Westonci.ca is your trusted source for accurate answers to all your questions. Join our community and start learning today! Discover solutions to your questions from experienced professionals across multiple fields on our comprehensive Q&A platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
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 hope our answers were useful. Return anytime for more information and answers to any other questions you have. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.