1. **State the problem:** We need to sketch one cycle of the function $$f(x) = 3\cos 2(x - 30^\circ) - 2$$ using transformations, then find its domain and range. 2. **Recall the general cosine function and transformations:** The general form is $$f(x) = A \cos(B(x - C)) + D$$ where: - $A$ is the amplitude (vertical stretch/compression), - $B$ affects the period, with period $$\frac{360^\circ}{B}$$, - $C$ is the horizontal shift (phase shift), - $D$ is the vertical shift. 3. **Identify parameters:** - Amplitude: $$|A| = |3| = 3$$ - Period: $$\frac{360^\circ}{2} = 180^\circ$$ - Horizontal shift: $$30^\circ$$ to the right - Vertical shift: $$-2$$ down 4. **Domain of one cycle:** Since the period is $$180^\circ$$ and the function is shifted right by $$30^\circ$$, one cycle runs from $$x = 30^\circ$$ to $$x = 30^\circ + 180^\circ = 210^\circ$$. 5. **Range:** Amplitude is 3 and vertical shift is -2, so: $$\text{Range} = [D - A, D + A] = [-2 - 3, -2 + 3] = [-5, 1]$$ 6. **Find the 5 key points:** Key points for cosine over one period are at: - Start (phase shift): $$x = 30^\circ$$ - Quarter period: $$30^\circ + \frac{180^\circ}{4} = 75^\circ$$ - Half period: $$30^\circ + \frac{180^\circ}{2} = 120^\circ$$ - Three quarters period: $$30^\circ + \frac{3 \times 180^\circ}{4} = 165^\circ$$ - End of cycle: $$210^\circ$$ Calculate function values: - $$f(30^\circ) = 3\cos 2(30^\circ - 30^\circ) - 2 = 3\cos 0^\circ - 2 = 3(1) - 2 = 1$$ - $$f(75^\circ) = 3\cos 2(75^\circ - 30^\circ) - 2 = 3\cos 90^\circ - 2 = 3(0) - 2 = -2$$ - $$f(120^\circ) = 3\cos 2(120^\circ - 30^\circ) - 2 = 3\cos 180^\circ - 2 = 3(-1) - 2 = -5$$ - $$f(165^\circ) = 3\cos 2(165^\circ - 30^\circ) - 2 = 3\cos 270^\circ - 2 = 3(0) - 2 = -2$$ - $$f(210^\circ) = 3\cos 2(210^\circ - 30^\circ) - 2 = 3\cos 360^\circ - 2 = 3(1) - 2 = 1$$ 7. **Summary:** - Domain: $$[30^\circ, 210^\circ]$$ - Range: $$[-5, 1]$$ - Key points: $$(30^\circ, 1), (75^\circ, -2), (120^\circ, -5), (165^\circ, -2), (210^\circ, 1)$$ This completes the analysis and sketch instructions for one cycle of the function.