Question
A 2-row table with 9 columns. The first row is labeled x with entries negative 5, negative 4, negative 3, negative 2, negative 1, 0, 1, 2, 3. The second row is labeled f of x with entries negative 14, 6, 0, negative 4, negative 6, negative 6, negative 4, 0, 6.
Based on the table, which statement best describes a prediction for the end behavior of the graph of f(x)?
As x → ∞, f(x) → –∞, and as x → –∞, f(x) → ∞
As x → ∞, f(x) → ∞, and as x → –∞, f(x) → ∞
As x → ∞, f(x) → ∞, and as x → –∞, f(x) → –∞
As x → ∞, f(x) → –∞, and as x → –∞, f(x) → –∞
Based on the table, which statement best describes a prediction for the end behavior of the graph of f(x)?
As x → ∞, f(x) → –∞, and as x → –∞, f(x) → ∞
As x → ∞, f(x) → ∞, and as x → –∞, f(x) → ∞
As x → ∞, f(x) → ∞, and as x → –∞, f(x) → –∞
As x → ∞, f(x) → –∞, and as x → –∞, f(x) → –∞
Answers
GPT-4o
To predict the end behavior of the function \( f(x) \) based on the given table, we should look at the general pattern of the function’s values as \( x \) increases and decreases.
Here is the table again for clarity:
\[
\begin{array}{c|ccccccccc}
x & -5 & -4 & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\
\hline
f(x) & -14 & 6 & 0 & -4 & -6 & -6 & -4 & 0 & 6 \\
\end{array}
\]
Let's analyze the behavior:
- As \( x \) goes from \( -5 \) to \( 3 \):
- When \( x = -5 \), \( f(x) = -14 \)
- When \( x = -4 \), \( f(x) = 6 \)
- When \( x = -3 \), \( f(x) = 0 \)
- When \( x = -2 \), \( f(x) = -4 \)
- When \( x = -1 \), \( f(x) = -6 \)
- When \( x = 0 \), \( f(x) = -6 \)
- When \( x = 1 \), \( f(x) = -4 \)
- When \( x = 2 \), \( f(x) = 0 \)
- When \( x = 3 \), \( f(x) = 6 \)
We observe that as \( x \) increases from \( -5 \) to \( 3 \), \( f(x) \) shows a pattern that doesn’t simply converge to a single point but rather shows a polynomial-like behavior. The values \( f(x) \) take as \( x \) increases from \( -4 \) to \( 3 \) increase to \( 6 \), and similarly as \( x \) goes from \( -5 \) to \( -4 \), \( f(x) \) also increases.
Given that this seems to be a repeated quadratic-like trend, likely resembling a cubic polynomial or higher, the end behavior can be inferred:
- For very large positive \( x \) (\( x \to \infty \)), \( f(x) \) seems to increase without bound, so \( f(x) \to \infty \).
- For very large negative \( x \) (\( x \to -\infty \)), as we see significant negative values starting at \( -5 \), it is likely that \( f(x) \) would decrease without bound, so \( f(x) \to -\infty \).
Therefore, the best statement that describes the end behavior is:
\[
\text{As } x \to \infty, f(x) \to \infty, \text{ and as } x \to -\infty, f(x) \to -\infty.
\]
Here is the table again for clarity:
\[
\begin{array}{c|ccccccccc}
x & -5 & -4 & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\
\hline
f(x) & -14 & 6 & 0 & -4 & -6 & -6 & -4 & 0 & 6 \\
\end{array}
\]
Let's analyze the behavior:
- As \( x \) goes from \( -5 \) to \( 3 \):
- When \( x = -5 \), \( f(x) = -14 \)
- When \( x = -4 \), \( f(x) = 6 \)
- When \( x = -3 \), \( f(x) = 0 \)
- When \( x = -2 \), \( f(x) = -4 \)
- When \( x = -1 \), \( f(x) = -6 \)
- When \( x = 0 \), \( f(x) = -6 \)
- When \( x = 1 \), \( f(x) = -4 \)
- When \( x = 2 \), \( f(x) = 0 \)
- When \( x = 3 \), \( f(x) = 6 \)
We observe that as \( x \) increases from \( -5 \) to \( 3 \), \( f(x) \) shows a pattern that doesn’t simply converge to a single point but rather shows a polynomial-like behavior. The values \( f(x) \) take as \( x \) increases from \( -4 \) to \( 3 \) increase to \( 6 \), and similarly as \( x \) goes from \( -5 \) to \( -4 \), \( f(x) \) also increases.
Given that this seems to be a repeated quadratic-like trend, likely resembling a cubic polynomial or higher, the end behavior can be inferred:
- For very large positive \( x \) (\( x \to \infty \)), \( f(x) \) seems to increase without bound, so \( f(x) \to \infty \).
- For very large negative \( x \) (\( x \to -\infty \)), as we see significant negative values starting at \( -5 \), it is likely that \( f(x) \) would decrease without bound, so \( f(x) \to -\infty \).
Therefore, the best statement that describes the end behavior is:
\[
\text{As } x \to \infty, f(x) \to \infty, \text{ and as } x \to -\infty, f(x) \to -\infty.
\]