The graph of a trigonometric function is a wave in which the same shape repeats at regular intervals forever.
Following the height of a point going round the unit circle traces the sine curve, which makes it a natural fit for anything that repeats in cycles, such as a pendulum, ocean waves or sound.
In y = A sin(Bx + C) + D, A sets the height of the wave, the amplitude; B sets the length of one repetition, the period; and C and D shift the graph sideways and up or down.
The first picture compares the shapes of sin, cos and tan; in the second, changing the four numbers stretches, squeezes and shifts the curve.
The basic graphs of sin, cos and tan
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The graph of y = sin x, y = cos x or y = tan x, chosen with the buttons (the red dashes for tan are asymptotes)
💡 How the three functions differ
①y = sin x is symmetric about the origin and has period 2π
②y = cos x is symmetric about the y-axis, has period 2π, and is y = sin x shifted π/2 to the left
③y = tan x is symmetric about the origin, has period π, and has asymptotes at x = π/2 + nπ
④sin and cos stay between −1 and 1, while tan takes every real value
Amplitude, period and shifts
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y = sin x (faint) and y = A sin(Bx + C) + D (gold) with the amplitude, period and shifts changed; the dashed lines mark the maximum and minimum
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The period and amplitude formulas
General Form
y = A sin(Bx + C) + D
Amplitude |A|, period 2π/|B|, horizontal shift −C/B, vertical shift D
Period Formula
period T = 2π|B|
The larger |B|, the shorter the period and the more tightly packed the wave
📐 What the four numbers do
①A sets the height of the wave (amplitude |A|); a negative A flips it upside down
②B sets the period 2π/|B|
③C shifts the graph by −C/B along the x-axis
④D raises or lowers the midline to y = D
Summary
Trig Graph Core
y = A sin(Bx + C) + D → amplitude |A|, period 2π|B|
The four numbers A, B, C, D fix the height, spacing and position of a sine curve
🎯 Exam Points
①sin and cos have period 2π, tan has period π
②y = A sin(Bx + C) + D has amplitude |A| and period 2π/|B|
③Since Bx + C = B(x + C/B), the horizontal shift is −C/B
④The maximum is |A| + D and the minimum is −|A| + D
⑤cos x = sin(x + π/2)
Worked Examples & Past Exam
Example 1
Find the amplitude and period of y = 3 sin 2x.
1
In the form y = A sin Bx, the amplitude is |A|, which is 3.
amplitude = |A| = |3| = 3
2
The period is 2π/|B|, and here B = 2.
period = 2π|2| = π
▸ amplitude 3, period π
Multiplying x by 2 makes the same wave go by twice as fast, so the period halves to π. The amplitude is set by the 3 in front.
Example 2
Find the maximum and minimum of y = 2 cos x + 1.
1
cos x takes values from −1 to 1.
-1 ≤ cos x ≤ 1
2
Put in the end values 1 and −1 to get the maximum and minimum.
max = 2(1)+1 = 3, min = 2(-1)+1 = -1
▸ max 3, min −1
The 2 doubles the height of the wave and the + 1 raises the midline by 1, so the maximum is 2 + 1 and the minimum is −2 + 1.
2022 KICE mock exam Math type, adapted
Find the period and maximum of y = 4 sin(2x − π) + 1.