To determine the function that matches the given end behavior, let's analyze each option step-by-step:
1. Option A: \( y = -3x^2 + 4 \)
- For large negative values of \( x \) (i.e., \( x \to -\infty \)):
[tex]\[
y = -3(-\infty)^2 + 4 \approx -\infty
\][/tex]
Thus, as \( x \) approaches negative infinity, \( y \) approaches negative infinity.
- For large positive values of \( x \) (i.e., \( x \to +\infty \)):
[tex]\[
y = -3(\infty)^2 + 4 \approx -\infty
\][/tex]
Hence, as \( x \) approaches positive infinity, \( y \) also approaches negative infinity.
2. Option B: \( y = 3(x-1)^2 \)
- For large negative values of \( x \) (i.e., \( x \to -\infty \)):
[tex]\[
y = 3(-\infty - 1)^2 \approx +\infty
\][/tex]
So, as \( x \) approaches negative infinity, \( y \) approaches positive infinity.
- For large positive values of \( x \) (i.e., \( x \to +\infty \)):
[tex]\[
y = 3(\infty - 1)^2 \approx +\infty
\][/tex]
Therefore, as \( x \) approaches positive infinity, \( y \) also approaches positive infinity.
3. Option C: \( y = (x+2)^3 - 9 \)
- For large negative values of \( x \) (i.e., \( x \to -\infty \)):
[tex]\[
y = (-\infty + 2)^3 - 9 \approx -\infty
\][/tex]
Hence, as \( x \) approaches negative infinity, \( y \) also approaches negative infinity.
- For large positive values of \( x \) (i.e., \( x \to +\infty \)):
[tex]\[
y = (\infty + 2)^3 - 9 \approx +\infty
\][/tex]
So, as \( x \) approaches positive infinity, \( y \) approaches positive infinity.
4. Option D: \( y = -2x^3 - 1 \)
- For large negative values of \( x \) (i.e., \( x \to -\infty \)):
[tex]\[
y = -2(-\infty)^3 - 1 \approx +\infty
\][/tex]
Hence, as \( x \) approaches negative infinity, \( y \) approaches positive infinity.
- For large positive values of \( x \) (i.e., \( x \to +\infty \)):
[tex]\[
y = -2(\infty)^3 - 1 \approx -\infty
\][/tex]
Therefore, as \( x \) approaches positive infinity, \( y \) approaches negative infinity.
Based on our analysis, the function that exhibits the correct end behavior (i.e., as \( x \) approaches negative infinity, \( y \) approaches positive infinity, and as \( x \) approaches positive infinity, \( y \) approaches negative infinity) is:
[tex]\[ \boxed{y = -2x^3 - 1} \][/tex]
So the correct answer is D.
