Sine Cosine Graphs

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Sine Cosine Graphs · Grade 12 · Worksheet 2

1. A sine wave is graphed on a coordinate plane with amplitude 3, period π, phase shift π/4 to the right, and vertical shift 2 units upward. The function passes through the point (π/4, 5). Write the equation of this sine function in the form y = A sin(B(x - C)) + D. Answer: y = 3 sin(2(x - π/4)) + 2 Solution: Sine function transformations follow specific patterns: amplitude multiplies the output, period relates to the horizontal stretch, phase shift moves the graph left or right, and vertical shift moves the entire graph up or down. The general form y = A sin(B(x - C)) + D shows how each parameter…
   Full step-by-step solution

   Sine function transformations follow specific patterns: amplitude multiplies the output, period relates to the horizontal stretch, phase shift moves the graph left or right, and vertical shift moves the entire graph up or down. The general form y = A sin(B(x - C)) + D shows how each parameter controls a different aspect of the transformation.

2. Emma is monitoring the water level in a tidal river near her research station. The depth of the water, in meters, varies sinusoidally over time due to tides. She models the depth with the function d(t) = 5 sin(π/6 (t - 3)) + 10, where t is the time in hours after midnight. During a 24-hour period, a cargo ship can only enter the river when the water depth is at least 12.5 meters. Determine the total number of hours in a 24-hour day that the ship can safely enter the river. Answer: 8 hours Solution: Set up the inequality: 5 sin(π/6 (t - 3)) + 10 >= 12.5. Subtract 10: 5 sin(π/6 (t - 3)) >= 2.5. Divide by 5: sin(π/6 (t - 3)) >= 0.5.
   Full step-by-step solution

   Step 1: Set up the inequality: 5 sin(π/6 (t - 3)) + 10 >= 12.5. Subtract 10: 5 sin(π/6 (t - 3)) >= 2.5. Divide by 5: sin(π/6 (t - 3)) >= 0.5. Step 2: Find the angles where sin θ = 0.5. On the unit circle, sin θ = 0.5 at θ = π/6 and θ = 5π/6 (in the first two quadrants). Since sin θ >= 0.5 between these angles, we have π/6 <= θ <= 5π/6. Step 3: Substitute θ = π/6 (t - 3): π/6 <= π/6 (t - 3) <= 5π/6. Multiply through by 6/π: 1 <= t - 3 <= 5. Add 3: 4 <= t <= 8. Step 4: The period of the function is 2π / (π/6) = 12 hours. So this pattern repeats every 12 hours. In the first 12 hours (0 to 12), the safe times are from t=4 to t=8, which is 4 hours. In the second 12 hours (12 to 24), add 12 to the bounds: t=16 to t=20, another 4 hours. Step 5: Total safe hours = 4 + 4 = 8 hours. The answer is 8 hours.

3. 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

4. ∫(3x² - 6x + 2)dx from 0 to 2 = ? Answer: 0 Solution: Find the antiderivative of 3x² - 6x + 2 Antiderivative = (3x³/3) - (6x²/2) + 2x = x³ - 3x² + 2x Evaluate F(2) - F(0) where F(x) = x³ - 3x² + 2x Calculate F(2) F(2) = (2)³ - 3(2)² + 2(2) = 8 - 12 + 4 = 0 Calculate F(0) F(0) = (0)³ - 3(0)² + 2(0) = 0 - 0 + 0 = 0 F(2) - F(0) = 0 - 0 = 0 The answer…
   Full step-by-step solution

   Step 1: Find the antiderivative of 3x² - 6x + 2 Antiderivative = (3x³/3) - (6x²/2) + 2x = x³ - 3x² + 2x Step 2: Apply the Fundamental Theorem of Calculus Evaluate F(2) - F(0) where F(x) = x³ - 3x² + 2x Step 3: Calculate F(2) F(2) = (2)³ - 3(2)² + 2(2) = 8 - 12 + 4 = 0 Step 4: Calculate F(0) F(0) = (0)³ - 3(0)² + 2(0) = 0 - 0 + 0 = 0 Step 5: Compute the definite integral F(2) - F(0) = 0 - 0 = 0 The answer is 0.

5. 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

6. Olivia is analyzing the motion of a Ferris wheel. The height of a passenger above the ground is modeled by a transformed sine function. The graph of the function on a coordinate plane shows a wave with an amplitude of 5, a period of 10 seconds, a phase shift of 2 seconds to the right, and a vertical translation of 15 units upward. The function is of the form y = A sin(B(x - C)) + D. What is the equation of this transformed sine function? Answer: y = 5 sin((π/5)(x - 2)) + 15 Solution: Identify the amplitude A. The amplitude is 5, so A = 5. Determine B using the period.
   Full step-by-step solution

   Step 1: Identify the amplitude A. The amplitude is 5, so A = 5. Step 2: Determine B using the period. The period is 10 seconds. For a sine function, period = 2π/B. So, 2π/B = 10. Solve for B: B = 2π/10 = π/5. Step 3: Identify the phase shift C. The phase shift is 2 seconds to the right. In the form y = A sin(B(x - C)), a shift to the right means C = 2. Step 4: Identify the vertical translation D. The vertical translation is 15 units upward, so D = 15. Step 5: Substitute all values into the general form y = A sin(B(x - C)) + D: y = 5 sin((π/5)(x - 2)) + 15. The final equation is y = 5 sin((π/5)(x - 2)) + 15.

7. y = 9sin(4(x - π/3)) + 2. Identify amplitude, period, phase shift, and vertical shift. Answer: Amplitude: 9, Period: π/2, Phase shift: π/3 right, Vertical shift: 2 up Solution: Write the equation in standard form: y = 9sin(4(x - π/3)) + 2 Identify amplitude (A): The coefficient of sine is 9, so amplitude = |9| = 9 Identify period: The coefficient B = 4, so period = 2π/B = 2π/4 = π/2 Identify phase shift: C = π/3, so phase shift = π/3 to the right Identify vertical…
   Full step-by-step solution

   Step 1: Write the equation in standard form: y = 9sin(4(x - π/3)) + 2 Step 2: Identify amplitude (A): The coefficient of sine is 9, so amplitude = |9| = 9 Step 3: Identify period: The coefficient B = 4, so period = 2π/B = 2π/4 = π/2 Step 4: Identify phase shift: C = π/3, so phase shift = π/3 to the right Step 5: Identify vertical shift: D = 2, so vertical shift = 2 units upward Step 6: Final answer: Amplitude = 9, Period = π/2, Phase shift = π/3 right, Vertical shift = 2 up

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

   Step 1: Identify the parameters from the equation y = 7sin(2(x - π/7)) + 2 A = 7, B = 2, C = π/7, D = 2 Step 2: Calculate the amplitude Amplitude = |A| = |7| = 7 Step 3: Calculate the period Period = 2π/|B| = 2π/|2| = 2π/2 = π Step 4: Identify the phase shift Phase shift = C = π/7 (to the right) Step 5: Identify the vertical shift Vertical shift = D = 2 (upward) Step 6: Final answer The amplitude is 7, period is π, phase shift is π/7, and vertical shift is 2.