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Chapter 2: Problem 24

Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. $$f(x)=-11 x^{4}-6 x^{2}+x+3$$

Short Answer

Expert verified
The end behavior of the given function \(f(x)=-11 x^{4}-6 x^{2}+x+3\) is: as \(x \rightarrow -\infty\) and \(x \rightarrow +\infty\), \(y \rightarrow -\infty\).

Step by step solution

01

Identify the leading coefficient and degree

In this polynomial \(f(x)=-11 x^{4}-6 x^{2}+x+3\), the term \( -11x^{4}\) is the term with highest degree (4) and its coefficient (-11) is the leading coefficient.
02

Determine if degree is odd or even

For the polynomial \(f(x)=-11 x^{4}-6 x^{2}+x+3\), the degree is 4 which is even; meaning the graph will have same end behavior on either sides.
03

Apply Leading Coefficient Test

Since the degree is even and the leading coefficient is negative, the rule states that: as \(x \rightarrow -\infty\) and \(x \rightarrow +\infty\), \(y \rightarrow -\infty\).

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Most popular questions from this chapter

a. Find the slant asymptote of the graph of each rational function and \(\mathbf{b} .\) Follow the seven-step strategy and use the slant asymptote to graph each rational function. $$f(x)=\frac{x^{3}-1}{x^{2}-9}$$

Use transformations of \(f(x)=\frac{1}{x}\) or \(f(x)=\frac{1}{x^{2}}\) to graph each rational function. $$g(x)=\frac{1}{x+1}-2$$

The equation for \(f\) is given by the simplified expression that results after performing the indicated operation. Write the equation for \(f\) and then graph the function. $$\frac{x}{2 x+6}-\frac{9}{x^{2}-9}$$

Use long division to rewrite the equation for \(g\) in the form $$\text {quotient }+\frac{\text {remainder}}{\text {divisor}}$$ Then use this form of the function's equation and transformations of \(f(x)=\frac{1}{x}\) to graph \(g.\) $$g(x)=\frac{3 x-7}{x-2}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The inequality \(\frac{x-2}{x+3}<2\) can be solved by multiplying both sides by \(x+3,\) resulting in the equivalent inequality \(x-2<2(x+3)\).
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