Trig model from its graph

Find a trig model's equation from its graph, a context or given points

In 14 of 20 recent exams
2% of the marks
Paper 1 or 2
1 to 7 marks

How it is set in the exam

Paper 1 (no calculator) and Paper 2 (calculator), 1–7 marks per part: usually the first parts of a Section B question, followed by its features and a level. Very common (14 of 20 exam versions: 5 on Paper 1 (no calculator), 9 on Paper 2 (calculator)), and with 35 parts the most frequent type in topic 3. Most parts ask for one constant at a time, for 1–2 marks each; a context with clock times asks for all four at once (up to 7 marks).

Spot it in the exam

Exam phrases to spot (5)

“find the values of $a$ , $b$ and $d$ ”
“the height is modelled by $h(t) = a\cos(bt) + d$ ; find …”
“show that $b = \frac{\pi}{6}$ ”
“high tide is at … and low tide at …”
“write $f(x)$ in the form $a\cos\left(b(x - c)\right)$ ”

In disguise (4)

a context with no graph: the greatest and least heights come from a radius and the height of a centre (a wheel $30$ m across whose lowest seat is $2$ m above the ground has $\lvert a \rvert = 15$ and $d = 17$ ), the period from a speed (a full turn every $4$ minutes), and the sign of $a$ from where the motion starts
a curve given only by two points and no extremes (a tan model): substitute both points, and the exact values give two linear equations
"the graph can also be written as $p\sin\left(b(x - q)\right) + d$ ": a second set of constants for a graph already fitted
a sine plus a cosine of the same argument, to be written as one cosine: from the GDC graph

Variants

from a context (the sign of $a$ from where it starts); from a graph, in radians or in degrees; clock times turned into decimal hours; all four constants of a shifted sine; $b$ as a show-that; an $x$ in terms of $b$ when $b$ is a letter (the period relation); a tan model through two points; a sine plus a cosine as one cosine.

Looks like it, but isn't (3)

the equation is given, and a value, an extreme, a time or a sketch is asked for: Read a sinusoid's features. (When $b$ is a letter and an $x$ is asked for in terms of it, it is this type: the period relation $\frac{2\pi}{b}$ , as in Example 1.2(c).)
one unknown constant at a time, each from one value at $0$ or a standard angle (whatever the formula, even $p + \tan x$ ): Exact values at a known angle
two sinusoids given, and the transformation between them (or its constants) asked for: Name the transformation

Worked example

A pedal that starts at its lowest point

Paper 1 (no calculator)

Question

A cyclist pedals at a steady rate. The height of one pedal above the ground, $h$ cm, $t$ seconds after the pedal is at its lowest point, is modelled by $h(t) = a\cos(bt) + d$ , where $b > 0$ . The least height of the pedal is $9$ cm, its greatest height is $43$ cm, and it makes one full turn every $0.8$ seconds.
(a) Find the value of $a$ . [2]
(b) Find the value of $d$ . [1]
(c) Find the value of $b$ . [2]

Try it yourself first, then compare.

Now try one like it

Write your working on paper or on screen. Paper Zero marks it step by step, like the worked example, and shows where each mark went. Free, with no account.

Try one like it

See it move: A pedal and its wave

$t = 0.10$ s: $h = 26 - 17\cos\left(\frac{5\pi}{2} \times 0.10\right) = 14.0$ cm.
Lowest 9 cm, highest 43 cm: the radius is $17 = \lvert a \rvert$ and the centre's height is $26 = d$ .
It starts at the bottom, but $\cos$ starts at the top, so $a = -17$ .
One turn takes 0.8 s, so $\frac{2\pi}{b} = 0.8$ and $b = \frac{5\pi}{2}$ .
Half a turn later, at $t = 0.4$ , the pedal is at the top: $h(0.4) = 43$ .

Which method?

