Question

considering the leading term of the polynomial function. what is the end behavior of the graph? describe the end behavior and provide the leading term. -3x^5+9x^4+5x^3+3

Answer 1

The leading term of polynomial function is the the term contain highest degree so here in the given question leading term is  $-3x^5$

and leading coefficient is the coefficient of the term with greatest exponent -3

RULES for End behaviour

we have following four cases

CASE1: Even degree and positive leading coefficient

$x\rightarrow -\infty$      $f(x)\rightarrow \infty$

$x\rightarrow \infty$        $f(x)\rightarrow \infty$

CASE2: Even degree and negative leading coefficient

$x\rightarrow -\infty$        $f(x)\rightarrow -\infty$

$x\rightarrow \infty$           $f(x)\rightarrow -\infty$

CASE3: Odd degree and positive leading coefficient

$x\rightarrow -\infty$          $f(x)\rightarrow -\infty$

$x\rightarrow \infty$              $f(x)\rightarrow \infty$

CASE4: Odd degree and negative leading coefficient

$x\rightarrow -\infty$              $x\rightarrow \infty$  

$x\rightarrow \infty$                  $f(x)\rightarrow -\infty$

Here in the given case we have odd degree and negative leading coefficient

$x\rightarrow -\infty$              $x\rightarrow -\infty$  

$x\rightarrow \infty$                  $f(x)\rightarrow -\infty$