To determine the correct function [tex]\( f(t) = a \sin(bt + c) + d \)[/tex], we need to match each parameter [tex]\( a \)[/tex], [tex]\( b \)[/tex], [tex]\( c \)[/tex], and [tex]\( d \)[/tex] with the characteristics of the given graph.
1. Amplitude ([tex]\(a\)[/tex]):
- The amplitude of a sine function is the coefficient of the sine term, which indicates the vertical stretch or compression of the graph.
- Here, we see that the amplitude must be [tex]\(-4\)[/tex]. The negative sign indicates that the sine function is reflected over the horizontal axis.
2. Frequency and Period ([tex]\(b\)[/tex]):
- The coefficient [tex]\( b \)[/tex] affects the period or the number of cycles the sine wave completes in a given interval.
- The standard sine function [tex]\(\sin(t)\)[/tex] has a period of [tex]\(2\pi\)[/tex]. When the argument of the sine function is [tex]\( bt \)[/tex], the period becomes [tex]\(\frac{2\pi}{b}\)[/tex].
- Given [tex]\( b = \frac{1}{3} \)[/tex], the period becomes [tex]\(\frac{2\pi}{1/3} = 6\pi\)[/tex].
- Thus, [tex]\( b \)[/tex] has to be [tex]\(\frac{1}{3}\)[/tex].
3. Horizontal Shift ([tex]\(c\)[/tex]):
- The phase shift of the sine function is determined by the value of [tex]\( c \)[/tex]. The function [tex]\( \sin(bt + c) \)[/tex] shifts horizontally by [tex]\(-\frac{c}{b}\)[/tex].
- Given [tex]\( c = \frac{\pi}{6} \)[/tex], the phase shift is [tex]\(\frac{\pi}{6}\)[/tex] units to the left.
4. Vertical Shift ([tex]\(d\)[/tex]):
- The vertical shift [tex]\( d \)[/tex] represents the displacement of the entire graph up or down.
- Here, [tex]\( d = 3 \)[/tex]. This shifts the graph upwards by 3 units.
After matching each aspect with the parameters:
- The correct amplitude is [tex]\(-4\)[/tex].
- The correct frequency coefficient is [tex]\(\frac{1}{3}\)[/tex].
- The correct phase shift coefficient is [tex]\(\frac{\pi}{6}\)[/tex].
- The correct vertical shift is [tex]\(3\)[/tex].
From the options provided, the function that matches all of these parameters is:
a. [tex]\( -4 \sin\left(\frac{1}{3} t + \frac{\pi}{6}\right) + 3 \)[/tex]
Therefore, the correct function is:
[tex]\[ f(t) = -4 \sin\left(\frac{1}{3} t + \frac{\pi}{6}\right) + 3 \][/tex]
