Sine Cosine Graphs

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Answer Key & Explanations

Sine Cosine Graphs · Grade 12 · Worksheet 2

1. y = 4sin(2(x - π/3)) + 1. Identify amplitude, period, phase shift, and vertical shift. Answer: 4, π, π/3, 1 Solution: Compare y = 4sin(2(x - π/3)) + 1 to the standard form y = A sin(B(x - C)) + D Amplitude A = |4| = 4 Period = 2π/B = 2π/2 = π Phase shift = C = π/3 (to the right) Vertical shift = D = 1 (upward) The parameters are: amplitude = 4, period = π, phase shift = π/3, vertical shift = 1 Final answer: 4,…
   Full step-by-step solution

   Step 1: Compare y = 4sin(2(x - π/3)) + 1 to the standard form y = A sin(B(x - C)) + D Step 2: Amplitude A = |4| = 4 Step 3: Period = 2π/B = 2π/2 = π Step 4: Phase shift = C = π/3 (to the right) Step 5: Vertical shift = D = 1 (upward) Step 6: The parameters are: amplitude = 4, period = π, phase shift = π/3, vertical shift = 1 Final answer: 4, π, π/3, 1

2. y = 4sin(2(x - π/3)) + 1. Find amplitude, period, phase shift, and vertical shift. Answer: Amplitude: 4, Period: π, Phase shift: π/3 right, Vertical shift: 1 up Solution: A = 4, so amplitude = |4| = 4 B = 2, so period = 2π/B = 2π/2 = π C = π/3, so phase shift = π/3 units to the right D = 1, so vertical shift = 1 unit upward Final answer: Amplitude: 4, Period: π, Phase shift: π/3 right, Vertical shift: 1 up
   Full step-by-step solution

   Step 1: Identify A (amplitude) A = 4, so amplitude = |4| = 4 Step 2: Identify B (affects period) B = 2, so period = 2π/B = 2π/2 = π Step 3: Identify C (phase shift) C = π/3, so phase shift = π/3 units to the right Step 4: Identify D (vertical shift) D = 1, so vertical shift = 1 unit upward Final answer: Amplitude: 4, Period: π, Phase shift: π/3 right, Vertical shift: 1 up

3. y = 5sin(3(x - π/5)) - 1. Identify amplitude, period, phase shift, and vertical shift. Answer: Amplitude: 5, Period: 2π/3, Phase Shift: π/5 right, Vertical Shift: -1 down Solution: Compare y = 5sin(3(x - π/5)) - 1 to standard form y = A sin(B(x - C)) + D Amplitude A = |5| = 5 Period = 2π/B = 2π/3 Phase shift C = π/5 (positive value indicates shift to the right) Vertical shift D = -1 (negative value indicates shift downward) Final answer: Amplitude: 5, Period: 2π/3, Phase…
   Full step-by-step solution

   Step 1: Compare y = 5sin(3(x - π/5)) - 1 to standard form y = A sin(B(x - C)) + D Step 2: Amplitude A = |5| = 5 Step 3: Period = 2π/B = 2π/3 Step 4: Phase shift C = π/5 (positive value indicates shift to the right) Step 5: Vertical shift D = -1 (negative value indicates shift downward) Final answer: Amplitude: 5, Period: 2π/3, Phase Shift: π/5 right, Vertical Shift: -1 down

4. y = 7sin(2(x - π/2)) - 3 = ? Answer: Amplitude: 7, Period: π, Phase Shift: π/2 right, Vertical Shift: 3 down Solution: The amplitude is the absolute value of the coefficient of sine: |7| = 7 Period = 2π/B where B = 2 Period = 2π/2 = π Phase shift = C = π/2 Since it's (x - π/2), the shift is π/2 units to the right D = -3, so the graph is shifted 3 units downward Amplitude: 7 Phase Shift: π/2 right Vertical Shift:…
   Full step-by-step solution

   Step 1: Identify the amplitude (A) The amplitude is the absolute value of the coefficient of sine: |7| = 7 Step 2: Identify the period Period = 2π/B where B = 2 Period = 2π/2 = π Step 3: Identify the phase shift (C) Phase shift = C = π/2 Since it's (x - π/2), the shift is π/2 units to the right Step 4: Identify the vertical shift (D) D = -3, so the graph is shifted 3 units downward Step 5: Final parameters Amplitude: 7 Period: π Phase Shift: π/2 right Vertical Shift: 3 down

