Transformed Trigonometric Graphs: Amplitude, Period & Tangent

BUKIT TIMAH TUTOR · ADDITIONAL MATHEMATICS SYNTHESIS GUIDE 32

A transformed trigonometric graph can be read as a small set of controls: vertical scale, horizontal scale, midline, phase and — for tangent — asymptote structure.

The graph of y=sinx is not a picture to memorise. It is a periodic structure that can be stretched, compressed, reflected and shifted. The same is true for cosine and tangent, but tangent has no finite amplitude and carries vertical asymptotes instead of maxima and minima.

This guide connects equation form to graph behaviour, parameter recovery and simple periodic modelling. It keeps degrees and radians explicit and treats graph features as consequences of the function rather than decorative sketching conventions.

Base graph → vertical scale → period → phase → midline/asymptotes → key points → model interpretation.

1. The three base graphs

y=sinx: range [−1,1], period 2π.
y=cosx: range [−1,1], period 2π.
y=tanx: range all real numbers, period π, undefined at π/2+kπ.

Use degree equivalents 360° and 180° when the problem is in degrees.

2. Amplitude belongs to bounded sine and cosine waves

For

y=a sinx

or y=a cosx, the amplitude is

|a|.

The range is [−|a|,|a|] before any vertical translation.

Tangent has no amplitude because it is unbounded.

3. Vertical translation creates a midline

For

y=a sinx+c,

the midline is y=c and the range is

[c−|a|, c+|a|].

The vertical shift changes the centre of oscillation, not the amplitude.

4. Worked amplitude and midline example

For

y=3sinx+5,

amplitude = 3;
midline y=5;
maximum = 8;
minimum = 2;
period = 2π.

5. Horizontal scaling changes period

For

y=sin(bx),

the period is

2π/|b|.

For tangent y=tan(bx), the period is

π/|b|.

A larger |b| means more cycles fit into the same x-interval.

6. Worked period example

For y=4cos3x−1:

amplitude 4;
midline y=−1;
period 2π/3;
range [−5,3].

Three complete cosine cycles occur across an x-length of 2π.

7. The alternative form sin(x/b)

For

y=a sin(x/b)+c, b≠0,

the period is

2π|b|.

This is the same horizontal-scaling principle written with reciprocal frequency.

8. Reflection from a negative vertical coefficient

Changing y=sinx to y=−sinx reflects the graph in its midline y=0.

For y=−2sinx+4, the amplitude remains 2 and midline remains y=4. The negative sign changes orientation, not amplitude.

9. Phase shifts

For

y=a sin[b(x−h)]+c,

the graph is shifted horizontally by h relative to the corresponding unshifted sine form.

Be careful with forms such as sin(bx−α): factor b to obtain b(x−α/b), so the horizontal shift is α/b.

10. Worked phase example

For y=2cos(3x−π/2)+1:

3x−π/2=3(x−π/6).

amplitude 2;
period 2π/3;
midline y=1;
horizontal shift π/6 to the right.

11. Key-point method for sine and cosine sketches

One sine or cosine cycle can be organised into quarter-period steps. For period T, move by T/4 through the familiar sequence of midline, extremum, midline, opposite extremum, midline.

This is more reliable than drawing a wave first and trying to force the equation onto it afterward.

12. Recover amplitude and midline from max/min

If a sinusoidal model has maximum M and minimum m, then

amplitude=(M−m)/2

and

midline=(M+m)/2.

For max 17 and min 5, amplitude is 6 and midline is 11.

13. Recover period from repeated features

The period is the horizontal distance between corresponding repeated states: peak to next peak, trough to next trough, or same-direction midline crossing to next same-direction crossing.

Adjacent maximum and minimum are separated by half a period, not one full period.

14. Tangent graph structure

For y=a tan(bx)+c:

period = π/|b|;
centre line y=c;
no maximum or minimum;
vertical asymptotes occur where bx=π/2+kπ.

The parameter a changes vertical steepness and orientation, but does not create an amplitude.

15. Worked tangent example

For

y=2tan(3x)−1,

period π/3;
centre line y=−1;
vertical asymptotes satisfy 3x=π/2+kπ, so x=π/6+kπ/3.

Between consecutive asymptotes, the branch passes through the centre point where tan(3x)=0.

16. A tangent graph must not be joined across asymptotes

At a vertical asymptote, the tangent function is undefined. The graph approaches without crossing that x-value as a continuous branch.

Drawing a connected line across an asymptote invents function values where none exist.

17. Graph transformations and equation solving

The equation a sin(bx)+c=k asks where the transformed graph meets the horizontal line y=k.

