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

Jul 15, 2018

End behavior : Down ( As $x \to - \infty , y \to - \infty$),

Up ( As $x \to \infty , y \to \infty$ )

#### Explanation:

$f \left(x\right) = {x}^{3} + 4 x$ 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 $x \to - \infty , y \to - \infty$),

Up ( As $x \to \infty , y \to \infty$),

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

Jul 15, 2018

${\lim}_{x \to \infty} f \left(x\right) = \infty$

${\lim}_{x \to - \infty} f \left(x\right) = - \infty$

#### Explanation:

To think about end behavior, let's think about what our function approaches as $x$ goes to $\pm \infty$.

To do this, let's take some limits:

${\lim}_{x \to \infty} {x}^{3} + 4 x = \infty$

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 $- \infty$?

${\lim}_{x \to - \infty} {x}^{3} + 4 x = - \infty$

Once again, as $x$ gets very negative, ${x}^{3}$ will dominate the end behavior. Since we have an odd exponent, our function will approach $- \infty$.

Hope this helps!