5. A cosine function is graphed on a coordinate plane. The graph shows a wave that has been vertically stretched by a factor of 4, horizontally compressed to have a period of π/2, reflected across the x-axis, shifted π/6 units to the left, and translated 3 units downward. Write the equation of this transformed cosine function in the form y = A cos(B(x - C)) + D. Answer: y = -4 cos(4(x + π/6)) - 3 Solution: Start with the parent function y = cos(x) Apply vertical stretch by factor 4: amplitude becomes 4, so y = 4 cos(x) Apply reflection across x-axis: multiply by -1, so y = -4 cos(x) Apply horizontal compression: period becomes π/2, so B = 2π/(π/2) = 4, so y = -4 cos(4x) Apply phase shift π/6 units…
   Full step-by-step solution

   Step 1: Start with the parent function y = cos(x) Step 2: Apply vertical stretch by factor 4: amplitude becomes 4, so y = 4 cos(x) Step 3: Apply reflection across x-axis: multiply by -1, so y = -4 cos(x) Step 4: Apply horizontal compression: period becomes π/2, so B = 2π/(π/2) = 4, so y = -4 cos(4x) Step 5: Apply phase shift π/6 units left: C = -π/6, so y = -4 cos(4(x - (-π/6))) = -4 cos(4(x + π/6)) Step 6: Apply vertical translation 3 units down: D = -3, so y = -4 cos(4(x + π/6)) - 3 The final equation is y = -4 cos(4(x + π/6)) - 3

6. An oceanographer is studying the vertical motion of a research buoy in wavy conditions. The buoy's height above its equilibrium position follows the function h(t) = 2sin(πt/3) + 3cos(πt/3), where h is height in meters and t is time in seconds. Determine the exact maximum height the buoy reaches above its equilibrium position during its motion. Answer: √13 Solution: For functions of the form A sin(ωt) + B cos(ωt), the maximum value can be found by treating it as a single sine or cosine function with a phase shift.
   Full step-by-step solution

   For functions of the form A sin(ωt) + B cos(ωt), the maximum value can be found by treating it as a single sine or cosine function with a phase shift. The amplitude of the resulting function is determined by the square root of the sum of the squares of the coefficients. This concept applies to many oscillatory systems in physics and engineering.

7. A marine biologist is studying the vertical motion of a dolphin relative to the water surface. The dolphin's depth d(t) in meters below the surface is modeled by the function d(t) = 4sin(πt/3 - π/2) + 6, where t is time in seconds. What is the maximum depth the dolphin reaches below the surface? Answer: 10 Solution: d(t) = 4 sin(πt/3 - π/2) + 6 The sine function, sin(θ), has a range from -1 to 1. -1 ≤ sin(πt/3 - π/2) ≤ 1 The amplitude is 4. Multiply the inequality by 4: -4 ≤ 4 sin(πt/3 - π/2) ≤ 4 The vertical shift is +6.
   Full step-by-step solution

   Let's go step-by-step. We are given: d(t) = 4 sin(πt/3 - π/2) + 6 --- **Step 1: Understand the sine function range** The sine function, sin(θ), has a range from -1 to 1. So: -1 ≤ sin(πt/3 - π/2) ≤ 1 --- **Step 2: Multiply by amplitude** The amplitude is 4. Multiply the inequality by 4: -4 ≤ 4 sin(πt/3 - π/2) ≤ 4 --- **Step 3: Add the vertical shift** The vertical shift is +6. Add 6 to all parts of the inequality: -4 + 6 ≤ 4 sin(πt/3 - π/2) + 6 ≤ 4 + 6 2 ≤ d(t) ≤ 10 --- **Step 4: Interpret the result** The minimum value of d(t) is 2 meters below the surface. The maximum value of d(t) is 10 meters below the surface. Since depth is measured below the surface, the maximum depth is the larger number, 10 meters. --- **Step 5: Conclusion** The dolphin’s maximum depth below the surface is 10 meters. --- **Final answer:** 10

8. y = 8sin(9(x - π/8)) + 7. Find amplitude, period, phase shift, and vertical shift. Answer: 8, 2π/9, π/8, 7 Solution: Identify the parameters from the equation y = 8sin(9(x - π/8)) + 7 A = 8, B = 9, C = π/8, D = 7 Amplitude = |A| = |8| = 8 Period = 2π/|B| = 2π/|9| = 2π/9 Phase shift = C = π/8 (to the right) Vertical shift = D = 7 (upward) Final answer: Amplitude = 8, Period = 2π/9, Phase shift = π/8, Vertical…
   Full step-by-step solution

   Step 1: Identify the parameters from the equation y = 8sin(9(x - π/8)) + 7 A = 8, B = 9, C = π/8, D = 7 Step 2: Calculate the amplitude Amplitude = |A| = |8| = 8 Step 3: Calculate the period Period = 2π/|B| = 2π/|9| = 2π/9 Step 4: Identify the phase shift Phase shift = C = π/8 (to the right) Step 5: Identify the vertical shift Vertical shift = D = 7 (upward) Final answer: Amplitude = 8, Period = 2π/9, Phase shift = π/8, Vertical shift = 7