Let's analyze the trigonometric equation given:
[tex]\[ y = \frac{1}{2} \sin(x + 7) \][/tex]
### Step 1: Determine the Amplitude
The amplitude of a sinusoidal function [tex]\( y = A \sin(B(x - C)) + D \)[/tex] is given by the coefficient [tex]\( A \)[/tex] in front of the sine function. Here, the equation is:
[tex]\[ y = \frac{1}{2} \sin(x + 7) \][/tex]
So, the amplitude is:
[tex]\[ \text{Amplitude} = \frac{1}{2} \][/tex]
### Step 2: Determine the Period
The period of the sine function [tex]\( y = A \sin(B(x - C)) + D \)[/tex] is determined by the value [tex]\( B \)[/tex] using the formula:
[tex]\[ \text{Period} = \frac{2\pi}{B} \][/tex]
In our equation, [tex]\( B \)[/tex] is 1 (since there's no coefficient directly multiplying [tex]\( x \)[/tex] inside the sine function), so the period is:
[tex]\[ \text{Period} = \frac{2\pi}{1} = 2\pi \][/tex]
### Step 3: Determine the Phase Shift
The phase shift is determined by the term inside the sine function. The general form [tex]\( y = A \sin(B(x - C)) + D \)[/tex] suggests a phase shift determined by [tex]\( C \)[/tex]:
[tex]\[ y = A \sin(B(x - C)) + D \][/tex]
For our equation [tex]\( y = \frac{1}{2} \sin(x + 7) \)[/tex], we can rewrite it as:
[tex]\[ y = \frac{1}{2} \sin \left( x - (-7) \right) \][/tex]
Here, it is evident that [tex]\( C = -7 \)[/tex]. This means the function is shifted to the left by 7 units.
### Summary
- Amplitude: [tex]\( \frac{1}{2} \)[/tex]
- Period: [tex]\( 2\pi \)[/tex]
- Phase Shift: shifted to the left by 7 units
So, the final results are:
- Amplitude: [tex]\( \frac{1}{2} \)[/tex]
- Period: [tex]\( 6.283185307179586 \)[/tex]
- Phase Shift: "shifted to the left"
