In this investigation the following will be associated with the given color in the graphs. We wish to consider the effects made on the graph as the values for a and b are changed.
When the values of a and b are both equal to 10 the graph will look like this:
Notice that when the powers of the equation are even numbers the graph remains in the first quadrant yet when the powers of the equation are odd the graph lies in all four quadrants. In the case that the powers are even the graph traces over itself four times. Also note that if the values of a and b are equal that the graph will always look similar to this just of a different size.
Now we wish to consider the effects on the graph if a<b. In the case that a=5 and b=8 the graph will look like this:
Note the in this case the graph appears to be squashed in from the sides so that it is an oval that is longer top to bottom than side to side. It is important to realize that the x values are ranging from -5 to 5 and the y values are ranging from -8 to 8. Still, only those equations with odd exponents are pictured in all four quadrants.
Now we wish to take a look at the effects on the graph when a>b. Take a look at this graph in which a=8 and b=5:
This case is very similar to the case in which a<b but now the oval is longer from side to side than it is from top to bottom. Once again, notice that the x values are effected by the value of a while the y values are effected by the values of b.
If you were to continue to increase the exponents used in these equations the graph would continue to bend closer and closer to a 90 degree angle. Keep in mind that if the exponents are even then the graph will only be realized in the first quadrant but that if the exponents are odd then the graph will show up in all four quadrants and appear almost like a four pointed star.
Here is a graph that includes the first 6 graphs plus these two:
As you can see, as the exponent gets larger the graph appears closer and closer to a right angle even though the exponent is still relatively small.