- Isolate \(f(x)\) to obtain the explicit polynomial: \(f(x) = -1.5x^4 + 10x^3 - 4x^2 - 0.5x + 0.5\).
- Identify the leading term \(-1.5x^4\), where the degree \(n=4\) is even and the leading coefficient \(a=-1.5\) is negative.
- Conclude that the graph opens downwards, meaning \(f(x) \rightarrow -\infty\) as \(x \rightarrow \infty\) and \(f(x) \rightarrow -\infty\) as \(x \rightarrow -\infty\).
- Select the matching choices: As \(x \rightarrow -\infty\), \(f(x) \rightarrow -\infty\) and as \(x \rightarrow \infty\), \(f(x) \rightarrow -\infty\).
Explanation
- Isolating the Function \(f(x)\)
To determine the end behavior of the polynomial function \(f(x)\), we must first isolate \(f(x)\) from the given equation:
\[ f(x) = -3x^4 - 3x^3 - f(x) + 23x^3 - 8x^2 - x + 1 \]
We begin by adding \(f(x)\) to both sides of the equation to group the \(f(x)\) terms together:
\[ 2f(x) = -3x^4 - 3x^3 + 23x^3 - 8x^2 - x + 1 \]
Next, we combine the like terms involving \(x^3\):
\[ -3x^3 + 23x^3 = 20x^3 \]
So the equation simplifies to:
\[ 2f(x) = -3x^4 + 20x^3 - 8x^2 - x + 1 \]
Finally, we divide both sides by \(2\) to express \(f(x)\) explicitly:
\[ f(x) = -1.5x^4 + 10x^3 - 4x^2 - 0.5x + 0.5 \]
- Analyzing the End Behavior
The end behavior of a polynomial function is determined primarily by its leading term, as \(x\) becomes very large in either the positive or negative direction. The leading term of our polynomial \(f(x) = -1.5x^4 + 10x^3 - 4x^2 - 0.5x + 0.5\) is \(-1.5x^4\).
Here, the degree of the polynomial is \(n=4\), which is an even number, and the leading coefficient is \(a=-1.5\), which is negative. According to the power function end behavior rules:
- When the degree is even and the leading coefficient is negative, the graph of the function opens downwards.
- As \(x\) approaches positive infinity (\(x \rightarrow \infty\)), the \(x^4\) term grows very large, and multiplied by the negative coefficient \(-1.5\), the function value \(f(x)\) approaches negative infinity (\(f(x) \rightarrow -\infty\)).
- As \(x\) approaches negative infinity (\(x \rightarrow -\infty\)), the \(x^4\) term (being an even power) also becomes very large and positive, which, when multiplied by the negative coefficient, causes \(f(x)\) to approach negative infinity (\(f(x) \rightarrow -\infty\)).
- Conclusion on End Behavior
Based on our analysis, the end behavior of the function \(f(x)\) is as follows:
- As \(x \rightarrow -\infty\), \(f(x) \rightarrow -\infty\).
- As \(x \rightarrow \infty\), \(f(x) \rightarrow -\infty\).
These findings correspond to the choices provided in the problem. The behavior in both directions is towards negative infinity.
Examples
End behavior analysis is essential in fields like engineering and economics to predict the long-term trends of systems. For example, if you are modeling the profit of a company over time where the profit is represented by a polynomial function, determining the end behavior helps you understand if the profit will eventually grow without bound or crash toward a loss in the long run. By identifying the leading term, analysts can quickly visualize whether a model is sustainable over large intervals.