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Chapter 2: Problem 26

Describe the right-hand and left-hand behavior of the graph of the polynomial function. \(f(x)=4 x^{5}-7 x+6.5\)

Short Answer

Expert verified
For the function \(f(x)=4x^{5}-7x+6.5\), the right-hand behavior is \(f(x) 鈫� 鈭瀄) as \(x 鈫� 鈭瀄) and the left-hand behavior is \(f(x) 鈫� -鈭瀄) as \(x 鈫� -鈭瀄)

Step by step solution

01

Identify Degree and Leading coefficient

The degree of the polynomial function \(f(x) = 4x^5 - 7x + 6.5\) is 5, because it is the highest power of \(x\). The leading coefficient is 4, because it is the coefficient of the term with highest power.
02

Determine Right-Hand Behavior

In general, if the degree of a polynomial function is odd and the leading coefficient is positive, then as \(x\) approaches positive infinity (鈭�), \(f(x)\) also approaches positive infinity (鈭�). Therefore, the end behavior of the function \(f(x)\) is \(f(x) 鈫� 鈭瀄) as \(x 鈫� 鈭瀄)
03

Determine Left-Hand Behavior

For a polynomial function with odd degree and a positive leading coefficient, as \(x\) approaches negative infinity (-鈭�), \(f(x)\) approaches negative infinity (-鈭�). Therefore, the end behavior of the function \(f(x)\) is \(f(x) 鈫� -鈭瀄) as \(x 鈫� -鈭瀄)

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