Equations of Straight Lines
UBC Calculus Online Course Notes

Powers of x: positive integer powers

On this page, we will review some properties of the functions which are powers of the independent variable $  x.  $ Many of these functions will probably be quite familiar, but there are several subtle points that are worth remembering early on in a "Calculus career". In fact, some of the features of these very simple building blocks will come in handy many times in our later discussions of properties of functions. We will begin the discussion by examining the simplest of these functions, those of the form $ y=x^n $ where n is a positive integer.

Below, we illustrate the graph of the function \[ 
y=a x^n 
 \] where $ n $ is a positive integer and $ a $ is a constant. By moving the sliders, you can vary the values of $  n 
 $ and $  a.  $


At this point, you should explore the properties of the function by seeing what happens when the values of $  a  $ and $  n  $ are changed. Then, answer the questions below. Your answers should be based exclusively on your observations here. Calculus, derivatives, and any other tools should not be aplied at this point since we want to help you develop your "powers of observation" (to use a pun) of information on a graph. This type of common sense (rather than mere calculations) will be very useful in many situations.

  1. Effect of the constant: Set n=2. Increase the parameter a. Explain what happens to the graph of the function. Is your observation true also for n=3 ? for bigger n values ? Explain in words what is the effect of the constant a on these graphs.

  2. Effect of the power: Now that we have explored the effect of the multiplicative constant, a, we will set it equal to 1 and investigate the effect of raising the power. Set a=1.
    • What is the shape of the graph when n is an even integer?
    • What is the shape when n is odd?
    • Can you summarize these observations in terms of symmetry of the graphs?

  3. Rates of growth: How can you explain the following statements in terms of the shape of the graphs?
    • From the point $  x=0  $ , the function $  x  $ grows more rapidly than $  x^3  $ .
    • From the point $  x=1  $ , the function $  x^3  $ grows more rapidly than $  x  $ .

    Generalize to other powers. How does the rate of growth at $  x 
= 0  $ and $  x = 1  $ depend on $  n  $ ?

  4. The symmetry of the graphs of the power functions lends an important terminology to functions in general. A function $  f(x)  $ is called even or odd if it shares the symmetry properties of the above odd and even power functions. Now that you know this little fact, match the description "odd" and "even" with the following properties of a function.

    • $  f(x)  $ is ____________ if $  f(x)=  f(-x)  $
    • $  f(x)  $ is ____________ if $  f(x)= - f(-x)  $

    Find other examples of even and odd functions. Not every function is either even or odd. (But it is an interesting fact that every function can be expressed as a sum of an even and an odd function.)

  5. Intersections Which point(s) do all the graphs of $ y=x^n $ have in common
    • for even values of n ?
    • for odd values of n ?
    • for all values of n ?

  6. Slope at the origin What is the slope of the graph of $ y=x^n $ at the origin? Is this slope the same or different for even and odd values of the power n?

  7. Changing slope Suppose you were an ant walking across the graph of the function $ y=x^n $ , starting at x=-1 and proceeding to x=1. Would you be walking uphill ? downhill ? Does it depend on the value of n? Translate this observation to a comment about how the slope of the function changes with increasing x values. Is this different for even and odd values of n?

  8. Minima Which of members of the above family of functions have a minimum value? Which do not?