Question
Predict the end behavior of the graph of the function.
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
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
Answers
GPT-4o mini
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.**
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.**