How do you find the end behavior of #f(x) = -2(x-1)(x+3)^3#?

Question

Anna Allen · Accepted Answer

#"see explanation"# #"To determine the end behaviour we only require to"# #"consider the term of highest degree, that is the leading"# #"term in standard form"# #"multiply the terms of highest degree from each factor"# #-2(x)(x^3)=-2x^4# #"the leading term of highest degree is "-2x^4# #"which is of even degree with negative coefficient"# #•color(white)(x)lim_(xto+-oo)=-oo# graph{-2(x-1)(x+3)^3 [-80, 80, -40, 40]}

Greyson Brown · Answer

undefined To find the end behavior of the function $f(x) = -2(x-1)(x+3)^3$, you examine the leading term, which is the term with the highest power of $x$. In this case, the leading term is $-2(x+3)^3$.

Since the highest power of $x$ is odd (3), and the coefficient of the leading term is negative (-2), the end behavior of the function is as follows:

As $x$ approaches negative infinity, $f(x)$ approaches negative infinity (falls to the left).
As $x$ approaches positive infinity, $f(x)$ approaches negative infinity (falls to the left).

Therefore, the end behavior of the function $f(x) = -2(x-1)(x+3)^3$ is that it decreases without bound as $x$ approaches both negative and positive infinity.