At Westonci.ca, we make it easy to get the answers you need from a community of informed and experienced contributors. Get detailed answers to your questions from a community of experts dedicated to providing accurate information. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
Let's analyze the given functions to determine what they have in common:
The functions are:
[tex]\( f(x) = 5 e^{x+5} - 5 \)[/tex]
[tex]\( g(x) = 0.5 (x - 5)^2 - 5 \)[/tex]
Step-by-step Analysis:
1. Vertical Stretch:
- [tex]\( f(x) = 5 e^{x+5} - 5 \)[/tex] involves an exponential term with a coefficient of 5.
- [tex]\( g(x) = 0.5 (x - 5)^2 - 5 \)[/tex] involves a quadratic term with a coefficient of 0.5.
Since the coefficients of the main terms (the terms involving [tex]\( x \)[/tex]) are different (5 for [tex]\( e^{x+5} \)[/tex] and 0.5 for [tex]\( (x-5)^2 \)[/tex]), they do not have the same vertical stretch.
2. Vertical Shift:
- Both functions involve a constant term of [tex]\(-5\)[/tex].
Because both functions subtract 5, they are both shifted downward by 5 units. Hence, they have the same vertical shift.
3. Horizontal Translation:
- The term [tex]\( e^{x+5} \)[/tex] in [tex]\( f(x) \)[/tex] implies a horizontal shift to the left by 5 units.
- The term [tex]\( (x - 5)^2 \)[/tex] in [tex]\( g(x) \)[/tex] implies a horizontal shift to the right by 5 units.
Therefore, their horizontal translations are not the same.
4. End Behavior:
- For large values of [tex]\( x \)[/tex], [tex]\( f(x) = 5 e^{x+5} - 5 \)[/tex] will increase exponentially, as exponential functions grow infinitely large.
- For large values of [tex]\( x \)[/tex], [tex]\( g(x) = 0.5 (x - 5)^2 - 5 \)[/tex], since it is a quadratic function, will also increase, but more slowly and symmetrically about the vertex.
Exponential functions and quadratic functions have different end behaviors.
Conclusion:
Based on the above analysis, the correct common property for the given functions is:
B. They have the same vertical shift.
The functions are:
[tex]\( f(x) = 5 e^{x+5} - 5 \)[/tex]
[tex]\( g(x) = 0.5 (x - 5)^2 - 5 \)[/tex]
Step-by-step Analysis:
1. Vertical Stretch:
- [tex]\( f(x) = 5 e^{x+5} - 5 \)[/tex] involves an exponential term with a coefficient of 5.
- [tex]\( g(x) = 0.5 (x - 5)^2 - 5 \)[/tex] involves a quadratic term with a coefficient of 0.5.
Since the coefficients of the main terms (the terms involving [tex]\( x \)[/tex]) are different (5 for [tex]\( e^{x+5} \)[/tex] and 0.5 for [tex]\( (x-5)^2 \)[/tex]), they do not have the same vertical stretch.
2. Vertical Shift:
- Both functions involve a constant term of [tex]\(-5\)[/tex].
Because both functions subtract 5, they are both shifted downward by 5 units. Hence, they have the same vertical shift.
3. Horizontal Translation:
- The term [tex]\( e^{x+5} \)[/tex] in [tex]\( f(x) \)[/tex] implies a horizontal shift to the left by 5 units.
- The term [tex]\( (x - 5)^2 \)[/tex] in [tex]\( g(x) \)[/tex] implies a horizontal shift to the right by 5 units.
Therefore, their horizontal translations are not the same.
4. End Behavior:
- For large values of [tex]\( x \)[/tex], [tex]\( f(x) = 5 e^{x+5} - 5 \)[/tex] will increase exponentially, as exponential functions grow infinitely large.
- For large values of [tex]\( x \)[/tex], [tex]\( g(x) = 0.5 (x - 5)^2 - 5 \)[/tex], since it is a quadratic function, will also increase, but more slowly and symmetrically about the vertex.
Exponential functions and quadratic functions have different end behaviors.
Conclusion:
Based on the above analysis, the correct common property for the given functions is:
B. They have the same vertical shift.
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.