Question

Answer parts (a)-(e) for the function shown below.

f(x)=x^3 + 3x^2 - x - 3

...

a. Use the leading coefficient test to determine
the graph's end behavior. Which statement describes
the behavior at the ends of f(x) =x3 +3x2 -x-3?

O A. The graph falls to the left and to the right.
O B. The graph rises to the left and to the right.
O C. The graph falls to the left and rises to the right.
D. The graph rises to the left and falls to the right.

Answer 1

To determine the end behavior of the function \( f(x) = x^3 + 3x^2 - x - 3 \), we can use the leading coefficient test.

1. **Identify the degree** and **leading coefficient** of the polynomial:
- The degree of the polynomial is 3 (since the highest power of \( x \) is \( x^3 \)).
- The leading coefficient is 1 (the coefficient of \( x^3 \)).

2. **Apply the leading coefficient test** for odd degree polynomials with a positive leading coefficient:
- For a polynomial of odd degree with a positive leading coefficient, the end behavior is that the graph falls to the left and rises to the right.

Based on this analysis, the correct answer is:

**C. The graph falls to the left and rises to the right.**