Home / IGCSE Add Maths 0606 / Trigonometry / Trigonometric graphs: amplitude and period
Cambridge IGCSE · Topic 10.3
IGCSE Add Maths 0606 / Topic 10 / IGCSE

Stretch it, squash it, lift it.

Every graph in this lesson is \(y = \sin x\), \(y = \cos x\) or \(y = \tan x\) after three simple changes. The number in front sets the height of the waves, the number multiplying \(x\) sets how often they repeat, and the number added on moves the whole curve up or down. Read those three numbers and you can sketch the graph, label its key points and count solutions without solving anything.

Full-screen labTrig graphs lab → Topic 10All trigonometry lessons →
The key idea

For \(y = a\sin bx + c\) and \(y = a\cos bx + c\): the amplitude is \(a\), the period is \(\dfrac{360^{\circ}}{b}\) (or \(\dfrac{2\pi}{b}\) radians), the curve oscillates about the line \(y = c\), with maximum \(c + a\) and minimum \(c - a\). For \(y = a\tan bx + c\) there is no amplitude, the period is \(\dfrac{180^{\circ}}{b}\) (or \(\dfrac{\pi}{b}\)), and the vertical asymptotes are where \(bx = 90^{\circ}, 270^{\circ}, \ldots\). The number of solutions of \(f(x) = k\) is the number of times the line \(y = k\) meets the curve.

x y 90° 180° 270° 360° −2 −1 1 2 3 4 y = 1 a = 3 period 180° (45°, 4) (225°, 4) (135°, −2) (315°, −2) y = 3 sin 2x + 1, 0° ≤ x ≤ 360°
Fig. 1: y = 3 sin 2x + 1 for 0° ≤ x ≤ 360°. The curve oscillates about the dashed line y = 1 with amplitude 3, so its maximum is 1 + 3 = 4 and its minimum is 1 - 3 = -2. Two full waves fit in 360°, so the period is 180°
Section 01

Amplitude, period and shift.

In the lab, drag the point P round the unit circle or press Play and watch its height trace out \(y = \sin x\); switch to cos and tan too. The three basic graphs: \(\sin x\) and \(\cos x\) are waves between \(-1\) and \(1\) that repeat every \(360^{\circ}\) (\(2\pi\) radians); \(\tan x\) has no maximum or minimum, repeats every \(180^{\circ}\) (\(\pi\) radians) and has vertical asymptotes at \(x = 90^{\circ}, 270^{\circ}, \ldots\). Now change the numbers \(a\), \(b\) and \(c\) in \(y = a\sin bx + c\) one at a time.

NumberEffect on the graphFor \(y = 3\sin 2x + 1\)
\(a\)stretches the height: amplitude \(a\) (sin and cos only)amplitude \(3\)
\(b\)squashes it sideways: period \(\dfrac{360^{\circ}}{b}\) or \(\dfrac{2\pi}{b}\); for tan \(\dfrac{180^{\circ}}{b}\) or \(\dfrac{\pi}{b}\)period \(\dfrac{360^{\circ}}{2} = 180^{\circ}\)
\(c\)moves it up by \(c\): the centre line is \(y = c\)centre line \(y = 1\)
max, min\(c + a\) and \(c - a\)\(4\) and \(-2\)
Worked example Read the numbers, then the turning points
For \(y = 3\sin 2x + 1\), \(0^{\circ} \le x \le 360^{\circ}\), state the amplitude, the period and the maximum and minimum values of \(y\), and find the \(x\)-coordinates where they occur.
  1. \(a = 3\), \(b = 2\), \(c = 1\): amplitude \(3\), period \(\dfrac{360^{\circ}}{2} = 180^{\circ}\).
  2. Maximum \(1 + 3 = 4\), minimum \(1 - 3 = -2\).
  3. \(\sin 2x = 1\) when \(2x = 90^{\circ}, 450^{\circ}\), so \(x = 45^{\circ}, 225^{\circ}\).
  4. \(\sin 2x = -1\) when \(2x = 270^{\circ}, 630^{\circ}\), so \(x = 135^{\circ}, 315^{\circ}\) (Fig. 1).
Answer: Amplitude \(3\), period \(180^{\circ}\); maximum \(4\) at \(x = 45^{\circ}, 225^{\circ}\); minimum \(-2\) at \(x = 135^{\circ}, 315^{\circ}\)
Worked example A fractional b, in radians
For \(y = 5\cos\dfrac{x}{2} - 2\), \(0 \le x \le 4\pi\), state the period in radians and the greatest and least values of \(y\), with the values of \(x\) where they occur.
  1. \(b = \dfrac{1}{2}\), so the period is \(2\pi \div \dfrac{1}{2} = 4\pi\) (that is \(720^{\circ}\)): the domain holds exactly one wave.
  2. Greatest value \(-2 + 5 = 3\), when \(\cos\dfrac{x}{2} = 1\): \(\dfrac{x}{2} = 0, 2\pi\), so \(x = 0, 4\pi\).
  3. Least value \(-2 - 5 = -7\), when \(\cos\dfrac{x}{2} = -1\): \(\dfrac{x}{2} = \pi\), so \(x = 2\pi\).
Answer: Period \(4\pi\); greatest value \(3\) at \(x = 0\) and \(x = 4\pi\); least value \(-7\) at \(x = 2\pi\)
Your turn
State the period of \(y = 2\tan 3x + 1\) in degrees, and explain why it has no amplitude.
Examiner trap

