To determine the correct function of the form [tex]\( f(t) = a \sin(bt + c) + d \)[/tex] from the given options, let's go through the process step-by-step:
We're given the graph of a sine function, and we have four different choices to consider. Let's analyze each choice individually and compare it with the general form of the function.
### 1. Analyze the form of the given function:
The general form is:
[tex]\[ f(t) = a \sin(bt + c) + d \][/tex]
Where:
- [tex]\( a \)[/tex] is the amplitude
- [tex]\( b \)[/tex] affects the period of the sine function
- [tex]\( c \)[/tex] is the phase shift
- [tex]\( d \)[/tex] is the vertical shift
### 2. Identify parameters from the choices:
Let's rewrite the given choices in the form [tex]\( f(t) = a \sin(bt + c) + d \)[/tex]:
- Choice a: [tex]\( -7 \sin \left(\frac{1}{3} t + \frac{\pi}{3}\right) - 3 \)[/tex]
- Amplitude [tex]\( (a) = -7 \)[/tex]
- Frequency [tex]\( (b) = \frac{1}{3} \)[/tex]
- Phase shift [tex]\( (c) = \frac{\pi}{3} \)[/tex]
- Vertical shift [tex]\( (d) = -3 \)[/tex]
- Choice b: [tex]\( 7 \sin \left(\frac{1}{3} t - \frac{\pi}{3}\right) = -2 \)[/tex]
- This choice seems incorrectly stated because the right-hand side is an equation, indicating an incorrect form.
- Choice c: [tex]\( -7 \sin \left(3 t - \frac{\pi}{3}\right) + 2 \)[/tex]
- Amplitude [tex]\( (a) = -7 \)[/tex]
- Frequency [tex]\( (b) = 3 \)[/tex]
- Phase shift [tex]\( (c) = -\frac{\pi}{3} \)[/tex]
- Vertical shift [tex]\( (d) = 2 \)[/tex]
- Choice d: [tex]\( -7 \sin \left(\frac{1}{3} t - \frac{\pi}{3}\right) + 2 \)[/tex]
- Amplitude [tex]\( (a) = -7 \)[/tex]
- Frequency [tex]\( (b) = \frac{1}{3} \)[/tex]
- Phase shift [tex]\( (c) = -\frac{\pi}{3} \)[/tex]
- Vertical shift [tex]\( (d) = 2 \)[/tex]
### 3. Compare the parameters to the general form:
Now let's compare these choices to see which function matches the description in the problem:
- Amplitude [tex]\( (a) \)[/tex]: The value of [tex]\( a = -7 \)[/tex]. Both choices a, c, and d have this characteristic.
- Frequency [tex]\( (b) \)[/tex]: The value of [tex]\( b = \frac{1}{3} \)[/tex]. Both choices a and d have this characteristic.
- Phase Shift [tex]\( (c) \)[/tex]: The value of [tex]\( c = -\frac{\pi}{3} \)[/tex]. Choice d matches this phase shift.
- Vertical Shift [tex]\( (d) \)[/tex]: The value of [tex]\( d = 2 \)[/tex]. Choice d matches this vertical shift.
### Conclusion:
After comparing all choices, the function [tex]$ f(t) = -7 \sin \left( \frac{1}{3} t - \frac{\pi}{3} \right) + 2 $[/tex] (Choice d) has the parameters [tex]\( a = -7 \)[/tex], [tex]\( b = \frac{1}{3} \)[/tex], [tex]\( c = -\frac{\pi}{3} \)[/tex], and [tex]\( d = 2 \)[/tex], which makes it identical to the given graph.
Therefore, the correct function is:
[tex]\[ \boxed{d. -7 \sin \left( \frac{1}{3} t - \frac{\pi}{3} \right) + 2} \][/tex]
