Basic functions: Power functions
Sums of power functions
We look at how power functions of the form
The diagrams below show the graphs of , and ; on the left-hand side for the interval and on the right-hand side for the interval .
By comparing the graphs we see that the graph of for near resembles the graph of and looks more like the graph of for large values of .
In other words, the function resembles for near and is more similar to the function for large values of .
This can also be seen by bringing out one of the powers as a factor.
It follows from that for small values of the term is very small (at least much smaller than ) and the expression between brackets is close to . Then looks like .
It follows from that for large values of the term is very small and the expression between brackets is close to . Then looks like .
In a sum of two or more power functions, the function resembles for large values of the power term with the largest exponent and resembles the power term with the smallest exponent for small values of .