1. **Problem Statement:** Find the amplitude, period, phase shift, vertical translation, and range of the function $y = -4 \sin(2x - \pi)$, then graph it over at least one period. 2. **General form and formulas:** The function is of the form $y = a \sin(bx - c) + d$. - Amplitude = $|a|$ - Period = $\frac{2\pi}{b}$ - Phase shift = $\frac{c}{b}$ (shift right if positive, left if negative) - Vertical translation = $d$ - Range = $[d - |a|, d + |a|]$ 3. **Identify parameters:** - $a = -4$ - $b = 2$ - $c = \pi$ - $d = 0$ 4. **Calculate amplitude:** $$\text{Amplitude} = |-4| = 4$$ 5. **Calculate period:** $$\text{Period} = \frac{2\pi}{2} = \pi$$ 6. **Calculate phase shift:** $$\text{Phase shift} = \frac{\pi}{2}$$ Since $c$ is positive, the graph shifts right by $\frac{\pi}{2}$. 7. **Vertical translation:** $$d = 0$$ No vertical shift. 8. **Calculate range:** $$[0 - 4, 0 + 4] = [-4, 4]$$ 9. **Summary:** - Amplitude: 4 - Period: $\pi$ - Phase shift: $\frac{\pi}{2}$ right - Vertical translation: 0 - Range: $[-4, 4]$ 10. **Graph function:** The function $y = -4 \sin(2x - \pi)$ can be rewritten as $y = -4 \sin\left(2\left(x - \frac{\pi}{2}\right)\right)$ to show the phase shift. --- **Next problem:** $y = 3 \cos(4x + \pi)$ 1. Parameters: - $a = 3$ - $b = 4$ - $c = -\pi$ (since $4x + \pi = 4(x + \frac{\pi}{4})$) - $d = 0$ 2. Amplitude: $$|3| = 3$$ 3. Period: $$\frac{2\pi}{4} = \frac{\pi}{2}$$ 4. Phase shift: $$-\frac{\pi}{4}$$ (shift left by $\frac{\pi}{4}$) 5. Vertical translation: $$0$$ 6. Range: $$[-3, 3]$$ 7. Rewrite function: $$y = 3 \cos\left(4\left(x + \frac{\pi}{4}\right)\right)$$ --- **Next problem:** $y = \frac{1}{2} \cos\left(\frac{1}{2}x - \frac{\pi}{4}\right)$ 1. Parameters: - $a = \frac{1}{2}$ - $b = \frac{1}{2}$ - $c = \frac{\pi}{4}$ - $d = 0$ 2. Amplitude: $$\frac{1}{2}$$ 3. Period: $$\frac{2\pi}{\frac{1}{2}} = 4\pi$$ 4. Phase shift: $$\frac{\pi/4}{1/2} = \frac{\pi}{2}$$ right 5. Vertical translation: $$0$$ 6. Range: $$\left[-\frac{1}{2}, \frac{1}{2}\right]$$ 7. Rewrite function: $$y = \frac{1}{2} \cos\left(\frac{1}{2}\left(x - \frac{\pi}{2}\right)\right)$$ --- **Next problem:** $y = -\frac{1}{4} \sin\left(\frac{3}{4}x + \frac{\pi}{8}\right)$ 1. Parameters: - $a = -\frac{1}{4}$ - $b = \frac{3}{4}$ - $c = -\frac{\pi}{8}$ - $d = 0$ 2. Amplitude: $$\frac{1}{4}$$ 3. Period: $$\frac{2\pi}{\frac{3}{4}} = \frac{8\pi}{3}$$ 4. Phase shift: $$-\frac{\pi/8}{3/4} = -\frac{\pi}{6}$$ left 5. Vertical translation: $$0$$ 6. Range: $$\left[-\frac{1}{4}, \frac{1}{4}\right]$$ 7. Rewrite function: $$y = -\frac{1}{4} \sin\left(\frac{3}{4}\left(x + \frac{\pi}{6}\right)\right)$$ --- **Next problem:** $y = 1 - \frac{2}{3} \sin\left(\frac{3}{4}x\right)$ 1. Parameters: - $a = -\frac{2}{3}$ - $b = \frac{3}{4}$ - $c = 0$ - $d = 1$ 2. Amplitude: $$\left| -\frac{2}{3} \right| = \frac{2}{3}$$ 3. Period: $$\frac{2\pi}{\frac{3}{4}} = \frac{8\pi}{3}$$ 4. Phase shift: $$0$$ 5. Vertical translation: $$1$$ 6. Range: $$\left[1 - \frac{2}{3}, 1 + \frac{2}{3}\right] = \left[\frac{1}{3}, \frac{5}{3}\right]$$ 7. Rewrite function: $$y = 1 - \frac{2}{3} \sin\left(\frac{3}{4}x\right)$$ --- **Next problem:** $y = -1 - 2 \cos(5x)$ 1. Parameters: - $a = -2$ - $b = 5$ - $c = 0$ - $d = -1$ 2. Amplitude: $$2$$ 3. Period: $$\frac{2\pi}{5}$$ 4. Phase shift: $$0$$ 5. Vertical translation: $$-1$$ 6. Range: $$[-1 - 2, -1 + 2] = [-3, 1]$$ 7. Rewrite function: $$y = -1 - 2 \cos(5x)$$ --- **Next problem:** $y = 1 - 2 \cos\left(\frac{1}{2}x\right)$ 1. Parameters: - $a = -2$ - $b = \frac{1}{2}$ - $c = 0$ - $d = 1$ 2. Amplitude: $$2$$ 3. Period: $$\frac{2\pi}{\frac{1}{2}} = 4\pi$$ 4. Phase shift: $$0$$ 5. Vertical translation: $$1$$ 6. Range: $$[1 - 2, 1 + 2] = [-1, 3]$$ 7. Rewrite function: $$y = 1 - 2 \cos\left(\frac{1}{2}x\right)$$ --- **Next problem:** $y = -3 + 3 \sin\left(\frac{1}{2}x\right)$ 1. Parameters: - $a = 3$ - $b = \frac{1}{2}$ - $c = 0$ - $d = -3$ 2. Amplitude: $$3$$ 3. Period: $$4\pi$$ 4. Phase shift: $$0$$ 5. Vertical translation: $$-3$$ 6. Range: $$[-3 - 3, -3 + 3] = [-6, 0]$$ 7. Rewrite function: $$y = -3 + 3 \sin\left(\frac{1}{2}x\right)$$ --- **Next problem:** $y = -3 + 2 \sin\left(x + \frac{\pi}{2}\right)$ 1. Parameters: - $a = 2$ - $b = 1$ - $c = -\frac{\pi}{2}$ - $d = -3$ 2. Amplitude: $$2$$ 3. Period: $$2\pi$$ 4. Phase shift: $$-\frac{\pi}{2}$$ left 5. Vertical translation: $$-3$$ 6. Range: $$[-3 - 2, -3 + 2] = [-5, -1]$$ 7. Rewrite function: $$y = -3 + 2 \sin\left(x + \frac{\pi}{2}\right) = -3 + 2 \sin\left(x - \left(-\frac{\pi}{2}\right)\right)$$ --- **Next problem:** $y = 4 - 3 \cos\left(x - \pi\right)$ 1. Parameters: - $a = -3$ - $b = 1$ - $c = \pi$ - $d = 4$ 2. Amplitude: $$3$$ 3. Period: $$2\pi$$ 4. Phase shift: $$\pi$$ right 5. Vertical translation: $$4$$ 6. Range: $$[4 - 3, 4 + 3] = [1, 7]$$ 7. Rewrite function: $$y = 4 - 3 \cos\left(x - \pi\right)$$ --- **Next problem:** $y = \frac{1}{2} + \sin\left(2\left(x + \frac{\pi}{4}\right)\right)$ 1. Parameters: - $a = 1$ - $b = 2$ - $c = -\frac{\pi}{4}$ - $d = \frac{1}{2}$ 2. Amplitude: $$1$$ 3. Period: $$\frac{2\pi}{2} = \pi$$ 4. Phase shift: $$-\frac{\pi}{4}$$ left 5. Vertical translation: $$\frac{1}{2}$$ 6. Range: $$\left[\frac{1}{2} - 1, \frac{1}{2} + 1\right] = \left[-\frac{1}{2}, \frac{3}{2}\right]$$ 7. Rewrite function: $$y = \frac{1}{2} + \sin\left(2\left(x + \frac{\pi}{4}\right)\right)$$ --- **Slug:** "trig function analysis" **Subject:** "trigonometry" **Desmos:** {"latex":"y=-4\sin(2x-\pi)","features":{"intercepts":true,"extrema":true}} **q_count:** 11