Question
What is the end behavior of the function f left-parenthesis x right-parenthesis equals negative 5 x cubed plus 2 x squared minus 4 x plus 6?
(1 point)
Responses
down to the left, up to the right
down to the left, up to the right
up to the left, down to the right
up to the left, down to the right
up to the left, up to the right
up to the left, up to the right
down to the left, down to the right
down to the left, down to the right
(1 point)
Responses
down to the left, up to the right
down to the left, up to the right
up to the left, down to the right
up to the left, down to the right
up to the left, up to the right
up to the left, up to the right
down to the left, down to the right
down to the left, down to the right
Answers
GPT-4o mini
To determine the end behavior of the function \( f(x) = -5x^3 + 2x^2 - 4x + 6 \), we need to look at the leading term, which is the term with the highest degree. In this case, the leading term is \( -5x^3 \).
The end behavior of a polynomial function is determined by the leading term as \( x \) approaches positive and negative infinity:
1. For \( x \to +\infty \) (as \( x \) goes to the right), the leading term \( -5x^3 \) will dominate, and since it has a negative coefficient (\(-5\)), the function \( f(x) \) will go to \(-\infty\).
2. For \( x \to -\infty \) (as \( x \) goes to the left), the leading term \( -5x^3 \) will again dominate. Because the degree is odd (3) and the leading coefficient is negative, \( f(x) \) will approach \( +\infty \).
Thus, the end behavior can be summarized as:
- As \( x \to -\infty \), \( f(x) \to +\infty \) (up to the left).
- As \( x \to +\infty \), \( f(x) \to -\infty \) (down to the right).
Therefore, the correct response is:
**up to the left, down to the right**.
The end behavior of a polynomial function is determined by the leading term as \( x \) approaches positive and negative infinity:
1. For \( x \to +\infty \) (as \( x \) goes to the right), the leading term \( -5x^3 \) will dominate, and since it has a negative coefficient (\(-5\)), the function \( f(x) \) will go to \(-\infty\).
2. For \( x \to -\infty \) (as \( x \) goes to the left), the leading term \( -5x^3 \) will again dominate. Because the degree is odd (3) and the leading coefficient is negative, \( f(x) \) will approach \( +\infty \).
Thus, the end behavior can be summarized as:
- As \( x \to -\infty \), \( f(x) \to +\infty \) (up to the left).
- As \( x \to +\infty \), \( f(x) \to -\infty \) (down to the right).
Therefore, the correct response is:
**up to the left, down to the right**.