DSE Maths | Trigonometric Graphs and Transformations: Tutor Lesson Plan

2026-11-03

Tutor lesson-prep guides · Secondary Maths

Four numbers in one expression fight each other and students guess. This lesson fixes an order: read the amplitude, compute the period, factor out the coefficient before reading the shift, then place five key points.

Objectives — Students can: (1) state that y = 3 sin(2x – 60) has amplitude 3 and period 180 degrees; (2) find that the shift is 30 degrees right, not 60; (3) locate the maximum at 75, the minimum at 165 and the zeros at 30 and 120; (4) apply the same order to y = -2 cos(3x) + 1, giving amplitude 2, period 120, maximum 3 and minimum -1.

01 Prerequisites (3-minute check)

If any of these fails, fix it before the lesson starts.

Skill	Quick check	Pass mark
Basic sine curve	Maximum and minimum of y = sin x	1 and -1
Meaning of period	How many cycles does y = sin 2x complete in 360 degrees	Two, so the period is 180
Solving simple equations	Smallest positive angle with sin equal to 1	90 degrees
Effect of a minus sign	How does y = -f(x) change a graph	It reflects in the x-axis

02 Lesson flow (60 minutes)

Time	Stage	Tutor does	Student does
0-8 min	Opening contrast	Sketch y = sin x and y = 3 sin 2x together	Say taller and more compressed
8-18 min	Amplitude	Amplitude is the number in front of sine	Read the amplitude of three expressions
18-30 min	Period	Show 360 divided by 2 equals 180	Compute the period for 3x themselves
30-44 min	Phase shift	Rewrite 2x – 60 as 2(x – 30)	Explain why the shift is 30 right
44-52 min	Five key points	Use 30, 75, 120 and 165	Mark maximum, minimum and zeros
52-60 min	Independent practice	Two questions in eight minutes	Name which parameter went wrong

03 Teaching script (read it out)

Scene 1: Amplitude is the size, the sign is only direction

For y = 3 sin(2x – 60) the amplitude is 3 because sine never exceeds 1 and the factor 3 scales that. With a leading minus, as in -2 cos 3x, the amplitude is still 2; the minus only flips the graph, so the maximum becomes +2 and the minimum -2. Students often call the amplitude negative and then every extreme value is wrong. Insist on writing amplitude equals the absolute value of a.

Ask for the amplitude of y = -4 sin x and where its maximum lies.

Scene 2: The period is 360 divided by the coefficient

The 2 in 2x means two full cycles fit into 360 degrees, so one cycle takes 180. Likewise 3x gives 120. This is independent of the amplitude, so the order matters: read a for amplitude, read b for period, and neither disturbs the other. Give one quick check: the period multiplied by b must equal 360.

Ask the student to check the period of y = sin 4x using the product rule.

Scene 3: Factor the bracket before reading the shift

Writing 2x – 60 as 2(x – 30) shows a shift of 30 degrees right, not 60. This is the most common lost mark, because the minus 60 is taken at face value. After factoring, the maximum sits where 2(x – 30) equals 90, so x is 75; the minimum where it equals 270, so x is 165; the zeros at 30 and 120. All four come from one instruction: factor, then solve.

Ask for the direction and size of the shift in y = sin(3x + 90).

04 Question types and worked steps

Type	Example	Steps	Answer
Read the parameters	y = 3 sin(2x – 60)	Amplitude from a, period from 360 over b, shift from the constant over b	Amplitude 3, period 180, shift 30 right
Locate an extreme	Maximum of the same curve	Set the bracket to 90, solve, check the domain	x = 75
Find the zeros	Zeros between 0 and 180	Set the bracket to 0 and 180, solve, order them	30 and 120
Negative with vertical shift	y = -2 cos(3x) + 1	Amplitude 2, period 120, then add 1 to the extremes	Maximum 3, minimum -1

Sanity check (teach this): Consistency check: maximum minus minimum must equal twice the amplitude, and 3 minus -3 is 6. Zeros are half a period apart, and 30 to 120 is 90, exactly half of 180. The maximum at 75 and the minimum at 165 are also 90 apart. If any of these fails, either the period or the shift is wrong.

05 Independent practice (3 questions)

Question	Answer	What to watch
Amplitude and period of y = 5 sin(4x)	Amplitude 5, period 90	Did you divide 360 by 4
Shift of y = sin(2x + 40)	20 degrees left	Did you divide by 2 first
Maximum of y = -2 cos(3x) + 1	3	Did you add the vertical shift

06 Common mistakes and what to say

Mistake	Student writes	Say this
Using the bracket constant as the shift	2x – 60 means 60 left	Factor out the 2 first: 2(x – 30) is a shift of 30 right, and you also reversed the direction.
Calling a negative amplitude	The amplitude of y = -2 cos x is -2	Amplitude is a distance and must be positive; the minus only flips the graph, so write the absolute value.
Multiplying 360 by b	The period of y = sin 2x is 720	Two cycles fit in 360, so one takes 180; period times b must equal 360.
Ignoring the given domain	The maximum is at 75, so it is the maximum on 0 to 60	75 lies outside your interval; compare interior critical points with the endpoints.

07 Homework (paper, section and question numbers)

Maths chapter index — Revise indices and logarithms, then questions 1 and 2
Indices and logarithms — Question 3 also uses graph transformations
Circle properties — Do the multiple choice only, for parameter reading

08 Exit check (last 5 minutes)

Question	Answer	Decision
Amplitude of y = 3 sin(2x – 60)	3	Error: read the sign or 60
Period of the same curve	180 degrees	Error: multiplied 360 by 2
Phase shift of the same curve	30 degrees right	Error: used 60
x of the maximum between 0 and 180	75	Error: set the bracket to zero

09 Tutor FAQ

Why factor out b before reading the shift?

The shift applies to x itself; 2x – 60 must be written as 2(x – 30) to show that x was reduced by 30.

Is the period of tan computed the same way?

No, the basic period of tangent is 180, so divide 180 by b.

Are five points enough for a sketch?

They fix one cycle; for an accurate graph also mark axis intercepts and asymptotes.

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A TutorZone lesson-prep guide. Written by the TutorZone team; questions and lesson flow are original and are not taken from any school or publisher material.