The period is \(360^{\circ}\) DIVIDED by \(b\), not multiplied: \(y = \sin 2x\) repeats twice as often, every \(180^{\circ}\), and \(y = \sin\dfrac{x}{2}\) half as often, every \(720^{\circ}\). And the amplitude is \(a\) alone: \(c\) moves the wave up but does not make it taller.

Section 02

Sketching over a domain.

A sketch needs the right shape and the right labels, not graph paper. Split each period into quarters: a sin or cos curve has a maximum, a minimum or a crossing of the centre line at every quarter-period. Then label the intercepts with the axes, the maximum and minimum points, and the end points of the domain. For tan, mark the asymptotes first, where \(bx = 90^{\circ}, 270^{\circ}, \ldots\) (or \(\dfrac{\pi}{2}, \dfrac{3\pi}{2}, \ldots\)), and label each one with its equation.

x y −360° −270° −90° 90° 270° 360° −4 −2 2 4 x = −180° x = 180° (90°, 2) (−90°, −2) y = 2 tan(x/2), −360° ≤ x ≤ 360°: period 360°
Fig. 2: y = 2 tan(x/2) for -360° ≤ x ≤ 360°. Here b = 1/2, so the period is 180° ÷ 1/2 = 360° and the asymptotes are where x/2 = ±90°, at x = -180° and x = 180°. The curve passes through (90°, 2) because tan 45° = 1
Worked example Intercepts of a sine curve
Sketch \(y = 3\sin 2x + 1\) for \(0^{\circ} \le x \le 360^{\circ}\), giving the coordinates of the points where it meets the axes.
  1. Quarter-period \(45^{\circ}\): \(y = 1, 4, 1, -2, 1\) at \(x = 0^{\circ}, 45^{\circ}, 90^{\circ}, 135^{\circ}, 180^{\circ}\), and the same again up to \(360^{\circ}\). Join with a smooth wave (Fig. 1).
  2. \(y\)-axis: \(x = 0\) gives \(y = 1\), the point \((0, 1)\).
  3. \(x\)-axis: \(\sin 2x = -\dfrac{1}{3}\). The principal value is \(-19.47^{\circ}\), so \(2x = 199.47^{\circ}, 340.53^{\circ}, 559.47^{\circ}, 700.53^{\circ}\) for \(0^{\circ} \le 2x \le 720^{\circ}\).
  4. Halve: \(x = 99.7^{\circ}, 170.3^{\circ}, 279.7^{\circ}, 350.3^{\circ}\).
Answer: Meets the \(y\)-axis at \((0, 1)\) and the \(x\)-axis at \(x = 99.7^{\circ}, 170.3^{\circ}, 279.7^{\circ}, 350.3^{\circ}\) (1 d.p.)
Worked example A tan curve in radians
Sketch \(y = \tan 2x + 1\) for \(0 \le x \le \pi\). Label the asymptotes and the points where the curve meets the axes.
  1. Period \(\dfrac{\pi}{2}\). Asymptotes where \(2x = \dfrac{\pi}{2}, \dfrac{3\pi}{2}\), so \(x = \dfrac{\pi}{4}\) and \(x = \dfrac{3\pi}{4}\).
  2. \(y\)-axis: \(\tan 0 + 1 = 1\), the point \((0, 1)\); the curve also returns to \(y = 1\) at \(x = \dfrac{\pi}{2}\) and \(x = \pi\).
  3. \(x\)-axis: \(\tan 2x = -1\), so \(2x = \dfrac{3\pi}{4}, \dfrac{7\pi}{4}\), giving \(x = \dfrac{3\pi}{8}, \dfrac{7\pi}{8}\).
  4. Draw \(y = \tan x\) shapes rising from left to right in each section, approaching but never touching the asymptotes.
Answer: Asymptotes \(x = \frac{\pi}{4}\), \(x = \frac{3\pi}{4}\); meets the \(y\)-axis at \((0, 1)\) and the \(x\)-axis at \(x = \frac{3\pi}{8}, \frac{7\pi}{8}\)
Examiner trap

For \(y = a\tan bx + c\) the asymptotes are at \(x = \dfrac{90^{\circ}}{b}\) and then every period \(\dfrac{180^{\circ}}{b}\) after that; \(c\) moves the curve up but does NOT move the asymptotes. Never let a tan curve cross or bend back from an asymptote.