Before inverse trigonometry, check whether k lies in the function’s range. If not, there are no real solutions.

Guide 28 develops the full principal-value and interval-solution process.

18. R-form connects graph amplitude to coefficients

An expression

a cosx+b sinx

can be rewritten as a single sinusoid with amplitude

√(a²+b²).

Thus R-form is also a graph-transformation tool: two same-frequency components become one shifted sinusoidal graph.

19. Periodic modelling

A simple model

H(t)=M+Acos(ωt−α)

has midline M, amplitude |A| and period 2π/|ω|.

The phase locates the cycle in time. The model is useful only over contexts where periodic behaviour is a reasonable assumption.

20. Worked model recovery

A periodic quantity varies between 4 and 16 and repeats every 8 hours.

amplitude = 6;
midline = 10;
ω=2π/8=π/4 radians per hour.

If a maximum occurs at t=0, one suitable model is

H(t)=10+6cos(πt/4).

Other equivalent phase representations may describe the same function.

21. Degree and radian parameters are not numerically interchangeable

A period of 360° and a period of 2π radians describe the same angular cycle but use different numerical coordinates.

When using a model such as sin(bt), b inherits the reciprocal unit needed to make the angle dimensionally meaningful. Keep the question’s angle convention consistent.

22. Common failure patterns

Calling a negative coefficient a negative amplitude.
Using 2π/b without taking |b| for a positive period.
Treating the vertical shift c as part of the amplitude.
Reading adjacent maximum-to-minimum distance as the full period.
Calling tangent’s vertical scale an amplitude.
Drawing tangent continuously across vertical asymptotes.
Reading the shift in sin(bx−α) as α instead of α/b.
Using a periodic model outside the context where its assumptions are justified.

23. A reliable transformed-graph routine

Identify sine, cosine or tangent.
Factor the angle if needed to expose horizontal shift.
Find vertical scale, midline and period.
For sine/cosine, determine range and extrema.
For tangent, determine vertical asymptotes and branch centres.
Use quarter-period or asymptote structure to place key points.
Check the graph against one or two exact function values.
For models, return parameters to units and physical meaning.

24. Practice set

State amplitude, period, midline and range of y=4sinx+2.
Do the same for y=−3cos2x+1.
Find the period of y=sin5x.
Find the period of y=cos(x/3).
Rewrite sin(4x−π) to expose its horizontal shift.
State the shift in Question 5.
A sinusoid has max 20 and min 8. Find amplitude and midline.
Peaks occur at x=1 and x=7. Find the period.
State the period of y=tan4x.
Find asymptotes of y=tan2x in general form.
Find period and centre line of y=3tan(2x)−4.
Why does tangent have no amplitude?
How many cycles of sin3x occur over 0≤x<2π?
State the range of y=5−2sinx.
For y=2cos(3x−π/2)+1, state amplitude, period, midline and shift.
A periodic model has max 15, min 5 and period 6. Find amplitude, midline and angular frequency.
If a maximum occurs at t=0 in Question 16, give one cosine model.
Why should tangent branches not be joined across asymptotes?
How can a graph check whether a trig equation has no real solution?
How does R-form determine the amplitude of a cosx+b sinx?

Answers

Amplitude 4; period 2π; midline 2; range [−2,6].
Amplitude 3; period π; midline 1; range [−2,4].
2π/5.
6π.
sin[4(x−π/4)].
π/4 right.
Amplitude 6, midline 14.
6.
π/4.
2x=π/2+kπ, so x=π/4+kπ/2.
Period π/2; centre line y=−4.
It is unbounded and has no finite maximum/minimum oscillation range.
Three.
[3,7].
Amplitude 2; period 2π/3; midline 1; shift π/6 right.
Amplitude 5; midline 10; ω=2π/6=π/3.
H(t)=10+5cos(πt/3).
The function is undefined at each vertical asymptote, so continuity cannot pass through it.
Compare the target horizontal level with the function’s range, or count graph intersections.
R=√(a²+b²).

25. What mastery looks like

Mastery means the learner reads transformed trigonometric equations as graph controls, distinguishes amplitude from reflection and midline, derives period from the angle coefficient, handles tangent through asymptotes rather than amplitude, and can recover simple model parameters from maxima, minima and repeated features.

The transfer test is to provide a graph without its equation, or an equation without a graph. If the learner can reconstruct the missing representation and verify it through exact features, transformed trigonometric graphs have become a working representation system.

Additional Mathematics Synthesis Guide 32: Transformed Trigonometric Graphs — Amplitude, Period, Midline, Tangent Structure and Modelling