1. **Problem Statement:** Find the period, amplitude, constants affecting the function, domain, and range of the sine function given in part 2a and graph the function in part 2b. 2. **Recall the sine function formula:** The general form of a sine function is $$y = A \sin(Bx + C) + D$$ where: - $A$ is the amplitude (height of the wave from the center line). - $B$ affects the period of the function. - $C$ is the phase shift (horizontal shift). - $D$ is the vertical shift. 3. **Analyze the sine graph in 2a:** - The standard sine function is $$y = \sin x$$. - Amplitude $A = 1$ (since sine ranges from -1 to 1). - Period $P = \frac{2\pi}{B} = 2\pi$ (since $B=1$ here). - Domain: all real numbers $(-\infty, \infty)$. - Range: $[-1, 1]$. - Constants affecting the function: amplitude $1$, no vertical or horizontal shifts. 4. **Graph the function in 2b:** Given $$y = \frac{4}{5} \sin\left(3x + \frac{7}{6}\pi\right) + 1$$ - Amplitude $A = \frac{4}{5}$. - $B = 3$, so period $$P = \frac{2\pi}{3}$$. - Phase shift $$= -\frac{C}{B} = -\frac{\frac{7}{6}\pi}{3} = -\frac{7\pi}{18}$$ (shift left). - Vertical shift $D = 1$ (shift up by 1). - Domain: all real numbers $(-\infty, \infty)$. - Range: from $1 - \frac{4}{5} = \frac{1}{5}$ to $1 + \frac{4}{5} = \frac{9}{5}$. 5. **Desmos LaTeX for graph 2b:** $$y=\frac{4}{5}\sin\left(3x + \frac{7}{6}\pi\right) + 1$$ 6. **Relation between graphs 2a and 2b:** - Graph 2b is a vertically shifted, horizontally compressed, and phase-shifted version of the standard sine wave in 2a. - Amplitude decreased from 1 to $\frac{4}{5}$. - Period decreased from $2\pi$ to $\frac{2\pi}{3}$. - Shifted left by $\frac{7\pi}{18}$ and up by 1. **Final answers:** - 2a: Amplitude = 1, Period = $2\pi$, Domain = $(-\infty, \infty)$, Range = $[-1,1]$. - 2b: Amplitude = $\frac{4}{5}$, Period = $\frac{2\pi}{3}$, Phase shift = $-\frac{7\pi}{18}$, Vertical shift = 1, Domain = $(-\infty, \infty)$, Range = $[\frac{1}{5}, \frac{9}{5}]$.