To determine the vertical shift of the function [tex]\( y = -4 + 5 \sin \left( \frac{2x}{3} + \frac{\pi}{3} \right) \)[/tex], we need to identify the constant term that is added to the sine function.
Let's break down the function step-by-step:
1. Identify the General Form of the Sine Function: The standard form of a sine function with a vertical shift is:
[tex]\[
y = a + b \sin (cx + d)
\][/tex]
Here, [tex]\(a\)[/tex] represents the vertical shift, [tex]\(b\)[/tex] represents the amplitude, [tex]\(c\)[/tex] determines the frequency, and [tex]\(d\)[/tex] is the phase shift.
2. Compare with Our Function: Our given function is:
[tex]\[
y = -4 + 5 \sin \left( \frac{2x}{3} + \frac{\pi}{3} \right)
\][/tex]
Comparing this with the standard form, we can see that:
[tex]\[
a = -4
\][/tex]
Hence, the vertical shift is [tex]\(a\)[/tex].
3. Vertical Shift: The constant term [tex]\(a\)[/tex] in the equation [tex]\( y = a + b \sin (cx + d) \)[/tex] indicates the vertical shift of the graph of the sine function. For our specific function:
[tex]\[
a = -4
\][/tex]
Therefore, the function [tex]\( y = -4 + 5 \sin \left( \frac{2x}{3} + \frac{\pi}{3} \right) \)[/tex] is shifted vertically downwards by 4 units.
4. Conclusion: The vertical shift of the function is [tex]\( -4 \)[/tex].
Thus, the correct answer is:
[tex]\[
-4
\][/tex]
