Sure, let's go through the function [tex]\( H(t) \)[/tex] step by step. The function is given by:
[tex]\[ H(t) = 4.8 \sin \left(\frac{\pi}{6}(t+3)\right) + 5.1 \][/tex]
Let's break this down:
1. Inside the Sine Function:
- First, consider the argument of the sine function: [tex]\( \frac{\pi}{6}(t+3) \)[/tex].
- This means we are scaling [tex]\( t \)[/tex] by [tex]\( \frac{\pi}{6} \)[/tex] and then shifting it horizontally by 3 units to the left.
2. Simplify the Argument:
- Distribute [tex]\( \frac{\pi}{6} \)[/tex] inside the parenthesis: [tex]\( \frac{\pi}{6} \cdot t + \frac{\pi}{6} \cdot 3 \)[/tex].
- This simplifies to [tex]\( \frac{\pi}{6} t + \frac{\pi}{2} \)[/tex].
3. Express in Simplified Form:
- So, [tex]\( \frac{\pi}{6}(t+3) \)[/tex] becomes [tex]\( \frac{\pi}{6} t + \frac{\pi}{2} \)[/tex].
4. Modified Function:
- Now substitute this back into the function: [tex]\( 4.8 \sin \left(\frac{\pi}{6} t + \frac{\pi}{2} \right) + 5.1 \)[/tex].
Thus, the function can be rewritten as:
[tex]\[ H(t) = 4.8 \sin \left( \frac{\pi}{6}t + \frac{\pi}{2} \right) + 5.1 \][/tex]
Now, we have our function [tex]\( H(t) \)[/tex] in a standard form:
[tex]\[ H(t) = 4.8 \sin \left( \frac{\pi}{6} t + \frac{\pi}{2} \right) + 5.1 \][/tex]
This form clearly shows the amplitude, phase shift, and vertical shift:
- The amplitude is 4.8.
- The phase shift is [tex]\( \frac{\pi}{2} \)[/tex], which means the graph of the sine function is shifted [tex]\( \frac{\pi}{2} \)[/tex] units to the left.
- The vertical shift is 5.1, which means the entire graph of the sine function is moved 5.1 units upward.
The simplified result for [tex]\( H(t) \)[/tex] is:
[tex]\[ 4.8 \sin \left( \frac{\pi}{6} t + \frac{\pi}{2} \right) + 5.1 \][/tex]
