Determine the end behavior of the following polynomial functions. (That is, does $f(x)$ increase or decrease without bound as $x \rightarrow \pm \infty$ ?)
a. If $f(x)=2.7 x^{4}-13 x-18$ then...
- As $x \rightarrow \infty, f(x) \rightarrow$
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As $x \rightarrow-\infty, f(x) \rightarrow$
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b. If $f(x)=1+5 x+x^{2}-x^{3}$ then...
As $x \rightarrow \infty, f(x) \rightarrow$
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As $x \rightarrow-\infty, f(x) \rightarrow$
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Answer
\(\boxed{\text{For the second function, } f(x)=1+5 x+x^{2}-x^{3}, \text{ as } x \rightarrow \infty, f(x) \rightarrow -\infty \text{ and as } x \rightarrow-\infty, f(x) \rightarrow \infty.}\)
Step 1 ：The end behavior of a polynomial function is determined by the degree and the leading coefficient of the polynomial. If the degree of the polynomial is even, then the end behavior of the function will be the same on both sides. If the degree is odd, then the function will have opposite end behavior on either side. The sign of the leading coefficient will determine whether the function increases or decreases as x approaches positive or negative infinity.
Step 2 ：For the first function, \(f(x)=2.7 x^{4}-13 x-18\), the degree is 4 (even) and the leading coefficient is 2.7 (positive). Therefore, as x approaches both positive and negative infinity, the function will increase.
Step 3 ：For the second function, \(f(x)=1+5 x+x^{2}-x^{3}\), the degree is 3 (odd) and the leading coefficient is -1 (negative). Therefore, as x approaches positive infinity, the function will decrease and as x approaches negative infinity, the function will increase.
Step 4 ：Final Answer: For the first function, \(f(x)=2.7 x^{4}-13 x-18\), as \(x \rightarrow \infty, f(x) \rightarrow \infty\) and as \(x \rightarrow-\infty, f(x) \rightarrow \infty\).
Step 5 ：For the second function, \(f(x)=1+5 x+x^{2}-x^{3}\), as \(x \rightarrow \infty, f(x) \rightarrow -\infty\) and as \(x \rightarrow-\infty, f(x) \rightarrow \infty\).
Step 6 ：\(\boxed{\text{For the first function, } f(x)=2.7 x^{4}-13 x-18, \text{ as } x \rightarrow \infty, f(x) \rightarrow \infty \text{ and as } x \rightarrow-\infty, f(x) \rightarrow \infty.}\)
Step 7 ：\(\boxed{\text{For the second function, } f(x)=1+5 x+x^{2}-x^{3}, \text{ as } x \rightarrow \infty, f(x) \rightarrow -\infty \text{ and as } x \rightarrow-\infty, f(x) \rightarrow \infty.}\)
