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学習ガイド > Precalculus II

Solutions to Graph's of the Sine and Cosine Function

Solutions to Try Its

1. 6π 2. 12\frac{1}{2} compressed 3. π2\frac{π}{2}; right 4. 2 units up 5. midline: y=0y=0; amplitude: |A|=12\frac{1}{2}; period: P=2π∣B∣=6π\frac{2π}{|B|}=6\pi; phase shift:CB=π\frac{C}{B}=\pi 6. f(x)=sin⁡(x)+2f(x)=\sin(x)+2 7. two possibilities: y=4sin⁡(π5x−π5)+4y=4\sin(\frac{π}{5}x−\frac{π}{5})+4 or y=−4sin(π5x+4π5)+4y=−4sin(\frac{π}{5}x+4\frac{π}{5})+4 8. midline: y=0; amplitude: |A|=0.8; period: P=2π∣B∣=π\frac{2π}{|B|}=\pi; phase shift: CB=0\frac{C}{B}=0 or none A graph of -0.8cos(2x). Graph has range of [-0.8, 0.8], period of pi, amplitude of 0.8, and is reflected about the x-axis compared to it's parent function cos(x). 9. midline:y=0;amplitude:∣A∣=2;period:P=2π∣B∣=6;phase shift:CB=−12\text{midline:}y=0;\text{amplitude:}|A|=2;\text{period:}\text{P}=\frac{2\pi}{|B|}=6;\text{phase shift:}\text{C}{B}=−\text{1}{2} A graph of -2cos((pi/3)x+(pi/6)). Graph has amplitude of 2, period of 6, and has a phase shift of 0.5 to the left. 10. 7 A graph of 7cos(x). Graph has amplitude of 7, period of 2pi, and range of [-7,7]. 11. y=3cos⁡(x)−4y=3\cos(x)−4 A cosine graph with range [-1,-7]. Period is 2 pi. Local maximums at (0,-1), (2pi,-1), and (4pi, -1). Local minimums at (pi,-7) and (3pi, -7).

Solutions to Odd-Numbered Exercises

1. The sine and cosine functions have the property that f(x+P)=f(x)f(x+P)=f(x) for a certain P. This means that the function values repeat for every P units on the x-axis. 3. The absolute value of the constant A (amplitude) increases the total range and the constant D (vertical shift) shifts the graph vertically. 5. At the point where the terminal side of t intersects the unit circle, you can determine that the sin t equals the y-coordinate of the point. 7. amplitude: 23\frac{2}{3}; period: 2π; midline: y=0y=0; maximum: y=23y=23 occurs at x=0x=0; minimum: y=−23y=−23 occurs at x=πx=\pi; for one period, the graph starts at 0 and ends at 2π A graph of (2/3)cos(x). Graph has amplitude of 2/3, period of 2pi, and range of [-2/3, 2/3]. 9. amplitude: 4; period: 2π; midline: y=0y=0; maximum y=4y=4 occurs at x=π2x=\frac{\pi}{2}; minimum: y=−4y=−4 occurs at x=3π2x=\frac{3\pi}{2}; one full period occurs from x=0x=0 to x=2πx=2π A graph of 4sin(x). Graph has amplitude of 4, period of 2pi, and range of [-4, 4]. 11. amplitude: 1; period: π; midline: y=0; maximum: y=1 occurs at x=πx=\pi; minimum: y=−1y=−1 occurs at x=π2x=\frac{\pi}{2}; one full period is graphed from x=0x=0 to x=πx=\pi A graph of cos(2x). Graph has amplitude of 1, period of pi, and range of [-1,1]. 13. amplitude: 4; period: 2; midline: y=0y=0; maximum: y=4y=4 occurs at x=0x=0; minimum: y=−4y=−4 occurs at x=1x=1 A graph of 4cos(pi*x). Grpah has amplitude of 4, period of 2, and range of [-4, 4]. 15. amplitude: 3; period: π4\frac{\pi}{4}; midline: y=5y=5; maximum: y=8y=8 occurs at x=0.12x=0.12; minimum: y=2y=2 occurs at x=0.516x=0.516; horizontal shift: −4; vertical translation 5; one period occurs from x=0x=0 to x=π4x=\frac{\pi}{4} A graph of 3sin(8(x+4))+5. Graph has amplitude of 3, range of [2, 8], and period of pi/4. 17. amplitude: 5; period: 2π5;midline:[latex]y=−2\frac{2\pi}{5}; midline: [latex]y=−2; maximum: y=3y=3 occurs at x=0.08x=0.08; minimum: y=−7y=−7 occurs at x=0.71x=0.71; phase shift:−4; vertical translation:−2; one full period can be graphed on x=0x=0 to x=2π5x=\frac{2\pi}{5} A graph of 5sin(5x+20)-2. Graph has an amplitude of 5, period of 2pi/5, and range of [-7,3]. 19. amplitude: 1; period: 2π; midline: y=1; maximum:y=2y=2 occurs at x=2.09x=2.09; maximum:y=2y=2 occurs att=2.09t=2.09; minimum:y=0y=0 occurs at t=5.24t=5.24; phase shift: −π3−\frac{\pi}{3}; vertical translation: 1; one full period is from t=0t=0 to t=2πt=2π A graph of -cos(t+pi/3)+1. Graph has amplitude of 1, period of 2pi, and range of [0,2]. Phase shifted pi/3 to the left. 21. amplitude: 1; period: 4π; midline: y=0y=0; maximum: y=1y=1 occurs at t=11.52t=11.52; minimum: y=−1y=−1 occurs at t=5.24t=5.24; phase shift: −10π3\frac{10\pi}{3}; vertical shift: 0 A graph of -sin((1/2)*t + 5pi/3). Graph has amplitude of 1, range of [-1,1], period of 4pi, and a phase shift of -10pi/3. 23. amplitude: 2; midline: y=−3y=−3; period: 4; equation: f(x)=2sin⁡(π2x)−3f(x)=2\sin(\frac{\pi}{2}x)−3 25. amplitude: 2; period: 5; midline: y=3y=3; equation: f(x)=−2cos⁡(2π5x)+3f(x)=−2\cos(\frac{2\pi}{5}x)+3 27. amplitude: 4; period: 2; midline: y=0y=0; equation: f(x)=−4cos⁡(π(x−π2))f(x)=−4\cos(\pi(x−\frac{\pi}{2})) 29. amplitude: 2; period: 2; midline y=1y=1; equation: f(x)=2cos⁡(πx)+1f(x)=2\cos(\frac{\pi}{x})+1 31. π6,5π6\frac{\pi}{6},\frac{5\pi}{6} 33. π4,3π4\frac{\pi}{4},\frac{3\pi}{4} 35. 3π2\frac{3\pi}{2} 37. π2,3π2\frac{\pi}{2},\frac{3\pi}{2} 39. π2,3π2\frac{\pi}{2},\frac{3\pi}{2} 41. π6,11π6\frac{\pi}{6},\frac{11\pi}{6} 43. The graph appears linear. The linear functions dominate the shape of the graph for large values of x. A sinusoidal graph that increases like the function y=x, shown from 0 to 100. 45. The graph is symmetric with respect to the y-axis and there is no amplitude because the function is not periodic. A sinusoidal graph that has increasing peaks and decreasing lows as the absolute value of x increases. 47. a. Amplitude: 12.5; period: 10; midline: y=13.5y=13.5; b. h(t)=12.5sin⁡(π5(t−2.5))+13.5;h(t)=12.5\sin(\frac{\pi}{5}(t−2.5))+13.5; c. 26 ft

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