Read a sinusoid's features

From its equation: values, extremes, times, cycles, sketch

In 14 of 20 recent exams
1% of the marks
Paper 1 or 2
1 to 5 marks

How it is set in the exam

Paper 1 (no calculator) and Paper 2 (calculator), 1–5 marks per part. Very common (14 of 20 exam versions: 5 on Paper 1 (no calculator), 9 on Paper 2 (calculator)). The short parts (the amplitude, the period, a value) open a question; the longer ones (a show-that about the greatest value, a sketch, a count) come once the equation is known, often after Find a trig model's equation.

Spot it in the exam

Exam phrases to spot (8)

“write down the amplitude”
“find the period”
“find the maximum height and the time at which it first occurs”
“how many times does it reach its greatest height …”
“sketch the graph of $f$ for $0 \le x \le \ldots$ ”
“show that the temperature never reaches …”
“find the value of $h$ when $t = \ldots$ ”
“find the time between high tide and the next low tide”

In disguise (6)

a sinusoidal velocity in a motion question, where a time or a count of extremes can be read from the period: no derivative is needed (with a derivative asked for, it is Greatest or least velocity, speed, acceleration or displacement on an interval)
" $g(x) \ge 4$ for all $x$ ; find the least value of $k$ ": the least value of $g$ , $d - \lvert a \rvert$ , set against the bound
the next point on the graph with the same height and the same gradient: one period later
the least positive $p$ for which $f(x + p) = f(x)$ for every $x$ (a horizontal shift that maps the graph onto itself): it is the period, although the word is never used
"at the start", "when the machine is switched on": the value at $t = 0$
two curves on the same axes whose meeting points come from an earlier equation: a sketch of both, with those points labelled

Variants

the amplitude and period of $\sin$ , $\cos$ or $\tan$ ; a value (: exact, at a standard angle in any quadrant; : on the GDC, in radians; at $t = 0$ ); never reaches a level; an unknown vertical shift from a lower bound, then the least $y$ -intercept; half a period in hours and minutes; the clock times of the maxima; a count of turns, maxima or minima; the first maximum after a known minimum; the next point with the same value (twice a cycle) or with the same value and gradient (once a period); a sketch of one period; two curves on one set of axes.

Looks like it, but isn't (7)

"find $a$ , $b$ , $c$ and $d$ " from a graph or a context: Find a trig model's equation
the model set equal to a level, or "for how long": Above or below a level
the greatest value with the derivative asked for, or a curve that is not a sinusoid (its extremes need $f'(x) = 0$ ): Stationary points
a sine curve stretched or translated from a given graph, with its new formula or sketch asked for (even when the new curve is given by its own formula, to be drawn on the same axes): Transform a graph
two sinusoids given, one a transformation of the other, and the transformation or a constant in it asked for (even when the constant is fixed by the greatest value): Name the transformation
a sine plus a cosine of the same argument: it is one sinusoid only through its graph, so Find a trig model's equation
an unknown constant found from the formula's value at $0$ or a standard angle, with no value of the model asked for: Exact values at a known angle

Worked example

Amplitude and period, then an exact value in another quadrant

Paper 1 (no calculator)

Question

Let $f(x) = 4\cos\left(\frac{2x}{3}\right) + 3$ .
(a) Write down the amplitude of $f$ . [1]
(b) Find the period of $f$ . [2]
(c) Find the exact value of $f\left(\frac{5\pi}{4}\right)$ . [3]

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See it move: Read the wave off its equation

$h(t) = 10 + 6\sin(4\pi t)$ : the greatest height is $d + a = 10 + 6 = 16$ cm and the least is $d - a = 4$ cm.
One cycle takes $\frac{2\pi}{4\pi} = \frac{1}{2}$ s. A sine with no shift starts on the midline going up, and the key points are a quarter period, $\frac{1}{8}$ s, apart: midline, maximum, midline, minimum, midline.
On $0 \le t \le \frac{1}{2}$ : 1 maximum ( $t = \frac{1}{8}$ ) and 1 minimum ( $t = \frac{3}{8}$ ). List them from the first, adding a period each time: $\frac{1}{2} \div \frac{1}{2} = 1$ counts cycles, not maxima.
The ends: $h(0) = 10$ , on the midline, and $h(\frac{1}{2}) = 10$ . A sketch stops at the ends of the domain.
(a) The greatest height is $10 + 6 = 16$ cm. (b) One cycle is one period, $\frac{2\pi}{4\pi} = \frac{1}{2}$ second.
(c) The domain is exactly one period: one smooth wave through $(0, 10)$ , the maximum $\left(\frac{1}{8}, 16\right)$ , $\left(\frac{1}{4}, 10\right)$ , the minimum $\left(\frac{3}{8}, 4\right)$ and $\left(\frac{1}{2}, 10\right)$ , where it stops.

