B. Trigonometric Functions · Radian Measure and Trigonometric Functions

Writing the Equation of a Sinusoidal Function

B2.6About 70 minutes

Learning goals

I can find a, k, d and c from the graph of a sinusoidal function, with angles in radians.
I can write a sinusoidal equation from a list of properties.
I can write both a sine and a cosine equation for the same graph.
I can check an equation by substituting a point.

Words to know

axis of the curve
The horizontal line y = c halfway between the maximum and minimum values: c = (max + min) ÷ 2.

amplitude
|a| = (max − min) ÷ 2.

phase shift
A horizontal translation d. For a cosine model with a > 0, d can be the x-coordinate of any maximum; for a sine model with a > 0, any point where the graph crosses its axis going up.

equivalent equations
Different equations that produce exactly the same graph.

Working backwards

You can now go from an equation to a graph. Today you go the other way: from a graph, or a description, to an equation. This is the key step in modelling, which comes in the next two lessons.

Four steps (for y = a sin(k(x − d)) + c or y = a cos(k(x − d)) + c):

c = (max + min) ÷ 2, the axis.
|a| = (max − min) ÷ 2, the amplitude.
Find the period (from one maximum to the next, or twice the distance from a maximum to the next minimum), then k = 2π ÷ period.
Choose d:

for cosine with a > 0: the x-coordinate of a maximum;
for cosine with a < 0: the x-coordinate of a minimum;
for sine with a > 0: an x-value where the graph crosses the axis going up.

Because the graph repeats, there are infinitely many correct equations. Pick the one that is easiest, usually the smallest positive d or d = 0.

Worked example

Example 1: From a graph

A sinusoidal graph has a maximum at (π/6, 5), and the next minimum is at (2π/3, −1). Write a cosine equation and a sine equation for it.

c = (5 + (−1)) ÷ 2 = 2. |a| = (5 − (−1)) ÷ 2 = 3.
Max to next min is half a period: 2π/3 − π/6 = 4π/6 − π/6 = π/2. So the period is π and k = 2π ÷ π = 2.
Cosine (a = 3): d is the x-coordinate of a maximum, π/6. So y = 3 cos(2(x − π/6)) + 2.
Sine (a = 3): the graph crosses the axis going up a quarter period before the maximum: π/6 − π/4 = −π/12. So y = 3 sin(2(x + π/12)) + 2.
Check both at x = 0: 3 cos(−π/3) + 2 = 3(1/2) + 2 = 3.5, and 3 sin(π/6) + 2 = 3(1/2) + 2 = 3.5 ✓.

y = 3 cos(2(x − π/6)) + 2 or y = 3 sin(2(x + π/12)) + 2 (other correct forms exist).

The graph from Example 1. Read c and a from the max and min, the period from max to max (π/6 to 7π/6), and d from a maximum (cosine) or an upward axis crossing (sine).

Worked example

Example 2: From properties

Write an equation for a sinusoidal function with amplitude 4, period 3π, a maximum at (π, 1), and range −7 ≤ y ≤ 1.

Max 1 and min −7: c = (1 + (−7)) ÷ 2 = −3. Check amplitude: (1 − (−7)) ÷ 2 = 4 ✓.
k = 2π ÷ 3π = 2/3.
Use cosine with a = 4 and d = π (the maximum).

y = 4 cos(⅔(x − π)) − 3

Worked example

Example 3: A graph starting at a minimum

A function has a minimum of 2 at x = 0, and its next maximum is 10, at x = 6. Write an equation.

c = (10 + 2) ÷ 2 = 6; |a| = (10 − 2) ÷ 2 = 4.
Min to max is half a period, 6, so the period is 12 and k = 2π/12 = π/6.
The graph starts at a minimum when x = 0: a reflected cosine with no phase shift does exactly that. Use a = −4, d = 0.

y = −4 cos(πx/6) + 6. Check: x = 6 gives −4 cos π + 6 = 4 + 6 = 10 ✓.

Example 3 on a decimal axis. A cycle that starts at a minimum suggests y = −a cos(...) with no phase shift.

Worked example

Example 4: A second equation with a < 0

Write another cosine equation for the graph in Example 1, this time with a negative.

With a = −3, the cosine cycle starts at a minimum.
A minimum is at x = 2π/3, so d = 2π/3.

y = −3 cos(2(x − 2π/3)) + 2. Check at x = π/6: −3 cos(2(−π/2)) + 2 = −3 cos(−π) + 2 = 3 + 2 = 5 ✓ (the maximum).

