Trig Graphs: Amplitude and Period

Sketch and read trig graphs such as y = 3 sin 2x + 1: amplitude, period, greatest and least values, and the asymptotes of tan graphs, in degrees or radians.

In the exam for IGCSE Additional Maths 0606, Edexcel Further Pure Maths 4PM1 · 11 real past-paper questions below

اقرأ هذا الدرس مشروحاً بالعربي ←

Waves that repeat

Sit on a big wheel and watch your height. You go up, come down, and do it all again, every turn. Trig graphs draw exactly this kind of repeating wave.

Each graph comes straight from the circle of radius 1: $sin x$ is the height of the point, $cos x$ is its across distance, and $tan x$ is sin over cos. Here are their key values.

$x$	$0^{\circ}$	$9 0^{\circ}$	$18 0^{\circ}$	$27 0^{\circ}$	$36 0^{\circ}$
$sin x$	$0$	$1$	$0$	$- 1$	$0$
$cos x$	$1$	$0$	$- 1$	$0$	$1$
$tan x$	$0$	none	$0$	none	$0$

$y = sin x$ starts at 0, rises to 1, falls through 0 to −1, and comes back to 0 at $36 0^{\circ}$ .
$y = cos x$ has the same shape, but starts at its top, 1. It is the sine wave moved $9 0^{\circ}$ to the left.
$y = tan x$ climbs forever as x gets near $9 0^{\circ}$ , then starts again from far below. It repeats every $18 0^{\circ}$ .

Rule

An asymptote is a line the curve gets closer and closer to but never touches. $y = tan x$ has vertical asymptotes wherever $cos x = 0$ : at $x = 9 0^{\circ}$ , $27 0^{\circ}$ , and so on.

Rule

The length of one full wave is the period. For $sin x$ and $cos x$ it is $36 0^{\circ}$ , or $2 π$ in radians. For $tan x$ it is $18 0^{\circ}$ , or $π$ .

Key idea

The electricity in your wall socket is a sine wave. In Egypt and in Qatar the voltage swings up and down 50 times every second, so its period is one fiftieth of a second.

Amplitude and the midline

Multiplying by a number stretches the wave up and down. Adding a number lifts the whole wave up. So $y = a sin x + c$ is a taller wave, moved up.

0606 and 4037 ask about amplitude and period by name. 4PM1 asks you to draw and use these graphs, often from a table of values, and the same ideas help you get the shape right.

Formula

y = a sin x + c

$a$ — the amplitude: the height from the middle of the wave to the top. In 0606 and 4037 graphs it is a positive whole number.
$c$ — the wave moves up and down around the midline $y = c$ .

Since $sin x$ goes from −1 to 1, the wave goes from $c - a$ to $c + a$ . The same is true for $cos x$ .

Formula

greatest value = c + a

$c$ — the midline.
$a$ — the amplitude, added on top.

Formula

least value = c - a

$c$ — the midline.
$a$ — the amplitude, taken away.

Worked example

1
For $y = 3 cos x + 1$ , find the amplitude, the midline, and the greatest and least values of y.
2
$a = 3$ — The number in front of cos.
3
$y = 1$ — The added number gives the midline.
4
$1 + 3 = 4$ — Greatest value: midline plus amplitude.
5
$1 - 3 = - 2$ — Least value: midline minus amplitude.

Worked example

1
The depth of water in a harbour is $d = 6 + 2 sin (30 t)^{\circ}$ metres, $t$ hours after midnight. Find the greatest depth, the least depth, and the first time the water is deepest.
2
$6 + 2 \times 1 = 8 m$ — Greatest depth when the sine is 1.
3
$6 + 2 \times (- 1) = 4 m$ — Least depth when the sine is −1.
4
$30 t = 90 \Rightarrow t = 3$ — Sin first reaches 1 at 90°, so the water is deepest at 3 am.

