Sine Cosine Graphs Worksheets Grade 12

1. 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.
2. ∫(3x² - 2x + 1) dx from 0 to 2 = ?
3. 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 4, 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.

…and 4 more problems

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.
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.
3. y = 4sin(2(x - π/3)) + 1. Identify amplitude, period, phase shift, and vertical shift.

…and 5 more problems

1. Emma is modeling the temperature variation in a chemical reaction. The temperature, in degrees Celsius, oscillates according to a transformed sine function. The graph of the function has an amplitude of 5, a period of 3π, a phase shift of π/2 units to the right, and a vertical translation of 7 units upward. Write the equation of this sine function in the form y = A sin(B(x - C)) + D.
2. Matiu is an engineer designing a wave energy converter. The height of the water wave relative to the mean sea level is modeled by the function h(t) = 4cos(πt/6 - π/2) + 8, where h(t) is the height in meters and t is the time in seconds. Determine the exact times during the first 12 seconds when the wave height is exactly 10 meters above the mean sea level.
3. A transformed sine function is graphed on a coordinate plane. The wave oscillates between y = -3 and y = 5. The first peak to the right of the y-axis occurs at x = π/9, and the next peak to the right occurs at x = 7π/9. The graph passes through the midline at x = π/18, moving upward. Write the equation of this sine function in the form y = A sin(B(x - C)) + D.

…and 4 more problems