Question

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`?

Answer 1

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->+-oo`

as `x->oo` , `\ \ \ \ \ \ \ \ \ \ \ -x^3->-oo`

as `x->-oo` , `\ \ \ \ \ \ -x^3->oo`

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

`y=-(0)^3-4(0)=0`

Coordinates:

`color(blue)( (0 ,0)`

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

`-x^3-4x=0`

`x^3+4x=0`

Factor:

`x(x^2+4)=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]}