Question

What is the end behavior of `f(x) = x^3 + 4x`?

Answer 1

End behavior : Down ( As `x -> -oo , y-> -oo`),

Up ( As `x -> oo , y-> oo` )

Explanation:

`f(x)= x^3 + 4 x` The end behavior of a graph describes far left

and far right portions. Using degree of polynomial and leading

coefficient we can determine the end behaviors. Here degree of

polynomial is `3` (odd) and leading coefficient is `+`.

For odd degree and positive leading coefficient the graph goes

down as we go left in `3` rd quadrant and goes up as we go

right in `1` st quadrant.

End behavior : Down ( As `x -> -oo , y-> -oo`),

Up ( As `x -> oo , y-> oo`),

graph{x^3 + 4 x [-20, 20, -10, 10]} [Ans]

Answer 2

`lim_(xtooo) f(x)=oo`

`lim_(xto-oo)f(x)=-oo`

Explanation:

To think about end behavior, let's think about what our function approaches as `x` goes to `+-oo`.

To do this, let's take some limits:

`lim_(xtooo) x^3+4x=oo`

To think about why this makes sense, as `x` balloons up, the only term that will matter is `x^3`. Since we have a positive exponent, this function will get very large quickly.

What does our function approach as `x` approaches `-oo`?

`lim_(xto-oo) x^3+4x=-oo`

Once again, as `x` gets very negative, `x^3` will dominate the end behavior. Since we have an odd exponent, our function will approach `-oo`.

Hope this helps!