What is the end behavior of the graph of #f(x)=-2x^4+7x^2+4x-4#?

Question

Elijah Allen · Accepted Answer

#lim_(xtooo) f(x)=-oo, lim_(xto-oo) f(x)=-oo#

To determine the end behavior, let's take the limit as #xtooo# and #xto-oo#.

In our polynomial, #f(x)#, the first term is what will dominate the end behavior, because it has the highest degree. So we can find the limit of that:

#lim_(xtooo) color(red)(-2)color(blue)(x^4)=-oo#

As #x# gets very large, the blue term will always be positive, but the #-2# (red) will turn it negative. This is why our limit evaluates to #-oo#.

#lim_(xto-oo) color(red)(-2)color(blue)(x^4)=-oo#

As #x# gets very negative, the even exponent will make the term positive, but the red #-2# on the outside will make it negative. Thus, this limit will also evaluate to #-oo#.

In general, the function is downward opening because of the negative coefficient on the #x^4# term.

Hope this helps!

Xavier Alonso · Answer

undefined The end behavior of the graph of $ f(x) = -2x^4 + 7x^2 + 4x - 4 $ can be determined by observing the leading term of the polynomial function, which is $ -2x^4 $.

As $ x $ approaches positive infinity ($ +\infty $), the leading term $ -2x^4 $ dominates the behavior of the function. Since the leading coefficient is negative and the exponent of $ x $ is even, the end behavior of the graph is as follows:

- As $ x $ approaches positive infinity, $ f(x) $ approaches negative infinity ($ -\infty $).

Similarly, as $ x $ approaches negative infinity ($ -\infty $), the leading term $ -2x^4 $ dominates the behavior of the function. Again, since the leading coefficient is negative and the exponent of $ x $ is even, the end behavior of the graph is as follows:

- As $ x $ approaches negative infinity, $ f(x) $ approaches negative infinity ($ -\infty $).