Analyse and transform sine, cosine, and tangent graphs

International Baccalaureate (IB) IB AA SL: Mathematics: Analysis and Approaches SL

Geometry and Trigonometry

Reading period, shifts, scale, and key features from equations and graphs

The graphs of sine, cosine, and tangent repeat in regular patterns. Their familiar shapes can be shifted, reflected, or stretched by changing parameters in an equation. To analyse a transformed graph, first identify its parent function and then connect each parameter to a visible feature. Angles in this lesson are in radians unless stated otherwise. Familiarity with the basic graphs and exact values such as

\sin 0=0

and

\cos 0=1

is useful.

What you will learn

Identify how parameters change the shape and position of sine, cosine, and tangent graphs.
Find periods, key points, midlines, ranges, and vertical asymptotes where appropriate.
Use equations, tables of values, graphs, and contextual information to describe a trigonometric model.
Use graphing technology to check a graph while supporting conclusions with mathematical reasoning.

1. Start with the parent graphs

The parent sine graph is

y=\sin x

. It repeats every

2\pi

, has midline

y=0

, and ranges from

-1

to

1

. One cycle passes through

(0,0)

,

(\frac{\pi}{2},1)

,

(\pi,0)

,

(\frac{3\pi}{2},-1)

, and

(2\pi,0)

.

The parent cosine graph is

y=\cos x

. It also repeats every

2\pi

and ranges from

-1

to

1

. Its cycle starts at a maximum:

(0,1)

,

(\frac{\pi}{2},0)

,

(\pi,-1)

,

(\frac{3\pi}{2},0)

, and

(2\pi,1)

.

The parent tangent graph is

y=\tan x

. It repeats every

\pi

and passes through

(0,0)

. It is undefined at

x=\frac{\pi}{2}+k\pi

, where

k

is any integer; these values produce vertical asymptotes. Its graph has no maximum or minimum and has range all real numbers.

Sine and cosine have period $2\pi$ ; tangent has period $\pi$ .
An asymptote is a line the graph approaches but does not meet at the excluded input.

2. Read the parameters in a transformed equation

A useful form for sine and cosine is

y=a\sin(b(x-c))+d

or

y=a\cos(b(x-c))+d

. Here

a

changes vertical scale and may reflect the graph;

b

changes horizontal scale and period;

c

shifts the graph horizontally; and

d

shifts it vertically. The horizontal shift is

c

to the right when

c>0

, and to the left when

c<0

.

For sine and cosine, the amplitude is |a|, the distance from the midline to a maximum or minimum. The midline is

y=d

, so the range is from

d-|a|

to

d+|a|

. The period is

\frac{2\pi}{|b|}

. If

a<0

, the graph is reflected vertically across its midline. If

b<0

, the direction of travel along the horizontal axis is reversed; the period still uses |b|.

For tangent, a corresponding form is

y=a\tan(b(x-c))+d

. Its period is

\frac{\pi}{|b|}

. The value

a

stretches or reflects the graph vertically, but tangent has no amplitude and its range remains all real numbers when

a\ne 0

. The central point is

(c,d)

. The vertical asymptotes occur when

b(x-c)=\frac{\pi}{2}+k\pi

, so their positions depend on both

b

and

c

.

To sketch a transformed sine or cosine graph, find the midline and period, then mark one cycle at five equally spaced horizontal positions. The spacing is one quarter-period. Use the parent graph’s sequence of highs, lows, and midline crossings, then apply the vertical scale and shift. For tangent, mark its central point and the two nearest asymptotes, then sketch the increasing or decreasing branch.

A context can help interpret the parameters. For example, a repeating height may have a midline representing average height, amplitude representing the size of the variation, and period representing the time for one complete cycle. The independent variable and its units must be stated; the equation’s shifts and period use those same units.

T_{\sin,\cos}=\frac{2\pi}{|b|},\qquad T_{\tan}=\frac{\pi}{|b|}

For sine and cosine: amplitude |a|, midline $y=d$ , period $\frac{2\pi}{|b|}$ .
For tangent: period $\frac{\pi}{|b|}$ ; locate asymptotes from the transformed input.
Use the graph’s domain restrictions, especially tangent’s excluded asymptote inputs.

3. Connect equations, tables, graphs, and technology

A table of values is a practical bridge between an equation and a graph. For a sine or cosine model, choose inputs one quarter-period apart and evaluate the function. Plot the resulting points, draw the midline, and join the points with a smooth repeating curve. The points show where the graph crosses its midline and reaches its turning points.

For tangent, make a table only at inputs where the function is defined. Do not join points across a vertical asymptote. The graph approaches the asymptote on either side, but the asymptote itself is not in the domain.

A graphing calculator can check a sketch and reveal whether the chosen viewing window shows a complete cycle or an asymptote. Enter the equation with parentheses around the transformed input, set the angle mode to radians when using radian values, and adjust the window to include the relevant interval. Then verify the period, midline, turning points, or asymptotes analytically. A calculator display is a check, not a substitute for explaining how those features follow from the equation.

