Question

What is the end behavior of the graph of the polynomial function f(x) = –x5 + 9x4 – 18x3?

Answer 1

Hint-

$f(x)=a_0+a_1x+a_2x^2+....+a_nx^n$

In the given function,

$Leading \ term=a_nx^n \ , \ Leading \ coefficient=a_n \ , \ degree = n$

Solution-

The given polynomial,

$f(x) = -x^5+9x^4-18x^3$

$Leading \ term=-x^5 \ , \ Leading \ coefficient=-1 \ , \ degree = 5$

The degree of the function is odd and the leading coefficient is negative. So, the end behavior is,

$As \ x \rightarrow -\infty,f(x)\rightarrow +\infty \ and \ x \rightarrow +\infty,f(x)\rightarrow -\infty$

So the graph will be in 2nd and 4th quadrant.

Answer 2

Because it's an odd function, the "tails" go off in different directions.  Also, because it's a negative function, the left starts from the upper left and the right goes down into negative infinity.  If it was a positive, the tails would be going in the other directions, meaning that the left would come up from negative infinity and the right would go up into positive infinity.