Review - Fall $$ 2024-5 \% $$ $$ f(x)=2 x-4 x^{3}-x^{5} $$ Use the Leading Coefficient Test to determine the end behavior of the polynomial function.

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To determine the end behavior of the polynomial function $$ f(x) = 2x - 4x^{3} - x^{5} $$ using the **Leading Coefficient Test**, follow these steps:

### 1. Identify the Leading Term
The **leading term** of a polynomial is the term with the highest power of $$ x $$. For $$ f(x) $$:

$$
f(x) = -x^5 - 4x^3 + 2x
$$

- **Leading Term:** $$ -x^5 $$
- **Degree:** 5 (which is odd)
- **Leading Coefficient:** -1 (negative)

### 2. Apply the Leading Coefficient Test
The **Leading Coefficient Test** helps determine the end behavior based on the degree and the leading coefficient:

- **If the degree is odd:**
  - **Leading Coefficient Positive:**
    - As $$ x \to +\infty $$, $$ f(x) \to +\infty $$
    - As $$ x \to -\infty $$, $$ f(x) \to -\infty $$
  - **Leading Coefficient Negative:**
    - As $$ x \to +\infty $$, $$ f(x) \to -\infty $$
    - As $$ x \to -\infty $$, $$ f(x) \to +\infty $$

### 3. Determine the End Behavior for $$ f(x) $$

Given that the degree is **5** (odd) and the leading coefficient is **-1** (negative):

- **As $$ x \to +\infty $$:** $$ f(x) \to -\infty $$
- **As $$ x \to -\infty $$:** $$ f(x) \to +\infty $$

### 4. Summary of End Behavior

$$
\begin{cases}
x \to +\infty & \Rightarrow f(x) \to -\infty \
x \to -\infty & \Rightarrow f(x) \to +\infty \
\end{cases}
$$

### 5. Visual Representation

Here’s a graphical illustration of the end behavior:

- **Left End ($$ x \to -\infty $$):** The graph rises to the **top left**.
- **Right End ($$ x \to +\infty $$):** The graph falls to the **bottom right**.

![Polynomial End Behavior](https://i.imgur.com/yourImageLink.png)

*Note: Replace the image link with an appropriate graph if needed.*

### Conclusion

Using the Leading Coefficient Test for $$ f(x) = 2x - 4x^{3} - x^{5} $$:

- **Left End:** The graph rises to positive infinity as $$ x $$ approaches negative infinity.
- **Right End:** The graph falls to negative infinity as $$ x $$ approaches positive infinity.

This behavior is characteristic of an odd-degree polynomial with a negative leading coefficient.
As $$ x $$ approaches positive infinity, $$ f(x) $$ approaches negative infinity, and as $$ x $$ approaches negative infinity, $$ f(x) $$ approaches positive infinity.

Review - Fall

Use the Leading Coefficient Test to determine the end behavior of the polynomial function.

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Review - Fall Use the Leading Coefficient Test to determine the end behavior of the polynomial function.