When a graph is provided instead, estimate its period by measuring the horizontal distance between matching points, such as consecutive maxima. For sine and cosine, the maximum and minimum values reveal the midline and amplitude: the midline is halfway between them, and the amplitude is half their difference. For tangent, use the spacing between equivalent branches or consecutive asymptotes to infer the period.

Use parentheses to preserve the input transformation in a calculator.
Check the window and angle mode before interpreting a displayed graph.
Read period and vertical features from the graph, then connect them to equation parameters.

4. A reliable analysis routine

For a sine or cosine equation, rewrite the input as

b(x-c)

when possible. Read off

a

,

b

,

c

, and

d

; calculate amplitude, period, midline, and range; then plot key points over one period. For tangent, identify the centre, period, and nearest asymptotes before sketching.

State the domain when it matters. Sine and cosine are defined for every real input. Tangent excludes inputs that make its angle an odd multiple of

\frac{\pi}{2}

. In a real situation, the context may further restrict the domain, for example to nonnegative time or a stated observation interval.

Finish by checking that the graph matches the equation’s features. A vertical shift changes the midline or centre, not the period. A horizontal scale changes the period, not the vertical range. These checks often expose a misplaced bracket or a shift applied in the wrong direction.

Separate horizontal changes from vertical changes.
State units and any context-specific domain restriction.
Use key features to confirm a sketch rather than relying on its visual appearance alone.

Worked example

Sine graph with a shift and vertical stretch

Analyse and sketch one cycle of

y=3\sin(2(x-\frac{\pi}{6}))+1

.

Identify the parameters
Compare the equation with $y=a\sin(b(x-c))+d$ . The amplitude is $3$ , the horizontal shift is $\frac{\pi}{6}$ to the right, and the midline is $y=1$ .
$a=3,\quad b=2,\quad c=\frac{\pi}{6},\quad d=1$
Find the period and range
The period is the parent period divided by the horizontal scale factor. The range extends three units above and below the midline.
$T=\frac{2\pi}{2}=\pi,\qquad -2\le y\le 4$
Mark the key points
A quarter-period is $\frac{\pi}{4}$ . Starting at the shifted sine crossing, use the usual sequence of midline, maximum, midline, minimum, and midline values.
$\left(\frac{\pi}{6},1\right),\left(\frac{5\pi}{12},4\right),\left(\frac{2\pi}{3},1\right),\left(\frac{11\pi}{12},-2\right),\left(\frac{7\pi}{6},1\right)$

Answer: One cycle runs from

x=\frac{\pi}{6}

to

x=\frac{7\pi}{6}

. The graph has midline

y=1

, amplitude

3

, period

\pi

, and range

[-2,4]

.

Check: The interval length is

\frac{7\pi}{6}-\frac{\pi}{6}=\pi

, matching the calculated period.

Worked example

Cosine model from graph features

A repeating temperature model has a maximum of

22

degrees at time

t=1

hour and a minimum of

10

degrees at time

t=7

hours. Find a cosine model with positive amplitude for one repeating cycle, and state its period.

Find the midline and amplitude
The midline is halfway between the maximum and minimum. The amplitude is half their difference.
$d=\frac{22+10}{2}=16,\qquad a=\frac{22-10}{2}=6$
Find the period
The maximum and minimum are half a cycle apart. Their time difference is six hours, so the full period is twelve hours.
$T=2(7-1)=12$
Build the model
A positive cosine graph begins at a maximum. Since the maximum occurs at $t=1$ , shift the cosine graph right by one hour. Choose $b$ so that its period is twelve hours.
$b=\frac{2\pi}{12}=\frac{\pi}{6},\qquad T(t)=6\cos\left(\frac{\pi}{6}(t-1)\right)+16$

Answer: One suitable model is

T(t)=6\cos(\frac{\pi}{6}(t-1))+16

, where

t

is in hours and temperature is in degrees. Its period is

12

hours.

Check: At

t=1

, the cosine input is

0

, giving the maximum

22

. At

t=7

, the input is

\pi

, giving the minimum

10

.

Worked example

Tangent period and asymptotes

For

y=2\tan(3(x+\frac{\pi}{6}))-1

, find the period, centre, and the two nearest vertical asymptotes.

Read the centre and period
The tangent centre occurs when its input is zero. The horizontal shift is left by $\frac{\pi}{6}$ , and the vertical shift is down one unit.
$\left(-\frac{\pi}{6},-1\right),\qquad T=\frac{\pi}{3}$
Locate asymptotes
Asymptotes occur when the tangent input is $\frac{\pi}{2}$ or $-\frac{\pi}{2}$ around the centre. Solve for $x$ at each value.
$3\left(x+\frac{\pi}{6}\right)=\pm\frac{\pi}{2}\quad\Longrightarrow\quad x=-\frac{\pi}{3},\ 0$
Describe the branch
The vertical factor is positive, so the branch between these asymptotes increases through the centre. The factor $2$ changes its vertical steepness but does not change the period.
$-\frac{\pi}{3}<x<0$

Answer: The period is

\frac{\pi}{3}

, the centre is

(-\frac{\pi}{6},-1)

, and the nearest asymptotes are

x=-\frac{\pi}{3}

and

x=0

.