Which method?

Illustrated with $y = 3 + 2\sin 4x$ , for $0 \le x \le \pi$ .

the amplitude, the period
The fact
$\lvert a \rvert$ ; $\frac{2\pi}{b}$ ( $\frac{\pi}{b}$ for $\tan$ )
Example
$2$ ; $\frac{\pi}{2}$
the greatest or least value; "never reaches $k$ "
The fact
$d \pm \lvert a \rvert$ , compared with $k$ , when a maximum (or minimum) lies inside the domain (any domain one period long has one); otherwise compare the values at the ends
Example
$5$ and $1$ ; it never reaches $6$ , because $5 < 6$
when an extreme happens
The fact
the bracket equals where $\sin$ or $\cos$ has that extreme (with $a > 0$ , a maximum of $\sin$ at $\frac{\pi}{2}$ , of $\cos$ at $0$ ); then add or subtract periods to reach the first one in the domain (a maximum to the next minimum is half a period)
Example
a maximum where $4x = \frac{\pi}{2}$ , at $x = \frac{\pi}{8}$ ; the next minimum at $\frac{3\pi}{8}$
how many
The fact
list them: the first one, then one period apart, to the end of the interval
Example
maxima at $\frac{\pi}{8}$ and $\frac{5\pi}{8}$ : $2$
a value
The fact
substitute (: reduce to a standard angle, with the quadrant's sign; : radian mode)
Example
at $x = \frac{7\pi}{24}$ : $3 + 2\sin\frac{7\pi}{6} = 3 - 1 = 2$
the next $x$ with the same value, but not the same gradient
The fact
each height between its least and greatest values comes twice a cycle, once rising and once falling: solve $f(x) =$ value on the GDC (zeros and intersections), or reflect in the extreme between them
Example
$y = 2$ at $x = \frac{7\pi}{24}$ and next at $x = \frac{11\pi}{24}$ , not a period later
the next point with the same height and gradient
The fact
add one period
Example
$\left(\frac{7\pi}{24} + \frac{\pi}{2}, 2\right) = \left(\frac{19\pi}{24}, 2\right)$
a sketch
The fact
the key points a quarter period apart, the ends of the domain, the maximum and the minimum
Example
$(0, 3)$ , $\left(\frac{\pi}{8}, 5\right)$ , $\left(\frac{\pi}{4}, 3\right)$ , $\left(\frac{3\pi}{8}, 1\right)$ , $\left(\frac{\pi}{2}, 3\right)$ , …

