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Sagot :
To determine which function has a [tex]\(y\)[/tex]-intercept at -1 and an amplitude of 2, let's analyze each of the given functions step by step.
### Function Analysis:
1. [tex]\( f(x) = -\sin(x) - 1 \)[/tex]
- Amplitude: The sine function, [tex]\(\sin(x)\)[/tex], has a natural amplitude of 1. Multiplying by -1 leaves the amplitude unchanged (still 1), since the negative sign only reflects the function over the x-axis.
- Y-intercept: To find the y-intercept, substitute [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = -\sin(0) - 1 = 0 - 1 = -1 \][/tex]
The y-intercept is -1.
- Since the amplitude is 1 (not 2), this function does not meet all the conditions.
2. [tex]\( f(x) = -2\sin(x) - 1 \)[/tex]
- Amplitude: The amplitude of the sine function is naturally 1. Here, it is multiplied by -2, which affects the amplitude but the amplitude is always taken as a positive value:
[tex]\[ \text{Amplitude} = 2 \][/tex]
- Y-intercept: To find the y-intercept, substitute [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = -2\sin(0) - 1 = 0 - 1 = -1 \][/tex]
The y-intercept is -1.
- Since both the amplitude is 2 and the y-intercept is -1, this function meets all the conditions.
3. [tex]\( f(x) = -\cos(x) \)[/tex]
- Amplitude: The cosine function, [tex]\(\cos(x)\)[/tex], has a natural amplitude of 1. Multiplying by -1 leaves the amplitude unchanged (still 1), as the negative sign only reflects the function over the x-axis.
- Y-intercept: To find the y-intercept, substitute [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = -\cos(0) = -1 \][/tex]
The y-intercept is -1.
- Since the amplitude is 1 (not 2), this function does not meet all the conditions.
4. [tex]\( f(x) = -2\cos(x) - 1 \)[/tex]
- Amplitude: The amplitude of the cosine function is naturally 1. Here, it is multiplied by -2, so the amplitude is:
[tex]\[ \text{Amplitude} = 2 \][/tex]
- Y-intercept: To find the y-intercept, substitute [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = -2\cos(0) - 1 = -2(1) - 1 = -2 - 1 = -3 \][/tex]
The y-intercept is -3.
- Since the y-intercept is -3 (and not -1), this function does not meet all the conditions.
### Conclusion:
The function that has a [tex]\( y \)[/tex]-intercept at -1 and an amplitude of 2 is:
[tex]\[ f(x) = -2 \sin(x) - 1 \][/tex]
Therefore, the correct function is:
[tex]\[ f(x) = -2\sin(x) - 1 \][/tex]
### Function Analysis:
1. [tex]\( f(x) = -\sin(x) - 1 \)[/tex]
- Amplitude: The sine function, [tex]\(\sin(x)\)[/tex], has a natural amplitude of 1. Multiplying by -1 leaves the amplitude unchanged (still 1), since the negative sign only reflects the function over the x-axis.
- Y-intercept: To find the y-intercept, substitute [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = -\sin(0) - 1 = 0 - 1 = -1 \][/tex]
The y-intercept is -1.
- Since the amplitude is 1 (not 2), this function does not meet all the conditions.
2. [tex]\( f(x) = -2\sin(x) - 1 \)[/tex]
- Amplitude: The amplitude of the sine function is naturally 1. Here, it is multiplied by -2, which affects the amplitude but the amplitude is always taken as a positive value:
[tex]\[ \text{Amplitude} = 2 \][/tex]
- Y-intercept: To find the y-intercept, substitute [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = -2\sin(0) - 1 = 0 - 1 = -1 \][/tex]
The y-intercept is -1.
- Since both the amplitude is 2 and the y-intercept is -1, this function meets all the conditions.
3. [tex]\( f(x) = -\cos(x) \)[/tex]
- Amplitude: The cosine function, [tex]\(\cos(x)\)[/tex], has a natural amplitude of 1. Multiplying by -1 leaves the amplitude unchanged (still 1), as the negative sign only reflects the function over the x-axis.
- Y-intercept: To find the y-intercept, substitute [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = -\cos(0) = -1 \][/tex]
The y-intercept is -1.
- Since the amplitude is 1 (not 2), this function does not meet all the conditions.
4. [tex]\( f(x) = -2\cos(x) - 1 \)[/tex]
- Amplitude: The amplitude of the cosine function is naturally 1. Here, it is multiplied by -2, so the amplitude is:
[tex]\[ \text{Amplitude} = 2 \][/tex]
- Y-intercept: To find the y-intercept, substitute [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = -2\cos(0) - 1 = -2(1) - 1 = -2 - 1 = -3 \][/tex]
The y-intercept is -3.
- Since the y-intercept is -3 (and not -1), this function does not meet all the conditions.
### Conclusion:
The function that has a [tex]\( y \)[/tex]-intercept at -1 and an amplitude of 2 is:
[tex]\[ f(x) = -2 \sin(x) - 1 \][/tex]
Therefore, the correct function is:
[tex]\[ f(x) = -2\sin(x) - 1 \][/tex]
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