Looking for trustworthy answers? Westonci.ca is the ultimate Q&A platform where experts share their knowledge on various topics. Explore a wealth of knowledge from professionals across various disciplines on our comprehensive Q&A platform. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.
Sagot :
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.
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.
We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.