Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
The first problems read features from equations. Later ones build models for a Ferris wheel, tides, daylight and temperature.
Standards: F-TF.B.5, F-IF.C.7e · Level: Grade 11 Algebra 2
Problem 2 Period of a compressed cosine ★★★
Find the period and amplitude of y = cos 3x.
Problem 3 Reflected sine ★★★
Describe y = −2 sin x: amplitude, period and what the negative sign does.
Problem 4 Vertical shift ★★★
Find the midline, maximum and minimum of y = sin x + 5.
Problem 5 Range of a cosine function ★★★
Find the maximum, minimum and range of y = 3 cos x − 1.
Problem 6 Period with π in the coefficient ★★★
Find the period of y = sin(πx/4).
Problem 7 Write an equation from a graph ★★★
The graph shows a sine wave through the origin. Write its equation.
Problem 8 Phase shift to the right ★★★
Describe the graph of y = sin(x − π/3) compared with y = sin x, and give the x-values where it first crosses the midline going up and its first maximum.
Problem 9 Shift left and down ★★★
State the amplitude, phase shift and midline of y = 2 cos(x + π/2) − 3.
Problem 10 Read features from a graph ★★★
The graph shows y = 2 cos(πx/3) + 4. Read the maximum, minimum, midline, amplitude and period.
Problem 11 Cosine from a description ★★★
Write a cosine function with midline y = 6, amplitude 4, period 12, and a maximum at x = 0.
Problem 12 Sine from a description ★★★
Write a sine function with midline y = 0, amplitude 2.5, period 10, that starts at the midline and rises.
Problem 13 All features at once ★★★
State the amplitude, period and phase shift of y = 3 sin(2x − π/2).
Problem 14 Locate maximum points ★★★
Find the x-coordinates of all maximum points of y = 5 sin 4x on the interval [0, π].
Problem 15 Ferris wheel model ★★★
A Ferris wheel has radius 20 feet, its center is 24 feet above the ground, and one rotation takes 40 seconds. A rider starts at the bottom, so h(t) = −20 cos(πt/20) + 24. Find the rider’s height at t = 15 seconds, to the nearest hundredth.
Problem 16 Tide model ★★★
At a harbor the water is 7.5 feet deep at high tide (t = 0) and 1.5 feet deep at low tide six hours later. Model the depth with a cosine function, then find the depth at t = 2 hours and the first time the depth is 4.5 feet.
Problem 17 Daylight hours ★★★
The hours of daylight in a city on day d of the year are modeled by D(d) = 2.8 sin(2π(d − 80)/365) + 12.1. Find the longest and shortest daylight, the period, and the daylight on day 120 to the nearest tenth of an hour.
Problem 18 Daily temperature ★★★
The temperature in °F, t hours after midnight, is T(t) = 14 sin(π(t − 9)/12) + 68. Find the highest and lowest temperatures, when the highest occurs, and T(12) to the nearest tenth.
Problem 19 Show two forms are equal ★★★
Show that cos x = sin(x + π/2) by comparing values at x = 0, π/2 and π, and explain with a graph shift.
Problem 20 Find a model from extreme values ★★★
A quantity has a maximum of 10 at t = 1 and the next minimum of −2 at t = 4. Write a cosine model.
