Question

choose the end behavior diagram that best describes the function.

\\(f(x) = -4.4x^4 + x^6 + 0.8x^7\\)

choose the correct diagram below.

Explanation:

Identify the leading term of the polynomial

To determine the end behavior of the polynomial function:

$$f(x) = -4.4x^4 + x^6 + 0.8x^7$$

we must find the term with the highest power of \(x\).
The terms are \(-4.4x^4\), \(x^6\), and \(0.8x^7\).
The term with the highest exponent is \(0.8x^7\).
Thus, the leading term is \(0.8x^7\).

Analyze the degree and leading coefficient

From the leading term \(0.8x^7\):

• The degree \(n = 7\) is odd.
• The leading coefficient \(a_n = 0.8\) is positive (\(a_n > 0\)).

Determine the end behavior

For any polynomial with an odd degree and a positive leading coefficient:

• As \(x \to -\infty\), \(f(x) \to -\infty\) (the graph falls to the left).
• As \(x \to \infty\), \(f(x) \to \infty\) (the graph rises to the right).

This is represented visually by a down-left arrow and an up-right arrow: \((\swarrow,
earrow)\).

Match with the given options

Let's evaluate the visible options:

• Option A shows both ends pointing downwards \((\swarrow, \searrow)\), which corresponds to an even degree with a negative leading coefficient.
• Option C shows both ends pointing upwards \((

warrow,
earrow)\), which corresponds to an even degree with a positive leading coefficient.

• Since the degree is odd and the leading coefficient is positive, the graph must fall to the left and rise to the right. This standard behavior is typically represented by option B or D (not fully visible, but logically must be the option showing a down-left and up-right behavior).

Answer:

• (A) down on the left, down on the right
• (C) up on the left, up on the right
• (B) down on the left, up on the right (Correct answer)