Asked for	The fact	Example
the amplitude, the period	$\lvert a \rvert$ ; $\frac{2\pi}{b}$ ( $\frac{\pi}{b}$ for $\tan$ )	$2$ ; $\frac{\pi}{2}$
the greatest or least value; "never reaches $k$ "	$d \pm \lvert a \rvert$ , compared with $k$ , when a maximum (or minimum) lies inside the domain (any domain one period long has one); otherwise compare the values at the ends	$5$ and $1$ ; it never reaches $6$ , because $5 < 6$
when an extreme happens	the bracket equals where $\sin$ or $\cos$ has that extreme (with $a > 0$ , a maximum of $\sin$ at $\frac{\pi}{2}$ , of $\cos$ at $0$ ); then add or subtract periods to reach the first one in the domain (a maximum to the next minimum is half a period)	a maximum where $4x = \frac{\pi}{2}$ , at $x = \frac{\pi}{8}$ ; the next minimum at $\frac{3\pi}{8}$
how many	list them: the first one, then one period apart, to the end of the interval	maxima at $\frac{\pi}{8}$ and $\frac{5\pi}{8}$ : $2$
a value	substitute (: reduce to a standard angle, with the quadrant's sign; : radian mode)	at $x = \frac{7\pi}{24}$ : $3 + 2\sin\frac{7\pi}{6} = 3 - 1 = 2$
the next $x$ with the same value, but not the same gradient	each height between its least and greatest values comes twice a cycle, once rising and once falling: solve $f(x) =$ value on the GDC (zeros and intersections), or reflect in the extreme between them	$y = 2$ at $x = \frac{7\pi}{24}$ and next at $x = \frac{11\pi}{24}$ , not a period later
the next point with the same height and gradient	add one period	$\left(\frac{7\pi}{24} + \frac{\pi}{2}, 2\right) = \left(\frac{19\pi}{24}, 2\right)$
a sketch	the key points a quarter period apart, the ends of the domain, the maximum and the minimum	$(0, 3)$ , $\left(\frac{\pi}{8}, 5\right)$ , $\left(\frac{\pi}{4}, 3\right)$ , $\left(\frac{3\pi}{8}, 1\right)$ , $\left(\frac{\pi}{2}, 3\right)$ , …

Steps

Read $a$ , $b$ , $c$ and $d$
from the equation, taking $b$ out of the bracket first. Note where the cycle starts (a negative $a$ turns it upside down) and whether the question is in degrees.
Pick the fact
from the table.
Times:
find the first one from the bracket (for $a\sin(\ldots) + d$ , a maximum is where the bracket is $\frac{\pi}{2}$ if $a > 0$ and $\frac{3\pi}{2}$ if $a < 0$ ; for $a\cos(\ldots) + d$ , where it is $0$ if $a > 0$ and $\pi$ if $a < 0$ ; each plus a whole number of $2\pi$ : take the first one that gives an $x$ inside the domain), or from the GDC's maximum on . Then add half periods or whole periods.
Counts:
list the times, starting from the first and adding a period each time, until the end of the interval. The interval divided by the period is the number of cycles, not the answer: where the first extreme falls, and what happens at the ends, decide the count.
Values:
substitute. On reduce the angle to a standard one and take the sign from its quadrant; on write the substitution, then the value (define a function and evaluate it), in radians.
Sketch:
plot the key points a quarter period apart (maximum, midline, minimum, midline), and the ends of the domain, and join them with a smooth wave that stops at the ends. Label what is asked (the maximum, the end points, intersections). On , graph it on the domain to check.
Answer in context:
units, clock times, a whole-number count, a sentence for "show that it never …".

Common mistakes

The period of $\tan$ as $\frac{2\pi}{b}$
it is $\frac{\pi}{b}$ .
Dividing the wrong way
for $\cos\left(\frac{2x}{3}\right)$ the period is $2\pi \div \frac{2}{3} = 3\pi$ , not $\frac{4\pi}{3}$ .
A count from the interval divided by the period,
rounded: list the times instead (Example 2.5).

More mistakes (8)

The sign of an exact value
$\cos\frac{5\pi}{6}$ is negative (second quadrant).
A sketch with straight lines or corners, or one that runs past the domain
the marks need a smooth wave that stops at the ends of the domain, with the points asked for labelled.
"Never reaches" without a reason
give the greatest value, its number and the comparison. The is only awarded after the and the .
Decimal hours as a time
$3.49$ hours is $3$ h $29$ min, not $3$ h $49$ min.
Degree mode
on : a temperature in °C is still modelled in radians. Check the settings.
$d + \lvert a \rvert$ on a domain that misses the peak
the greatest value is $d + \lvert a \rvert$ only if a maximum point lies in the domain. On an interval shorter than a period, find the times of the extremes and compare the ends. (Differentiating also scores, but for a sinusoid it is slower.)
A negative amplitude
the amplitude of $3 - 2\cos\left(\frac{\pi}{4}x\right)$ is $2$ , not $-2$ : it is $\lvert a \rvert$ .
The next time a height repeats taken as one period later
a sinusoid reaches each height between its least and greatest values twice in a cycle, once rising and once falling. Only the same height and the same gradient come back one period later (Your turn 3).

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