Question

How do you describe the end behavior of `f(x)=x^3-4x^2+7`?

Answer 1

Increasingly negative to the left and increasingly positive to the right.

Explanation:

In order to determine end behavior, only the highest degree term (term with the highest exponent) matters. Because `f(x)`'s highest degree term is `x^3`, it will determine the end behavior.

We then look for two key factors in determining the end behavior:

1. Power of the exponent:
If the power is even (`x^2`, `x^4`, etc.) then both ends will go in the same direction; either the graph will be positive at both ends or negative at both ends.

If the power is odd (`x`, `x^3`, etc.), then one end be positive and one will be negative.

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 `x^3` has a coefficient of `1`, which is positive, and has an odd power, the end behavior will be negative to the right and positive to the left. Hope this helps!