1. **State the problem:** We need to determine which statements are true for the function $$f(x) = -2\sin(x) - 3$$. 2. **Recall the general form and properties:** The general sine function is $$y = A\sin(Bx + C) + D$$ where: - Amplitude = $$|A|$$ - Period = $$\frac{2\pi}{|B|}$$ - Vertical shift = $$D$$ 3. **Analyze the given function:** Here, $$A = -2$$, $$B = 1$$, and $$D = -3$$. - Amplitude = $$|-2| = 2$$ - Period = $$\frac{2\pi}{1} = 2\pi$$ - Vertical shift = $$-3$$ (shift down 3 units) 4. **Check each statement:** - "The range of the function is the set of real numbers $$-2 \leq y \leq 2$$." - The sine function ranges from $$-1$$ to $$1$$. - Multiply by $$-2$$: range becomes $$[-2, 2]$$ but reversed because of negative sign, so actually $$[ -2, 2 ]$$ but flipped vertically. - Then shift down by 3: range becomes $$[-2 - 3, 2 - 3] = [-5, -1]$$. - So the statement is **false**. - "The graph of the function is the graph of $$f(x) = -2\sin(x)$$ shifted 3 units up." - The function is $$-2\sin(x) - 3$$, which is shifted **down** 3 units, not up. - So this statement is **false**. - "The amplitude of the function is 2." - Amplitude is the absolute value of $$A$$, which is $$2$$. - This statement is **true**. - "The period of the function is $$4\pi$$." - Period is $$\frac{2\pi}{|B|} = 2\pi$$. - So this statement is **false**. **Final answer:** The only true statement is "The amplitude of the function is 2."