Check before you move on

Always check an equation with a point you know, preferably one that is not the point you used for d. If the y-values do not match, re-check the sign of a and the direction of d.

Just for fun

Teachers write "answers may vary" on these questions for a reason. Every sinusoidal graph has infinitely many correct equations, so this is the one test where everyone could be right in a different way.

Hands-on activity

Roll a can and model the sticker

You need: a can or a round container, a small sticker or a dot of tape, a ruler, a pencil and paper, and a table or the floor.

Measure the radius r of the can in centimetres.
Put the sticker on the rim of one end of the can. Lay the can on its side on the table with the sticker touching the table. Mark the starting position on the table.
Roll the can slowly in a straight line. Stop every so often (about every eighth of a turn), and measure the distance x rolled and the height h of the sticker above the table. Do at least one full turn.
Plot h against x. Find the max, min and period from your data.
Write an equation for h in terms of x. Because the sticker starts at the bottom, use a reflected cosine.

What to look for: the period should be close to the circumference 2πr, so k = 2π ÷ (2πr) = 1/r: the angle turned is x/r radians, the definition of a radian. Your equation should be close to h = −r cos(x/r) + r (max 2r, min 0). For a can with r = 4 cm: h = −4 cos(x/4) + 4.

Your turn

A sinusoidal graph has max 7 and min −1. Find the amplitude and the axis.

Two consecutive maximum points are at x = π/4 and x = 5π/4. Find the period and k.

A graph has a maximum at (0, 5) and the next minimum at (π, −1). Write a cosine equation.

Write a sine function with amplitude 2, period π, phase shift π/3 right, and axis y = −1.

A graph has a maximum at (π/2, 4) and the next minimum at (3π/2, −2). Write a cosine equation, then a simpler sine equation.

A function has a minimum of 1 at x = 0 and its next maximum of 9 at x = 4. Write an equation.

Is (π/3, 4) on the graph of y = 2 sin(x − π/6) + 3?

More practice: warm-up

Max 10, min 4. What is the axis?

Max 10, min 4. What is the amplitude?

The period is 4π. What is k?

A cosine graph with a > 0 and no phase shift starts at which kind of point?

A sine graph with a > 0 and no phase shift starts at which kind of point?

More practice: on your own

A graph has a maximum at (π/8, 3) and the next minimum at (5π/8, −3). Write a cosine equation and a sine equation.

Write an equation for a sinusoidal function with range −2 ≤ y ≤ 6, period 6π and a maximum at x = π.

For the graph in the first question, write a cosine equation with a < 0.

A sinusoid crosses its axis y = 2 going up at x = π/6, and its next maximum is 5, at x = π/3. Write an equation.

In a model of daily temperature, the maximum is 18 °C at t = 15 (3 p.m.) and the minimum is 6 °C at t = 3 (3 a.m.), with t in hours. Write a cosine model and check it at t = 3.

Challenge

Explain why y = 2 sin(x + π/2) and y = 2 cos x have the same graph. How far apart are the "starting points" of the sine and cosine forms of any sinusoid, as a fraction of the period?

A sinusoidal graph with period 2π, maximum 3 and minimum −1 passes through (0, 1) and is increasing there. Write its equation.

A maximum is at (π/12, 5) and the next minimum is at (π/4, 1). Write an equation and use it to find y when x = 0.

Check your understanding

Quiz

A sinusoid has maximum 9 and minimum 1. What are its amplitude and axis?

Two consecutive maxima are at x = π/6 and x = 5π/6. What is k?

In a cosine model with a > 0, the phase shift d can be the x-coordinate of...

Which equation has a maximum at (π/4, 3), a minimum value of −1 and period π?

A sinusoidal model has a period of 10. What is k?

Which equation has the same graph as y = cos x?

For parents and teachers

Big idea (B2.6): to write an equation from a graph, find c = (max + min)/2, |a| = (max − min)/2, k = 2π/period, then choose d at a maximum (cosine) or an upward axis crossing (sine). Many equations are correct.

How to help: Draw any smooth wave on graph paper with a clear max and min, and ask your teen for two different equations. Then pick a point on the graph and have them check that both equations give the right y-value.

Common difficulties:

Using the distance from max to min as the full period (it is half).
Writing k = period instead of k = 2π/period.
Choosing d at a minimum for a cosine with a > 0.
Not checking with a second point.