Predict the end behavior of the graph of the function.

Question

​f(x)equals3 left parenthesis x minus 3 right parenthesis left parenthesis x squared plus 3 right parenthesis
Question content area bottom
Part 1
A.
Up on the right​ side, down on the left side
B.
Up on both sides
C.
Down on both sides
D.
Down on the right​ side, up on the left side

Answer 1

To determine the end behavior of the function \( f(x) = 3(x - 3)(x^2 + 3) \), we can analyze the degrees of the polynomial.

1. The function is a product of a linear term \( (x - 3) \) and a quadratic term \( (x^2 + 3) \).
2. The degree of \( (x - 3) \) is 1, and the degree of \( (x^2 + 3) \) is 2.
3. Therefore, the overall degree of \( f(x) \) is \( 1 + 2 = 3 \), meaning it is a cubic polynomial.
4. Since the leading coefficient (from the term \( 3 \)) is positive, we know how the ends of the graph behave based on the degree.
5. For a cubic polynomial with a positive leading coefficient, the end behavior is:
   • As \( x \to -\infty \) (left side), \( f(x) \to -\infty \).
   • As \( x \to +\infty \) (right side), \( f(x) \to +\infty \).

Based on this analysis, the correct choice is:

D. Down on the left side, up on the right side.