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End Behavior Graphically

User S J
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We will investigate how to determine the end behaviours of polynomial functions.

The function given to us is:


f(x)=123x^3+9x^4-786x-3x^{5^{}}-189x^2\text{ + 1260}

Whenever we try to determine the end-behaviour of any function. We are usually looking for value of f ( x ) for the following two cases:


x\to\infty\text{ and x}\to-\infty

The most important thing to note when dealing with end-behaviour of polynomial functions is that the behaviour is pre-dominantly governed by the highest order term of a polynomial. The rest of the terms are considered small or negligible when considering end-behaviours of polynomials.

The highest order terms in the given function can be written as:


f(x)=-3x^5

Then the next step is to consider each case for the value of ( x ) and evaluate the value of f ( x ) respectively.


\begin{gathered} x\to\infty \\ f\text{ ( }\infty\text{ ) = -3}\cdot(\infty)^5 \\ f\text{ ( }\infty\text{ ) = -3}\cdot\infty \\ f\text{ ( }\infty\text{ ) = -}\infty \end{gathered}

Similarly repeat the process for the second case:


\begin{gathered} x\to-\infty \\ f\text{ ( -}\infty\text{ ) = -3}\cdot(-\infty)^5 \\ f\text{ ( -}\infty\text{ ) = 3}\cdot\infty \\ f\text{ ( -}\infty\text{ ) = }\infty \end{gathered}

Combining the result of two cases we get the following solution:


As\text{ x}\to\text{ }\infty\text{ , y}\to\text{ -}\infty\text{ and as x}\to-\infty\text{ , y}\to\text{ }\infty

Correct option is:


\text{Option C}

User Ghandhikus
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