Predict the end behavior of the graph of the function.

Question

​f(x)equals6 left parenthesis x minus 9 right parenthesis left parenthesis x squared plus 9 right parenthesis
Question content area bottom
Part 1
A.
Down on both sides
B.
Down on the right​ side, up on the left side
C.
Up on both sides
D.
Up on the right​ side, down on the left side

Answer 1

To determine the end behavior of the function \( f(x) = 6(x - 9)(x^2 + 9) \), we need to analyze the leading term as \( x \) approaches positive and negative infinity.

1. Identify the degree of the polynomial:

   • The term \( x - 9 \) is a linear term, and \( x^2 + 9 \) is a quadratic term.
   • The degree of \( (x - 9) \) is 1, and the degree of \( (x^2 + 9) \) is 2. Therefore, the overall degree of the polynomial is \( 1 + 2 = 3 \).

2. Leading coefficient:

   • The leading term (when fully expanded) will be \( 6x^3 \) since the leading term from \( (x - 9) \) is \( x \) and from \( (x^2 + 9) \) is \( x^2 \). Therefore, we multiply the leading coefficients: \( 6 \times 1 \times 1 = 6 \).

3. End behavior analysis:

   • Since the degree is odd (3) and the leading coefficient is positive (6), the end behavior will be:
     • As \( x \to +\infty \), \( f(x) \to +\infty \)
     • As \( x \to -\infty \), \( f(x) \to -\infty \)

Thus, the end behavior is "Up on the right side and down on the left side."

The correct answer is: D. Up on the right side, down on the left side.