Sine Cosine Graphs

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Sine Cosine Graphs · Grade 12 · Worksheet 1

1. A cosine function is graphed on a coordinate plane. The function has been transformed from the parent function y = cos(x) by a vertical stretch of factor 2, a horizontal compression such that the period becomes 2π/3, a phase shift of π/6 units to the left, and a vertical translation of 1 unit downward. Write the equation of the transformed function in the form y = a cos(b(x - c)) + d. Answer: y = 2 cos(3(x + π/6)) - 1 Solution: Identify the amplitude (a). The vertical stretch factor is 2, so a = 2. Determine the period coefficient (b).
   Full step-by-step solution

   Step 1: Identify the amplitude (a). The vertical stretch factor is 2, so a = 2. Step 2: Determine the period coefficient (b). The period is given as 2π/3. For cosine functions, period = 2π/b, so 2π/b = 2π/3. Solving for b: b = 3. Step 3: Identify the phase shift (c). A phase shift of π/6 units to the left means c = -π/6. Step 4: Identify the vertical translation (d). A translation of 1 unit downward means d = -1. Step 5: Substitute all values into the general form y = a cos(b(x - c)) + d: y = 2 cos(3(x - (-π/6))) - 1 = 2 cos(3(x + π/6)) - 1. The equation is y = 2 cos(3(x + π/6)) - 1.

2. ∫(3x² - 4x + 2)dx from 0 to 2 = ? Answer: 4 Solution: We are finding the definite integral of (3x² - 4x + 2) from 0 to 2. Find the antiderivative (indefinite integral). The antiderivative of 3x² is 3 * (x³/3) = x³.
   Full step-by-step solution

   We are finding the definite integral of (3x² - 4x + 2) from 0 to 2. Step 1: Find the antiderivative (indefinite integral). The antiderivative of 3x² is 3 * (x³/3) = x³. The antiderivative of -4x is -4 * (x²/2) = -2x². The antiderivative of 2 is 2x. So the antiderivative F(x) = x³ - 2x² + 2x. Step 2: Apply the Fundamental Theorem of Calculus. We evaluate F(2) - F(0). First, compute F(2): F(2) = (2)³ - 2*(2)² + 2*(2) = 8 - 2*4 + 4 = 8 - 8 + 4 = 4. Next, compute F(0): F(0) = (0)³ - 2*(0)² + 2*(0) = 0 - 0 + 0 = 0. Step 3: Subtract F(0) from F(2). F(2) - F(0) = 4 - 0 = 4. Thus, the value of the definite integral is 4.

3. A marine biologist is studying the vertical motion of a dolphin swimming near a buoy. The dolphin's depth relative to the surface follows the function d(t) = 4sin(πt/3 - π/2) + 6, where d is depth in meters and t is time in seconds. At what times during the first 6 seconds does the dolphin reach its maximum depth? Answer: t = 3 seconds and t = 9 seconds Solution: Sine functions of the form Asin(Bx - C) + D have maximum and minimum values determined by the amplitude and vertical shift. The minimum occurs when the sine component equals -1, which creates the lowest point of the oscillation.
   Full step-by-step solution

   Sine functions of the form Asin(Bx - C) + D have maximum and minimum values determined by the amplitude and vertical shift. The minimum occurs when the sine component equals -1, which creates the lowest point of the oscillation. Phase shifts move these critical points horizontally along the x-axis, while the period determines how frequently they repeat.

4. The function f(x) = 4cos(3x - π/3) undergoes a transformation to become g(x) = 4cos(3x + π/6). If the phase shift of f(x) is φ₁ and the phase shift of g(x) is φ₂, what is the absolute difference |φ₁ - φ₂| in radians? Express your answer as a simplified exact value. Answer: π/2 Solution: Write f(x) in standard form: f(x) = 4cos(3x - π/3) = 4cos(3(x - π/9)) The phase shift of f(x) is φ₁ = π/9 Write g(x) in standard form: g(x) = 4cos(3x + π/6) = 4cos(3(x + π/18)) The phase shift of g(x) is φ₂ = -π/18 Calculate the absolute difference: |φ₁ - φ₂| = |π/9 - (-π/18)| = |π/9 + π/18|…
   Full step-by-step solution

   Step 1: Write f(x) in standard form: f(x) = 4cos(3x - π/3) = 4cos(3(x - π/9)) Step 2: The phase shift of f(x) is φ₁ = π/9 Step 3: Write g(x) in standard form: g(x) = 4cos(3x + π/6) = 4cos(3(x + π/18)) Step 4: The phase shift of g(x) is φ₂ = -π/18 Step 5: Calculate the absolute difference: |φ₁ - φ₂| = |π/9 - (-π/18)| = |π/9 + π/18| Step 6: Find a common denominator: π/9 = 2π/18, so |2π/18 + π/18| = |3π/18| Step 7: Simplify: 3π/18 = π/6 Step 8: The absolute difference is π/6 The answer is π/6.

5. y = 3sin(5(x - π/3)) - 1. Find amplitude, period, phase shift, and vertical shift. Answer: Amplitude: 3, Period: 2π/5, Phase shift: π/3 right, Vertical shift: -1 down Solution: Identify the parameters from y = 3sin(5(x - π/3)) - 1 A = 3, B = 5, C = π/3, D = -1 Amplitude = |A| = |3| = 3 Period = 2π/B = 2π/5 Phase shift = C = π/3 (since it's (x - π/3), the shift is π/3 units to the right) Vertical shift = D = -1 (1 unit downward) Final answer: Amplitude = 3, Period =…
   Full step-by-step solution

   Step 1: Identify the parameters from y = 3sin(5(x - π/3)) - 1 A = 3, B = 5, C = π/3, D = -1 Step 2: Calculate amplitude Amplitude = |A| = |3| = 3 Step 3: Calculate period Period = 2π/B = 2π/5 Step 4: Determine phase shift Phase shift = C = π/3 (since it's (x - π/3), the shift is π/3 units to the right) Step 5: Determine vertical shift Vertical shift = D = -1 (1 unit downward) Final answer: Amplitude = 3, Period = 2π/5, Phase shift = π/3 right, Vertical shift = -1 down

6. A Ferris wheel at an amusement park has a diameter of 50 meters and completes one full rotation every 4 minutes. The height of a passenger above the ground, in meters, can be modeled by a transformed cosine function: h(t) = A cos(B(t - C)) + D, where t is time in minutes after boarding. If passengers board at the lowest point, which is 5 meters above the ground, and reach their maximum height after 2 minutes, determine the exact values of A, B, C, and D in the function h(t). Answer: A=25, B=π/2, C=2, D=30 Solution: For trigonometric functions modeling circular motion, the amplitude represents half the total range of motion, the coefficient B relates to the period through the formula period = 2π/B, the phase shift C horizontally translates the function to match timing conditions, and the vertical shift D…
   Full step-by-step solution

   For trigonometric functions modeling circular motion, the amplitude represents half the total range of motion, the coefficient B relates to the period through the formula period = 2π/B, the phase shift C horizontally translates the function to match timing conditions, and the vertical shift D centers the function at the average height. These transformations allow cosine functions to model real-world periodic phenomena like Ferris wheel motion.

7. A sine wave is graphed on a coordinate plane with amplitude 3, period π, phase shift π/4 to the right, and vertical translation 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 translation shifts the entire graph up or down. The general form y = a sin(b(x - c)) + d helps organize these…
   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 translation shifts the entire graph up or down. The general form y = a sin(b(x - c)) + d helps organize these transformations systematically.