1. **State the problem:** We are given the function $y = 3\sin\theta - 2$ and need to find its domain, range, amplitude, period, phase shift, and vertical slide. 2. **Recall the general sine function form:** $$y = A \sin(B(\theta - C)) + D$$ where: - $A$ is the amplitude - $\frac{2\pi}{B}$ is the period - $C$ is the phase shift - $D$ is the vertical shift 3. **Identify parameters from the given function:** - $A = 3$ - $B = 1$ (since no coefficient is shown before $\theta$) - $C = 0$ (no horizontal shift) - $D = -2$ 4. **Calculate each property:** - **Domain:** The sine function is defined for all real numbers, so domain is $(-\infty, \infty)$. - **Amplitude:** $|A| = |3| = 3$ - **Period:** $$\text{Period} = \frac{2\pi}{B} = \frac{2\pi}{1} = 2\pi$$ - **Phase shift:** Since $C=0$, phase shift is $0$. - **Vertical slide:** $D = -2$ 5. **Find the range:** The sine function oscillates between $-1$ and $1$, so: $$y = 3\sin\theta - 2$$ oscillates between: $$3(-1) - 2 = -3 - 2 = -5$$ and $$3(1) - 2 = 3 - 2 = 1$$ Thus, range is $[-5, 1]$. **Final answers:** - Domain: $(-\infty, \infty)$ - Range: $[-5, 1]$ - Amplitude: $3$ - Period: $2\pi$ - Phase shift: $0$ - Vertical slide: $-2$