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What is the period of \(y = 4\sin 3x\), in degrees?

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What are the greatest and least values of \(y = 2\cos x - 3\)?

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Where are the asymptotes of \(y = \tan 2x\) for \(0^{\circ} \le x \le 180^{\circ}\)?

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What is the period of \(y = 3\tan\dfrac{x}{2}\), in radians?

Section 03

Reading a graph and counting solutions.

Working backwards from a graph uses the same three facts: \(a = \dfrac{\max - \min}{2}\), \(c = \dfrac{\max + \min}{2}\) and \(b = \dfrac{360^{\circ}}{\text{period}}\). To find how many solutions \(f(x) = k\) has, draw the horizontal line \(y = k\) on the sketch and count where it meets the curve; you do not have to solve anything. The same idea works for \(y = |f(x)|\) from topic 1.4: reflect every part of the curve below the \(x\)-axis in that axis, then count.

Worked example Find a, b and c
The graph of \(y = a\cos bx + c\), for \(0^{\circ} \le x \le 360^{\circ}\), has a maximum point at \((0^{\circ}, 5)\), its next minimum point at \((120^{\circ}, -1)\) and its next maximum at \((240^{\circ}, 5)\). Find \(a\), \(b\) and \(c\).
  1. \(a = \dfrac{5 - (-1)}{2} = 3\) and \(c = \dfrac{5 + (-1)}{2} = 2\).
  2. Maximum to maximum is one period: \(240^{\circ}\). So \(b = \dfrac{360^{\circ}}{240^{\circ}} = \dfrac{3}{2}\).
  3. Check at \(x = 120^{\circ}\): \(3\cos 180^{\circ} + 2 = -3 + 2 = -1\). Correct.
Answer: \(a = 3\), \(b = \frac{3}{2}\), \(c = 2\): \(y = 3\cos\frac{3x}{2} + 2\)
Worked example Count, do not solve
Use Fig. 1 to state the number of solutions of \(3\sin 2x + 1 = k\) for \(0^{\circ} \le x \le 360^{\circ}\) when (a) \(k = 2.5\) (b) \(k = 4\) (c) \(k = -3\).
  1. (a) \(y = 2.5\) lies between the minimum \(-2\) and the maximum \(4\); it cuts each of the two waves twice: \(4\) solutions.
  2. (b) \(y = 4\) only touches the two maximum points: \(2\) solutions (\(x = 45^{\circ}, 225^{\circ}\)).
  3. (c) \(y = -3\) is below the minimum \(-2\): it misses the curve, \(0\) solutions.
Answer: (a) \(4\)   (b) \(2\)   (c) \(0\)
Your turn
How many solutions does \(|3\sin 2x + 1| = 3\) have for \(0^{\circ} \le x \le 360^{\circ}\)?
Examiner trap

A full period runs from one maximum to the NEXT maximum. Maximum to minimum is only half a period; taking it as the whole period makes \(b\) twice too big. Also check the ends of the domain: a solution exactly at \(x = 0^{\circ}\) or \(x = 360^{\circ}\) counts.

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The curve \(y = a\sin bx + c\), where \(a\) and \(b\) are positive, has a maximum value of \(7\), a minimum value of \(-1\) and a period of \(90^{\circ}\). Find \(a\), \(b\) and \(c\).

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How many solutions does \(2\sin 3x + 1 = 2\) have for \(0^{\circ} \le x \le 360^{\circ}\)?

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Find the equation of each asymptote of \(y = 5\tan\dfrac{x}{3} + 2\) for \(0^{\circ} \le x \le 720^{\circ}\).

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How many solutions does \(|2\cos x - 1| = 2\) have for \(0^{\circ} \le x \le 360^{\circ}\)?

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Six original Cambridge-style questions
Amplitude, period and turning points in degrees and radians, a tan sketch with asymptotes, finding \(a\), \(b\) and \(c\) from a graph, and counting solutions, including one with a modulus.