To determine which function's graph passes through the point [tex]\((0,3)\)[/tex] and has an amplitude of 3, we need to analyze each option step-by-step:
1. Option 1: [tex]\( f(x) = \sin(x) + 3 \)[/tex]
- Amplitude: The coefficient of the sine function here is 1, so the amplitude is 1.
- Passes through [tex]\((0, 3)\)[/tex]: When [tex]\(x = 0\)[/tex], [tex]\(f(0) = \sin(0) + 3 = 0 + 3 = 3\)[/tex].
Conclusion: This function passes through [tex]\((0, 3)\)[/tex], but its amplitude is not 3. It is 1. Therefore, this is not the correct function.
2. Option 2: [tex]\( f(x) = \cos(x) + 3 \)[/tex]
- Amplitude: The coefficient of the cosine function here is 1, so the amplitude is 1.
- Passes through [tex]\((0, 3)\)[/tex]: When [tex]\(x = 0\)[/tex], [tex]\(f(0) = \cos(0) + 3 = 1 + 3 = 4\)[/tex].
Conclusion: This function does not pass through [tex]\((0, 3)\)[/tex], and its amplitude is not 3. It is 1. Therefore, this is not the correct function.
3. Option 3: [tex]\( f(x) = 3 \sin(x) \)[/tex]
- Amplitude: The coefficient of the sine function here is 3, so the amplitude is 3.
- Passes through [tex]\((0, 3)\)[/tex]: When [tex]\(x = 0\)[/tex], [tex]\(f(0) = 3 \sin(0) = 3 \times 0 = 0\)[/tex].
Conclusion: This function has the correct amplitude of 3, but it does not pass through [tex]\((0, 3)\)[/tex]. Therefore, this is not the correct function.
4. Option 4: [tex]\( f(x) = 3 \cos(x) \)[/tex]
- Amplitude: The coefficient of the cosine function here is 3, so the amplitude is 3.
- Passes through [tex]\((0, 3)\)[/tex]: When [tex]\(x = 0\)[/tex], [tex]\(f(0) = 3 \cos(0) = 3 \times 1 = 3\)[/tex].
Conclusion: This function has the correct amplitude of 3, and it passes through [tex]\((0, 3)\)[/tex]. Therefore, this is the correct function.
Thus, the correct function is [tex]\(\boxed{4}\)[/tex].
