Question

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Part 1
Answer parts ​(a)dash​(e) for the function shown below.
​f(x)equals x cubed plus 3 x squared minus x minus 3
Question content area bottom left
Part 1
a. Use the leading coefficient test to determine the​ graph's end behavior. Which statement describes the behavior at the ends of ​f(x)equals x cubed plus 3 x squared minus x minus 3​?
A.
The graph falls to the left and rises to the right.
B.
The graph rises to the left and to the right.
C.
The graph rises to the left and falls to the right.
D.
The graph falls to the left and to the right.
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Answer 1

To determine the end behavior of the function \( f(x) = x^3 + 3x^2 - x - 3 \) using the leading coefficient test, we can follow these steps:

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

2. **Apply the leading coefficient test**:
- Since the degree (3) is odd, and the leading coefficient (1) is positive, we can use the following rules:
- 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)

Based on the analysis, the end behavior of the function can be summarized as:

- The graph falls to the left and rises to the right.

Therefore, the correct statement that describes the end behavior of \( f(x) \) is:

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