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Chapter 2: Problem 24

Describe the right-hand and left-hand behavior of the graph of the polynomial function. \(h(x)=1-x^{6}\)

Short Answer

Expert verified
The graph of the given polynomial function \(h(x)=1-x^{6}\) falls to the left and falls to the right.

Step by step solution

01

Analyze the degree of the polynomial

The degree of the given polynomial function \(h(x)=1-x^{6}\) is 6. Remember that the degree is determined by the highest power of x.
02

Analyze the leading coefficient of the polynomial

The leading coefficient of the given polynomial function \(h(x)=1-x^{6}\) is -1. The leading coefficient is determined by the coefficient of the term with the highest power.
03

Determine the end behavior of the polynomial

For a polynomial function, if the degree is even and the leading coefficient is negative, then the graph falls to the left and falls to the right. This means that as x approaches positive infinity (\(x \to +\infty\)) the function approaches negative infinity (\(h(x) \to -\infty\)) and as x approaches negative infinity (\(x \to -\infty\)), the function also approaches negative infinity (\(h(x) \to -\infty\)).

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Most popular questions from this chapter

The mean salaries \(S\) (in thousands of dollars) of classroom teachers in the United States from 2000 through 2007 are shown in the table. $$\begin{array}{|c|c|}\hline \text { Year } & \text { Salary, } S \\\\\hline 2000 & 42.2 \\ 2001 & 43.7 \\\2002 & 43.8 \\ 2003 & 45.0 \\\2004 & 45.6 \\\2005 & 45.9 \\\2006 & 48.2 \\\2007 & 49.3 \\\\\hline\end{array}$$ A model that approximates these data is given by \(S=\frac{42.6-1.95 t}{1-0.06 t}\) where \(t\) represents the year, with \(t=0\) corresponding to 2000\. (Source: Educational Research Service, Arlington, VA) (a) Use a graphing utility to create a scatter plot of the data. Then graph the model in the same viewing window. (b) How well does the model fit the data? Explain. (c) According to the model, in what year will the salary for classroom teachers exceed \(\$ 60,000 ?\) (d) Is the model valid for long-term predictions of classroom teacher salaries? Explain.

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The maximum safe load uniformly distributed over a one-foot section of a two- inch-wide wooden beam is approximated by the model Load \(=168.5 d^{2}-472.1,\) where \(d\) is the depth of the beam. (a) Evaluate the model for \(d=4, d=6, d=8\), \(d=10,\) and \(d=12\). Use the results to create a bar graph. (b) Determine the minimum depth of the beam that
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