Question

question 3
describe the long run behavior of f(x)=x^3
as x→−∞, f(x)→?
as x→∞, f(x)→?
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question 4
what is the least possible degree of the polynomial graphed above?
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Explanation:

Step1: Analyze $x\to-\infty$ for $f(x)=x^3$

When $x$ approaches negative - infinity, we have $f(x)=x^3$. Since the exponent 3 is odd and the leading - coefficient is 1 (positive), as $x\to-\infty$, $x^3\to-\infty$.

Step2: Analyze $x\to\infty$ for $f(x)=x^3$

When $x$ approaches positive infinity, since the exponent 3 is odd and the leading - coefficient is 1 (positive), as $x\to\infty$, $x^3\to\infty$.

Step3: Determine the degree of the polynomial from the graph

The number of turning points of a polynomial function $y = P(x)$ of degree $n$ is at most $n - 1$. The given graph has 2 turning points. The least possible degree $n$ of a polynomial with $k$ turning points satisfies $n=k + 1$. So, if $k = 2$, then $n=3$.

Answer:

As $x\to-\infty$, $f(x)\to-\infty$; As $x\to\infty$, $f(x)\to\infty$; The least possible degree of the polynomial is 3.