Start from y = sin x
The graph of y = sin x is a wave. It starts at 0, climbs to its highest value, 1, at 90°, comes back to 0 at 180°, falls to its lowest value, −1, at 270°, and is back at 0 at 360°. Then it repeats.
The wave swings evenly about the x-axis: it rises 1 above it and falls 1 below it. Two numbers describe that swing, the level it swings about and how far it swings. Multiplying and adding change each of them separately.
Multiplying stretches the heights
Compare y = sin x with y = 2 sin x. At every angle the second height is twice the first: at 90° it is 2 × 1 = 2, at 270° it is 2 × (−1) = −2, and at 30° it is 2 × ½ = 1. This is a vertical stretch by a factor of 2, the same stretch that multiplies every height of any graph.
The points on the x-axis stay where they are, because 2 × 0 = 0. So y = 2 sin x still crosses the axis at 0°, 180° and 360°, and it still repeats every 360°. Only the heights change: the wave now rises 2 above the axis and falls 2 below it.
The distance from the middle level of a wave to its highest point is called the amplitude. The amplitude of y = sin x is 1 and the amplitude of y = 2 sin x is 2.
y = 2 sin x in gold, over the plain y = sin x. One square across is 90°. Every height is doubled, so the amplitude is 2, and the crossings at 0°, 180° and 360° do not move.
A negative multiplier
In y = −2 sin x every height is multiplied by −2. That is the stretch by 2 together with a reflection in the x-axis, so the wave falls to −2 at 90° and rises to 2 at 270°. It still rises and falls 2 either side of the axis, so its amplitude is 2, not −2.
An amplitude is a distance, so it is never negative. For y = a sin x the amplitude is |a|, the size of a without its sign.
Adding moves the principal axis
Now add 3: y = 2 sin x + 3. Every height of y = 2 sin x goes up by 3. That is a translation of the whole graph 3 units up, the change outside the function that moves a graph up or down.
The wave no longer swings about the x-axis. It swings about the horizontal line y = 3, rising 2 above it and falling 2 below it. The horizontal line a wave swings about, halfway between its highest and lowest values, is called the principal axis. The principal axis of y = 2 sin x + 3 is y = 3.
The crest and the trough
The highest point of a wave is called a crest and the lowest point a trough. For y = 2 sin x + 3 the crest comes where sin x = 1, at 90°: y = 2 × 1 + 3 = 5. The trough comes where sin x = −1, at 270°: y = 2 × (−1) + 3 = 1.
So the maximum value is 3 + 2 = 5 and the minimum value is 3 − 2 = 1: the principal axis plus and minus the amplitude.
y = 2 sin x + 3, with one square across for every 90°. It swings about the dashed principal axis y = 3, up to 5 at 90° and down to 1 at 270°.
Reading them off a graph
Going the other way, the principal axis and the amplitude can be read from the maximum and minimum values. The principal axis is halfway between them: (5 + 1) ÷ 2 = 3. The amplitude is half the gap between them: (5 − 1) ÷ 2 = 2.
A wave that rises to 7 and falls to −1 has its principal axis at (7 + (−1)) ÷ 2 = 3 and an amplitude of (7 − (−1)) ÷ 2 = 4. If it is a sine wave that starts on its principal axis and rises, its equation is y = 4 sin x + 3.
The general rule
In y = a sin x + d, the number a is multiplied by the sine and the number d is added on. The amplitude is |a| and the principal axis is the line y = d. The maximum value is d + |a| and the minimum value is d − |a|.
The same holds for cosine. For y = −3 cos x + 1 the amplitude is 3 and the principal axis is y = 1, so the values run from 1 − 3 = −2 to 1 + 3 = 4. Because a is negative the wave is reflected: at 0°, y = −3 × 1 + 1 = −2, the minimum, and at 180°, y = −3 × (−1) + 1 = 4, the maximum.
Neither a nor d changes where the wave repeats. The period is still 360°. A number multiplying x inside the sine, as in y = sin 2x, is what changes the period.
The usual mistakes
Giving the maximum as the amplitude. For y = 2 sin x + 3 the maximum is 5, but the amplitude is 2, the distance from the principal axis up to the crest.
Taking the whole distance from trough to crest as the amplitude. That distance is 5 − 1 = 4, twice the amplitude; the amplitude is half of it.
Giving a negative amplitude. For y = −2 sin x the amplitude is 2: the minus sign reflects the wave but does not change how far it swings.
Putting the principal axis at y = 0 when d is not 0. The + 3 moves every point up by 3, the principal axis included.
How long one cycle takes
The application below also asks how long one cycle takes. Its model is d = 3 sin (30x)° + 5, where d is the depth of water in meters and x is the number of hours after midnight. Here the amplitude is 3 and the principal axis is d = 5.
The angle in the model is 30x, not x. The sine repeats when its angle grows by 360°, and 30x grows by 360 when x grows by 360 ÷ 30 = 12. So the wave repeats every 12 hours: its period is 12 hours.
Its crest comes where the sine is 1, and on the sine graph that happens first where the angle is 90°: 30x = 90, so x = 3.
Worked example: The Depth of Water at a Harbor Mouth Through the Day
Question At a harbor mouth the depth of water, d meters, x hours after midnight is modeled by d = 3 sin (30x)° + 5 for 0 ≤ x ≤ 24. (a) What is the depth at high water, the depth at low water, and the mean depth about which the water rises and falls? (b) How long is it from one high water to the next, and at what time is the first high water after midnight?
1.The sine of any angle lies between −1 and 1, so 3 sin (30x)° lies between −3 and 3. The amplitude is 3 m: the water rises 3 m above its middle level and falls 3 m below it.
The sine lies between −1 and 1, so the water rises and falls 3 m about its middle level: the amplitude is 3 m. 2.The principal axis is d = 5, from the + 5 in the model. (a) The mean depth is 5 m, the depth at high water is 5 + 3 = 8 m and the depth at low water is 5 − 3 = 2 m.
(a) The principal axis is d = 5: the mean depth is 5 m, high water 8 m, low water 2 m. 3.The sine repeats every 360°, and 30x grows by 360 when x grows by 36030 = 12. The period is 12 hours, the time from one high water to the next.
The angle 30x grows by 360 in 36030 = 12 hours: the period is 12 hours. 4.High water is where sin (30x)° = 1, first when 30x = 90, so x = 3. (b) It is 12 hours from one high water to the next, and the first high water is at 3 a.m. Check: at x = 3, d = 3 sin 90° + 5 = 3 + 5 = 8 m, and the next high water is at x = 3 + 12 = 15, which is 3 p.m.
(b) High water first comes where 30x = 90, at x = 3, which is 3 a.m.; the next is at 3 p.m.
Answer: (a) 8 m at high water, 2 m at low water, and a mean depth of 5 m; (b) 12 hours, and the first high water is at 3 a.m.
Common mistakes
- Reading the amplitude 3 as the depth at high water. The amplitude is how far the water rises ABOVE the mean level, so the depth at high water is 5 + 3 = 8 m.
- Taking the period as 30 hours, from the 30 in the model. The 30 is how many degrees the angle turns in one hour, and a full cycle of 360° takes 360 ÷ 30 = 12 hours.