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.
