How do you describe the end behavior of #f(x)=x^3-4x^2+7#?
Increasingly negative to the left and increasingly positive to the right.
We then look for two key factors in determining the end behavior:
1. Power of the exponent:
If the power is even (
If the power is odd (
But how do we know if the ends are positive or negative? We have to check the...
2. Sign of the coefficient:
If the coefficient is positive, then
- If the highest power is even then both ends go up (think of the graph
#x^2#; all end behavior for functions with an even highest power will look like that).
- If the highest power is odd, then when
#x#is very negative, then #y#will also become very negative and when #x#is very positive, #y#becomes very positive (think #x^3#).
If the coefficient is negative, then reverse what is above: if the highest power is even, both ends go down and if the highest power is odd, the left goes up and the right goes down.
For your function, since