Questions: Select one of the four sketches (A) (D) that follow to describe the end behavior of the graph of the function. P(x) = -x^6 + x^5 - x^4 - x + 3 Choose the behavior which best models the polynomial.

Solution Steps

To determine the end behavior of the polynomial function \( P(x) = -x^6 + x^5 - x^4 - x + 3 \), we need to consider the leading term, which is the term with the highest power of \( x \). The leading term in this polynomial is \( -x^6 \). The end behavior of the polynomial will be dominated by this term.

For a polynomial with a leading term \( -x^6 \):

• As \( x \) approaches \( \infty \), \( -x^6 \) approaches \( -\infty \).
• As \( x \) approaches \( -\infty \), \( -x^6 \) approaches \( -\infty \).

Thus, the end behavior of the polynomial \( P(x) \) is that it goes to \( -\infty \) as \( x \) goes to both \( \infty \) and \( -\infty \).

1. Identify the leading term of the polynomial.
2. Determine the end behavior based on the leading term.

We are given the polynomial function: \[ P(x) = -x^6 + x^5 - x^4 - x + 3 \]

The leading term of the polynomial is the term with the highest power of \( x \). In this case, the leading term is: \[ -x^6 \]

The leading term \( -x^6 \) will dominate the behavior of the polynomial for large values of \( |x| \). Since the coefficient of \( x^6 \) is negative, the polynomial will behave as follows:

• As \( x \to \infty \), \( -x^6 \to -\infty \)
• As \( x \to -\infty \), \( -x^6 \to -\infty \)

Based on the leading term analysis, the end behavior of the polynomial \( P(x) \) is:

• As \( x \to \infty \), \( P(x) \to -\infty \)
• As \( x \to -\infty \), \( P(x) \to -\infty \)

We need to choose the sketch that best represents the end behavior of the polynomial. The correct sketch will show the graph of the polynomial going to negative infinity as \( x \) approaches both positive and negative infinity.

Final Answer