What you know	Route
the greatest and least values and the period (from a graph or a context)	$\lvert a \rvert = \frac{\max - \min}{2}$ , with the sign from where the cycle starts; $d = \frac{\max + \min}{2}$ ; $b = \frac{2\pi}{\text{period}}$ ; then $c$ from a known point
only points on the curve, no extremes (a tan model, or any curve whose two unknowns are the multiplier and the added constant, with $b$ known)	substitute each point; exact values give linear equations; solve them
a sine plus a cosine of the same argument, $p\sin(bx) + q\cos(bx)$	graph it on the GDC: the amplitude is its greatest value, $b$ is unchanged, and for $a\cos\left(b(x - c)\right)$ with $a > 0$ the shift $c$ is the $x$ of a maximum (maximum or minimum of a graph); for a sine, $c$ is where it crosses $0$ going up

Steps

Sine or cosine, and the sign of $a$ .
Use the form the question gives. A cosine that starts at a maximum has $a > 0$ ; one that starts at a minimum has $a < 0$ . A sine that starts on the midline going up has $a > 0$ ; going down, $a < 0$ .
$a$ and $d$ from the extremes:
$\lvert a \rvert = \frac{\max - \min}{2}$ and $d = \frac{\max + \min}{2}$ . For a wheel, these are the radius and the height of the centre.
The period, then $b$ .
Two consecutive maxima (or minima) are one period apart; a maximum and the next minimum are half a period apart. $b = \frac{2\pi}{\text{period}}$ , or $\frac{360°}{\text{period}}$ if the question is in degrees. In $a\cos(bx) + d$ or $a\sin(bx) + d$ (no shift), a point at $x = 0$ can't give $b$ , because $b \times 0 = 0$ for every $b$ .
$c$ from a known point.
For a cosine with $a > 0$ , $c$ is the $x$ of a maximum (with $a < 0$ , of a minimum). For a sine with $a > 0$ , $c$ is where it crosses the midline going up, halfway between a minimum and the next maximum, or substitute a maximum: $b(x - c) = \frac{\pi}{2}$ ( $90°$ in degrees). With $a < 0$ , $c$ is where the sine crosses the midline going down. Adding a period to $c$ gives the same graph, so keep $c$ inside any stated range.
Points only
(no extremes): substitute each point, use exact values, and solve the linear equations: by hand on , or with simultaneous equations on .
Check
with a point you haven't used: the model must give its value.

Common mistakes

A positive $a$ for a cosine that starts at a minimum, or for a sine that starts on the midline going down
the sign has its own mark, and $17$ for $-17$ scores . A cosine that starts at a maximum also goes down first, and its $a$ is positive (Example 1.4): compare the start with how $\sin$ or $\cos$ starts, not with the direction.
The amplitude as the whole drop
$\lvert a \rvert$ is half of max $-$ min.
Half a period taken for a whole one
a maximum and the next minimum are half a period apart.

More mistakes (8)

$b$ from the point at $x = 0$
$\cos(b \times 0) = 1$ for every $b$ , so that point says nothing about $b$ (and earns nothing).
$b$ in degrees in a radian question
a period of $12$ gives $b = \frac{2\pi}{12} = \frac{\pi}{6}$ , not $\frac{360}{12} = 30$ . Degrees only when the question shows them; markschemes reject $30$ in a radian model.
Clock times read as decimals
09:35 is $9.583$ hours, not $9.35$ .
The shift with the wrong sign, or outside the stated range
$\sin\left(b(t - c)\right)$ moves the graph $c$ to the right, and $c$ plus a period is the same graph but may break " $0 < c < 12$ ".
Rounding $b$ before using it for $c$
on : use the stored value.
A show-that worked backwards
for "show that $b = 2$ ", start from the period ( $\frac{2\pi}{b} = \pi$ ) and end on $b = 2$ . Substituting $b = 2$ and checking it proves nothing and scores nothing.
Adding the coefficients
of a sine plus a cosine for the amplitude: $2 + 3 \ne 3.61$ . Graph it.
Degree mode
on with a model in radians: check the settings.

Easy to confuse with

Next typeRead a sinusoid's features

Now try one like it

Keep going

Easy to confuse with

More in Trig graphs and sinusoidal models