)
  • As $n$n increases, within the interval $\left[-1,1\right]$[1,1] the graphs lie closer to the $x$x-axis, 'flatter' about the point of inflection
  • As $n$n increases and $x$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 lower powers    

Even degree power functions all have the following properties:

  • The graph is symmetric about the $y$y-axis, that is $f(-x)=f(x)$f(x)=f(x) for all $x$x 
  • The graph moves in same directions at the extremities
  • They all pass through $\left(0,0\right)$(0,0), $\left(-1,1\right)$(1,1) and $\left(1,1\right)$(1,1)
  • Domain is and $\left(-\infty,\infty\right)$(,)and the range is$\left[0,\infty\right)$[0,)
  • As $n$n increases, within the interval $\left[-1,1\right]$[1,1] the graphs lie closer to the $x$x-axis, 'flatter' about the turning point
  • As $n$n increases and $x$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 lower powers        

 

 

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