From Jim Wilson’s website: Examine:
See the graph.What happens if the 4 is replaced by other numbers (not necessarily integers)? Try 5, 3, 2, 1, 1.1, 0.9, -3. Any unusual event? Interpret.
What equation would give the following graph:
What happens if a constant is added to one side of the equation? Try several graphs in some systematic way. Click here for one set of graphs.
By exploring the graph of for various values of n, we discover that there are three classes of graphs: n<0, 0< n < 1, n=1, and n>1 .
For each of these classes, the y-intercepts will remain the same. The y-intercepts occur when x = 0, so the y-intercepts are solutions to . By factoring, we have . Every graph of the form will have will have y-intercepts 0, 1, and -1.
When n > 1, we have a graph such
below. (Here with n = 4)
The most apparent characteristics of this graph are it’s intercepts. Note that x-intercepts occur when y = 0, so the x-intercepts are solutions to . By recognizing the difference of two squares, we can solve this equation easily by transforming it to When n > 1, we have three x-intercepts: .
When n = 1, we have:
This graph appears to be the union of an ellipse and the line y=x. Algebraically, we have: . So . By distributing and combining like terms, we have . By factoring, we arrive at . The zero product property states that if the product of two numbers is zero, then one of the two numbers must be zero. So we have or . In the first case, means and we the line y = x appears graphically as expected. In the second case, we have , which is an ellipse and confirms the second dominant shape in the graphical representation.
When n < 1, we have a graph such
below. (Here with n = -4)
There is only one x-intercept in the case when n < 1. In factored form we have . For integers less than 1, this means n is negative, or n is zero. If n is zero, the only solution to is x = 0. If n is negative, one solution remains zero, while the remaining two solutions are complex.
is between zero and 1, a case similar to when n > 1
occurs. This is the
result of three real solutions to the equation . The graph below
shows when n = .5