determine the end behavior of the following function. y=(x^4 - 9x^3 - 6x^5)/(3x^2)
determine the end behavior of the following function. y=(x^4 - 9x^3 - 6x^5)/(3x^2)
Answer
Explanation:
Step1: Simplify the function
Divide each term in the numerator by $3x^{2}$: $y=\frac{x^{4}}{3x^{2}}-\frac{9x^{3}}{3x^{2}}-\frac{6x^{5}}{3x^{2}}=\frac{1}{3}x^{2}- 3x - 2x^{3}$
Step2: Identify the leading - term
The leading - term of the polynomial function $y = - 2x^{3}+\frac{1}{3}x^{2}-3x$ is $-2x^{3}$ (the term with the highest degree).
Step3: Determine the end - behavior
For a polynomial function $y = ax^{n}$, when $n$ is odd and $a<0$ (here $n = 3$ and $a=-2$): As $x\to+\infty$, $y\to-\infty$; as $x\to-\infty$, $y\to+\infty$. This corresponds to the green arrow option (where the left - hand side of the graph goes up and the right - hand side goes down).