Sinusoidal Graphs Worksheet with Answers – Precalculus (Free PDF)

Problem 1 Amplitude and Period ★★★

Find the amplitude and the period of each function. (a) y = 3 sin x   (b) y = −2 cos(4x)   (c) y = (1/2) sin(x/3)   (d) y = 5 cos(πx)

Problem 2 Midline and Range ★★★

Give the midline, the maximum and the minimum. (a) y = 2 sin x + 3   (b) y = −cos x − 2

Problem 3 Key Points of a Sine Wave ★★★

Complete a table of the five key points of y = 3 sin x for x = 0, π/2, π, 3π/2 and 2π. The graph below shows the curve.

Problem 4 Read the Phase Shift ★★★

Describe the horizontal shift of each graph compared with y = sin x or y = cos x. (a) y = sin(x − π/3)   (b) y = cos(x + π/2)   (c) y = sin(2x − π)

Problem 5 Write an Equation from a Graph ★★★

The graph is a sine curve through the origin with peak (π/4, 3), trough (3π/4, −3) and a zero at (π, 0). Find the amplitude, the period and an equation of the form y = A sin(Bx).

Problem 6 Tangent Properties ★★★

For y = tan x on the interval [−π, π], list the vertical asymptotes and the zeros, and give the value at x = π/4.

Problem 7 Write a Cosine from a Description ★★★

A cosine function has amplitude 2.5, period 6, midline y = −1 and a maximum at x = 0. Write its equation.

Problem 8 Equation from a Maximum and a Minimum ★★★

One period of a cosine-shaped wave has a maximum at (π/6, 5) and the next minimum at (7π/6, −1). Find A, D, the period, and an equation y = A cos(B(x − C)) + D.

Problem 9 Reflect, Shift and Compress ★★★

Write y = −2 sin(3x − π) + 4 as y = A sin(B(x − C)) + D and list the amplitude, period, shift, midline, maximum and minimum.

Problem 10 Tangent Transformations ★★★

Find the period and the vertical asymptotes nearest the y-axis. (a) y = 5 tan(x/6)   (b) y = tan(x + π/6) − 1

Problem 11 Pier Water Depth ★★★

The water depth at a fishing pier is d(t) = 4 sin(πt/6) + 9 feet, where t is the number of hours since the first reading. Find the period, the greatest and least depths, and d(2) to the nearest tenth.

Problem 12 Ferris Wheel Height ★★★

A rider’s height on a Ferris wheel is H(t) = 30 − 25cos(πt/20) feet, t in seconds. The graph shows one turn. Find H(0), H(10) and H(20), and the first time the rider is 42.5 feet high.

Problem 13 Model from a Table ★★★

A sinusoidal function has these values.

Find its amplitude, midline, period and an equation using cosine.

Problem 14 Graph of a Compressed Tangent ★★★

The graph shows y = tan(5x) with its asymptotes on the interval [−π/5, π/5]. State the period, the asymptotes shown, the zeros on this interval, and the value of y at x = π/20.

Problem 15 Describe the Transformations ★★★

Describe, in order, how the graph of y = sin x is transformed to give y = −2 sin(x/2) + 3. State the amplitude, period and range.

Problem 16 Pedal Height ★★★

A bicycle pedal moves in a circle of radius 8 inches whose center is 14 inches above the ground. The pedal starts at its lowest point and makes one turn every 1.2 seconds, so h(t) = 14 − 8cos(5πt/3). Find the lowest and highest heights, h(0.3), and the first time the pedal is 18 inches high.

Problem 17 Hours of Daylight ★★★

A city has 15.4 hours of daylight on day 172 and 8.6 hours on day 355. Model the daylight with D(d) = Acos(2π(d − 172)/365) + 12.0 after finding A. Find D(80) and the number of days with at least 14 hours of daylight.

Problem 18 Mass on a Spring ★★★

A mass on a spring has displacement y = 6 sin(4πt) centimeters, t in seconds. Find the amplitude, period and frequency, and the two times in the first cycle when the displacement is 3 centimeters.

Problem 19 Household Voltage ★★★

Voltage in a house circuit is V(t) = 170 sin(120πt) volts, t in seconds. Find the peak voltage, the period, the voltage at t = 1/240, and the first two times the voltage is 85 volts.

Problem 20 Model the Tide from Data ★★★

High tide is 7.8 feet at 3:00 a.m. and the next low tide is 0.6 feet at 9:12 a.m. Write a cosine model h(t) with t in hours after midnight, then find the height at 6:00 a.m. and at noon.