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Sagot :
Let's address Question 8 step-by-step:
### Step-by-Step Solution
1. Identify the Growth Rate:
- The exponential function increases at a rate of [tex]\(30\%\)[/tex]. This growth rate does not directly affect the horizontal asymptote but is an important feature of the exponential function.
2. Determine the Initial Value:
- The function passes through the point [tex]\((0, -1)\)[/tex]. Therefore, when [tex]\(x = 0\)[/tex], [tex]\(t(x) = -1\)[/tex]. This is the initial value before any shifts are considered.
3. Apply the Vertical Shift:
- The exponential function is shifted down by 5 units. A vertical shift affects the entire graph of the function. Since the function is shifted downward by 5 units, the horizontal asymptote is also shifted downward.
4. Determine the Horizontal Asymptote:
- The horizontal asymptote of an exponential function [tex]\(f(x) = a(b^x) + c\)[/tex] is determined by the value [tex]\(c\)[/tex]. This is the value the function approaches as [tex]\(x\)[/tex] approaches infinity.
- Since the function is shifted down by 5 units, the horizontal asymptote will also be translated down by the same amount. If the original horizontal asymptote was [tex]\(y = 0\)[/tex] for a typical exponential function, shifting it down by 5 units results in a new horizontal asymptote of [tex]\(y = -5\)[/tex].
5. Conclusion:
- The horizontal asymptote of the exponential function, after considering the vertical shift, is [tex]\(y = -5\)[/tex].
Therefore, the correct answer is:
[tex]\[ y = -5 \][/tex]
Answer choice:
[tex]\[ \boxed{y = -5} \][/tex]
This solution addresses all the factors influencing the horizontal asymptote including the growth rate, initial value, and the vertical shift.
### Step-by-Step Solution
1. Identify the Growth Rate:
- The exponential function increases at a rate of [tex]\(30\%\)[/tex]. This growth rate does not directly affect the horizontal asymptote but is an important feature of the exponential function.
2. Determine the Initial Value:
- The function passes through the point [tex]\((0, -1)\)[/tex]. Therefore, when [tex]\(x = 0\)[/tex], [tex]\(t(x) = -1\)[/tex]. This is the initial value before any shifts are considered.
3. Apply the Vertical Shift:
- The exponential function is shifted down by 5 units. A vertical shift affects the entire graph of the function. Since the function is shifted downward by 5 units, the horizontal asymptote is also shifted downward.
4. Determine the Horizontal Asymptote:
- The horizontal asymptote of an exponential function [tex]\(f(x) = a(b^x) + c\)[/tex] is determined by the value [tex]\(c\)[/tex]. This is the value the function approaches as [tex]\(x\)[/tex] approaches infinity.
- Since the function is shifted down by 5 units, the horizontal asymptote will also be translated down by the same amount. If the original horizontal asymptote was [tex]\(y = 0\)[/tex] for a typical exponential function, shifting it down by 5 units results in a new horizontal asymptote of [tex]\(y = -5\)[/tex].
5. Conclusion:
- The horizontal asymptote of the exponential function, after considering the vertical shift, is [tex]\(y = -5\)[/tex].
Therefore, the correct answer is:
[tex]\[ y = -5 \][/tex]
Answer choice:
[tex]\[ \boxed{y = -5} \][/tex]
This solution addresses all the factors influencing the horizontal asymptote including the growth rate, initial value, and the vertical shift.
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