Check: The centre lies halfway between the asymptotes, and their separation is

\frac{\pi}{3}

, equal to one tangent period.

Common mistakes and how to avoid them

Using

2\pi b

as the sine or cosine period.

Correction: The period is divided by the magnitude of the coefficient of

x

:

\frac{2\pi}{|b|}

. For tangent, use

\frac{\pi}{|b|}

.

Treating a shift written as

(x+q)

as a shift right by

q

.

Correction: Rewrite the input as

b(x-c)

. A positive

c

shifts right; a negative

c

shifts left.

Calling the factor outside tangent its amplitude.

Correction: Tangent has no maximum or minimum, so it has no amplitude. The factor changes vertical scale and the range remains all real numbers when the factor is nonzero.

Drawing through a tangent asymptote or including its input in the domain.

Correction: Find the excluded inputs from the transformed angle and sketch separate branches on either side.

Lesson summary

Sine and cosine have period $\frac{2\pi}{|b|}$ ; tangent has period $\frac{\pi}{|b|}$ .
For sine and cosine, |a| is the amplitude and $y=d$ is the midline.
Use shifts, key points, and asymptotes to connect each equation to its graph.
Use technology to check a carefully reasoned analysis, with appropriate angle mode and viewing window.

Check your understanding

Question 1

What is the period of

y=\cos(4x)-2

?

$\frac{\pi}{2}$
$4\pi$
$2\pi$
$\frac{\pi}{4}$

Show answer and explanation

\frac{\pi}{2}

For cosine, the period is

\frac{2\pi}{|b|}

. With

b=4

, it is

\frac{\pi}{2}

; the vertical shift does not change the period.

Question 2

For

y=-2\sin(x)+3

, what are the range and midline?

Range $[1,5]$ , midline $y=3$
Range $[-2,2]$ , midline $y=0$
Range $[3,5]$ , midline $y=-2$
Range $[-5,-1]$ , midline $y=-3$

Show answer and explanation

Range

[1,5]

, midline

y=3

The amplitude is

2

and the midline is

y=3

, so the range is

[3-2,3+2]=[1,5]

. The negative sign reflects the graph but does not change these values.

Question 3

Which inputs are vertical asymptotes of

y=\tan(2x)

in the interval

0\le x\le\pi

?

$x=\frac{\pi}{4},\frac{3\pi}{4}$
$x=\frac{\pi}{2}$ only
$x=\frac{\pi}{2},\pi$
$x=\frac{\pi}{8},\frac{5\pi}{8}$

Show answer and explanation

x=\frac{\pi}{4},\frac{3\pi}{4}

Set

2x=\frac{\pi}{2}+k\pi

. In the stated interval, this gives

x=\frac{\pi}{4}

and

x=\frac{3\pi}{4}

.

Key terms

Amplitude: For a sine or cosine graph, the distance from its midline to a maximum or minimum.
Midline: The horizontal line halfway between the maximum and minimum of a sine or cosine graph.
Period: The horizontal length of one complete repeating cycle.
Vertical asymptote: A vertical line that a graph approaches but does not meet; tangent is undefined at its asymptote inputs.

Continue through IB AA SL

View the complete IB AA SL International Baccalaureate (IB) IB AA SL: Mathematics: Analysis and Approaches SL curriculum and lessons

Tutor support for SL 3.7 · Analyse and transform sine, cosine, and tangent graphs

Working on SL 3.7 · Analyse and transform sine, cosine, and tangent graphs in IB AA SL? Bring the question and your own attempt. In a live session, you can label the axes and connect each feature of the graph to the equation or situation. Ask about the first step that does not make sense, then try a similar question yourself.

Explore math tutoring · Request an eligible free 30-minute introduction. Your request is matched by subject and language; a tutor must accept before a time is confirmed.

برای مبحث SL 3.7 · تحلیل و تبدیل نمودارهای سینوس، کسینوس و تانژانت در درس IB AA SL، سؤال و تلاش خود را همراه بیاورید. در جلسه می‌توانید محورها را برچسب بزنید و هر ویژگی نمودار را به معادله یا موقعیت مسئله ربط دهید. راهنمای معلم ریاضی را ببینید یا جلسهٔ معارفهٔ رایگان ۳۰ دقیقه‌ای درخواست کنید. زمان جلسه پس از پذیرش معلم قطعی می‌شود.

About this lesson and its review

Published by DoAssignment. This reviewed lesson follows International Baccalaureate (IB) IB AA SL: Mathematics: Analysis and Approaches SL, study topic SL 3.7. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

Official curriculum reference

Report a correction or ask a question