# How do you find the end behavior and state the possible number of x intercepts and the value of the y intercept given y=-x^3-4x?

Apr 10, 2018

See below.

#### Explanation:

To find the end behaviour of a polynomial, we only need to look at the degree and leading coefficient of the polynomial. The degree is the highest power of $x$ in this case.

$- {x}^{3}$

We now see what happens as $x \to \pm \infty$

as $x \to \infty$ , $\setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus - {x}^{3} \to - \infty$

as $x \to - \infty$ , $\setminus \setminus \setminus \setminus \setminus \setminus - {x}^{3} \to \infty$

$y$ axis intercepts occur where $x = 0$:

$y = - {\left(0\right)}^{3} - 4 \left(0\right) = 0$

Coordinates:

color(blue)( (0 ,0)

$x$ axis intercepts occur where $y = 0$

$- {x}^{3} - 4 x = 0$

${x}^{3} + 4 x = 0$

Factor:

$x \left({x}^{2} + 4\right) = 0$

$x = 0$

${x}^{2} + 4 = 0$ ( this has no real solutions ).

coordinates:

color(blue)(( 0 , 0 )

The graph confirms these findings:

graph{y=-x^3-4x [-16.01, 16.02, -20,20]}