Common mistake

Calling the greatest value the amplitude. For $y = 3 cos x + 1$ the top of the wave is 4, but the amplitude is 3: it is measured from the midline, not from the x-axis.

Key idea

The midline is where the wave lives. The amplitude is how far it reaches above and below it.

The period

A number multiplying x squeezes the wave sideways. In $y = sin 2 x$ , the angle grows twice as fast, so one full wave is finished by $18 0^{\circ}$ instead of $36 0^{\circ}$ .

Formula

period = \frac{36 0 ^{\circ}}{b}

$b$ — the number multiplying x in $sin b x$ or $cos b x$ . It can be a fraction such as $\frac{1}{2}$ .

Formula

period = \frac{2 π}{b}

$b$ — the same number. Use this form when x is in radians.

Tan repeats every $18 0^{\circ}$ , not $36 0^{\circ}$ , so its formula has half the top:

Formula

period of tan b x = \frac{18 0 ^{\circ}}{b} = \frac{π}{b}

$b$ — the number multiplying x inside the tan.

Rule

No board prints these. Learn them: they come straight from the period of the basic graph.

Worked example

1
Find the period of each graph.
2
$y = 4 sin 3 x : \frac{36 0 ^{\circ}}{3} = 12 0^{\circ}$ — Three full waves fit into 360°.
3
$y = 2 cos \frac{x}{2} + 1 : \frac{36 0 ^{\circ}}{\frac{1}{2}} = 72 0^{\circ}$ — b is a half, and dividing by a half doubles. In radians that is 4π.
4
$y = tan 4 x : \frac{18 0 ^{\circ}}{4} = 4 5^{\circ}$ — Tan uses 180°. In radians that is π/4.

Common mistake

Multiplying instead of dividing. The period of $sin 3 x$ is $12 0^{\circ}$ , not $108 0^{\circ}$ . A bigger b means a shorter wave.

Common mistake

Using $36 0^{\circ}$ for tan. The period of $tan 4 x$ is $4 5^{\circ}$ , not $9 0^{\circ}$ .

Key idea

a and c move the wave up and down. b changes how long each wave is. Read all three before you draw.

Sketching step by step

To sketch $y = a sin b x + c$ or $y = a cos b x + c$ , you do not need a table of values. Find a few key points and join them with a smooth wave.

Steps

1
1. Find the amplitude a, the midline $y = c$ and the period.
2. Cut one period into 4 equal parts.
3. Cos starts at the top: top, middle, bottom, middle, top. Sin starts in the middle and goes up: middle, top, middle, bottom, middle.
4. Repeat until you fill the whole domain given in the question.
5. Join with a smooth, rounded curve and label the greatest and least values.

Worked example

1
Sketch $y = 2 cos 3 x + 1$ for $0^{\circ} \leq x \leq 24 0^{\circ}$ .
2
$a = 2, y = 1, period = 12 0^{\circ}$ — Amplitude, midline, and 360° ÷ 3.
3
$y_{m a x} = 3, y_{m i n} = - 1$ — Midline plus and minus the amplitude.
4
$12 0^{\circ} \div 4 = 3 0^{\circ}$ — Key points every 30°.
5
$(0, 3), (30, 1), (60, - 1), (90, 1), (120, 3)$ — Cos pattern: top, middle, bottom, middle, top.
6
$(150, 1), (180, - 1), (210, 1), (240, 3)$ — The domain is 240°, two periods, so the pattern repeats once.

For tan, the asymptotes come first. They sit where the angle inside the tan is $9 0^{\circ}$ , $27 0^{\circ}$ , and so on.

