QuestionAnalyze the polynomial $g(x)=-10 x^{7}-9 x^{5}+7 x^{3}-8 x$ to determine its end behavior.

Studdy Solution

STEP 1

Assumptions1. The function $g(x)=-10 x^{7}-9 x^{5}+7 x^{3}-8 x function. . The end behavior of a polynomial function is determined by the degree and the leading coefficient of the polynomial.

STEP 2

First, we need to identify the degree and the leading coefficient of the polynomial. The degree of the polynomial is the highest power of $x$ in the polynomial, and the leading coefficient is the coefficient of the term with the highest power.
In the given function $g(x)=-10 x^{7}-9 x^{5}+7 x^{}-8 x$ , the degree is7 and the leading coefficient is -10.

STEP 3

Now, we can determine the end behavior of the graph of $g$ . For a polynomial of odd degree- If the leading coefficient is positive, the graph falls to the left and rises to the right. - If the leading coefficient is negative, the graph rises to the left and falls to the right.
Since our polynomial has an odd degree (7) and a negative leading coefficient (-10), the graph of $g$ rises to the left and falls to the right.

STEP 4

We can express the end behavior using limit notation. As $x$ approaches negative infinity ( $x \rightarrow -\infty$ ), the value of $g(x)$ approaches positive infinity ( $g(x) \rightarrow +\infty$ ). As $x$ approaches positive infinity ( $x \rightarrow +\infty$ ), the value of $g(x)$ approaches negative infinity ( $g(x) \rightarrow -\infty$ ).
Therefore, the end behavior of the graph of $g$ isAs $x \rightarrow -\infty$ , $g(x) \rightarrow +\infty$ and as $x \rightarrow +\infty$ , $g(x) \rightarrow -\infty$ .

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