Worked example

1
Sketch $y = 3 tan 2 x + 1$ for $0 \leq x \leq π$ , where x is in radians.
2
$period = \frac{π}{2}$ — Tan: π ÷ 2.
3
$2 x = \frac{π}{2}, \frac{3 π}{2} \Rightarrow x = \frac{π}{4}, \frac{3 π}{4}$ — Asymptotes where the inside is a quarter turn or three quarter turns.
4
$x = 0 : y = 3 tan 0 + 1 = 1$ — The y-intercept is 1.
5
$x = \frac{π}{2} : y = 3 tan π + 1 = 1$ — Halfway between the asymptotes the curve crosses its midline y = 1.
6
$3 pieces: 0 to \frac{π}{4}, \frac{π}{4} to \frac{3 π}{4}, \frac{3 π}{4} to π$ — Each piece climbs from left to right and bends towards the asymptotes.

Rule

For a tan graph, draw each asymptote as a dashed vertical line and write its x-value. Mark schemes give marks for the asymptotes and for the intercept, as well as for the shape.

Common mistake

Drawing the wrong number of waves. Check: domain ÷ period = number of full waves. Here 240° ÷ 120° = 2.

Common mistake

Pointed tops like a V, or a flat top. Trig curves are smooth and rounded at every peak and dip.

Key idea

Amplitude, midline, period, then quarter points. Four numbers give you the whole graph.

Reading a graph back

Some questions show you the graph and ask for a, b and c. Work backwards from the highest and lowest points.

Formula

a = \frac{max - min}{2}

$max$ — the greatest y-value on the graph.
$min$ — the least y-value. Half the gap between them is the amplitude.

Formula

c = \frac{max + min}{2}

$max$ and min — the midline is exactly halfway between them, their mean.

Then find the period on the graph, and use $b = \frac{36 0 ^{\circ}}{period}$ .

Worked example

1
The graph of $y = a sin b x + c$ has greatest value 7 and least value −1. One full wave takes $9 0^{\circ}$ . Find a, b and c.
2
$a = \frac{7 - ( - 1 )}{2} = 4$ — Half the gap between the top and the bottom.
3
$c = \frac{7 + ( - 1 )}{2} = 3$ — Halfway between the top and the bottom.
4
$b = \frac{36 0 ^{\circ}}{9 0 ^{\circ}} = 4$ — Four waves fit into 360°.
5
$y = 4 sin 4 x + 3$ — Check: the top is 3 + 4 = 7 and the bottom is 3 − 4 = −1.

A graph also tells you how many solutions an equation has. The solutions of $f (x) = k$ are where the curve meets the flat line $y = k$ .

Worked example

1
For $0^{\circ} \leq x < 36 0^{\circ}$ , find the values of k for which $2 cos x + 3 = k$ has no solutions, and the values for which it has two.
2
$y_{m a x} = 5, y_{m i n} = 1$ — Midline 3, amplitude 2.
3
$k > 5 or k < 1$ — The line y = k misses the curve completely: no solutions.
4
$1 < k < 5$ — The line cuts the wave twice: two solutions.

Rule

When you complete a table of values for a trig graph in radians, set the calculator to radians first. Plot the points and join them with a smooth curve.

Common mistake

Reading the period from a peak to the next dip. That is only half a wave. The period runs from a peak to the next peak.

Key idea

Top and bottom give a and c. The length of one wave gives b.

Which method? Read the question

The wording tells you which number to look for first.

You see	You do
“Write down the amplitude”	The number a in front of sin or cos
“Write down the period”	$36 0^{\circ} \div b$ (or $2 π \div b$ ); for tan, $18 0^{\circ} \div b$
“Greatest / least value”	$c + a$ and $c - a$ , using sin or cos between −1 and 1
“Sketch … for $0^{\circ} \leq x \leq \dots$ ”	Amplitude, midline, period, quarter points; count the waves
A tan graph	Asymptotes where the inside is $9 0^{\circ}$ , $27 0^{\circ}$ …; label them
A graph given, find a, b, c	Half the gap, the middle, then $36 0^{\circ} \div$ period
“How many solutions”	Draw $y = k$ and count where it meets the curve

Common mistake

Mixing degrees and radians. If the domain is in radians, the period must be in radians too: $2 π \div b$ .

Key idea

Every trig graph is the basic wave, stretched by a, squeezed by b and lifted by c.

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