The graphs of $y = \sin x$ and $y = \cos x$ model everything from sound waves to planetary orbits. In this lesson you will learn the key features, amplitude, period, and intercepts, and discover how to transform these graphs into more general sinusoidal functions.
Today's hook, A pure musical tone is a sine wave. When you play middle C on a piano, the air pressure oscillates 262 times per second in a perfect sinusoidal pattern. But what happens if you double the amplitude? Or compress the period? These transformations are exactly what you'll master today.
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Orient and recall
Meet the destination, bring back what you already know, and gather the terms and formulas this lesson leans on.
Worksheets
Practise this lesson
Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.
You know that $\sin 0 = 0$, $\sin \frac{\pi}{2} = 1$, $\sin \pi = 0$, $\sin \frac{3\pi}{2} = -1$, and $\sin 2\pi = 0$. If you plot these points and join them with a smooth curve, what shape do you expect? And how do you think the graph of $y = \cos x$ will differ?
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Two moves, general form at a glance
Work through the core explanation before applying it.
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Two moves, general form at a glance
+5 XP to read
Every sine or cosine graph you will meet in this course can be written in one of these two forms. Learn to read the parameters and you can sketch any transformed wave in under a minute.
The general forms are $y = a\sin(bx) + d$ and $y = a\cos(bx) + d$. The parameter $a$ controls amplitude (and reflects if negative), $b$ controls horizontal stretching via the period formula, and $d$ shifts everything vertically. The range is always centred on the midline $y = d$ and extends $|a|$ units above and below.
General Form
Amplitude
Amplitude = $|a|$. This is always positive. If $a < 0$, the graph is reflected in the $x$-axis.
Period
Period = $\frac{2\pi}{|b|}$ (radians) or $\frac{360^\circ}{|b|}$ (degrees). Larger $b$ means shorter period.
Vertical Shift
$d$ shifts the midline from $y = 0$ to $y = d$. Range = $[d - |a|, \, d + |a|]$.
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What you'll master
Meet the destination, bring back what you already know, and gather the terms and formulas this lesson leans on.
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What you'll master
Know
Key facts
The shape and key features of $y = \sin x$ and $y = \cos x$
How to find amplitude, period, and vertical shift from an equation
The relationship between degrees and radians in graphing
Understand
Concepts
Why sine and cosine are periodic with period $2\pi$
How the parameter $b$ affects horizontal stretching/compressing
Why the cosine graph is a horizontal translation of the sine graph
Can do
Skills
Sketch the graphs of $y = \sin x$ and $y = \cos x$
Sketch transformed sine and cosine graphs
Find amplitude, period, and range from an equation
Read key features from a graph
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Key terms
PeriodThe length of one complete cycle of a periodic function.
AmplitudeThe maximum displacement from the centre line of a periodic function.
Sine CurveThe graph of $y = \sin x$; oscillates between $-1$ and $1$ with period $2\pi$.
Cosine CurveThe graph of $y = \cos x$; oscillates between $-1$ and $1$ with period $2\pi$.
MaximumThe highest point on a graph in a given interval.
MinimumThe lowest point on a graph in a given interval.
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Basic graphs of $y = \sin x$ and $y = \cos x$
Work through the core explanation before applying it.
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Basic graphs of $y = \sin x$ and $y = \cos x$
core concept
The graph of $y = \sin x$ is a smooth wave that passes through the origin, reaches a maximum of $1$ at $x = \frac{\pi}{2}$, returns to $0$ at $x = \pi$, hits a minimum of $-1$ at $x = \frac{3\pi}{2}$, and completes one full cycle at $x = 2\pi$.
The graph of $y = \cos x$ has the identical wave shape, but starts at a maximum of $1$ when $x = 0$. It crosses zero at $x = \frac{\pi}{2}$, reaches $-1$ at $x = \pi$, and returns to $1$ at $x = 2\pi$.
Sine (blue) and cosine (orange) waves, both have period $2\pi$, amplitude 1, and are phase-shifted by $\frac{\pi}{2}$
The sine-cosine shift. The cosine graph is exactly the sine graph shifted $\frac{\pi}{2}$ units to the left. This is expressed algebraically as $\cos x = \sin\left(x + \frac{\pi}{2}\right)$. In signal processing, this phase difference determines whether a wave is leading or lagging.
$y = \sin x$: starts at 0, max 1 at $\frac{\pi}{2}$, returns to 0 at $\pi$, min $-1$ at $\frac{3\pi}{2}$, back to 0 at $2\pi$; $y = \cos x$: starts at 1, drops to 0 at $\frac{\pi}{2}$, min $-1$ at...
Pause, copy the five key points of $y = \sin x$ (0, $\frac{\pi}{2}$, $\pi$, $\frac{3\pi}{2}$, $2\pi$) and the fact that $y = \cos x$ starts at 1 (same shape shifted $\frac{\pi}{2}$ left) into your book.
Did you get this? True or false: the graph of $y = \cos x$ starts at 1 when $x = 0$, whereas $y = \sin x$ starts at 0.
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Transformations of sine and cosine
We just saw that $y = \sin x$ and $y = \cos x$ each complete one cycle in $2\pi$ and oscillate between $-1$ and $1$. That raises a question: how does the graph change if we scale the height, change the speed of oscillation, or shift the midline? This card answers it → the general form $y = a\sin(bx) + d$: amplitude $= |a|$, period $= \frac{2\pi}{|b|}$, midline $y = d$.
For functions of the form $y = a\sin(bx) + d$ and $y = a\cos(bx) + d$:
Parameter
Effect
$a$
Amplitude = $|a|$. If $a < 0$, the graph is reflected in the $x$-axis.
$b$
Period = $\frac{2\pi}{|b|}$ (radians) or $\frac{360^\circ}{|b|}$ (degrees). If $|b| > 1$, the graph is compressed horizontally. If $0 < |b| < 1$, it is stretched.
$d$
Vertical shift. The midline of the wave moves from $y = 0$ to $y = d$.
Finding the Range
The maximum value of $a\sin(bx) + d$ is $|a| + d$, and the minimum value is $-|a| + d$. Therefore:
$$\text{Range: } [d - |a|, \, d + |a|]$$
General form: $y = a\sin(bx) + d$ or $y = a\cos(bx) + d$; Amplitude = $|a|$; Period = $\frac{2\pi}{|b|}$; midline = $y = d$
Pause, copy the general form $y = a\sin(bx) + d$ with amplitude $= |a|$, period $= \frac{2\pi}{|b|}$, and midline $y = d$ into your book.
Quick check: What is the period of $y = \cos(3x)$?
Worked examples · 3 in a row, reveal as you go
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Worked Example, Sketching $y = 2\sin x$
+5 XP for trying first
Sketch one cycle of $y = 2\sin x$ for $0 \leq x \leq 2\pi$ and state its amplitude and range.
Your turn first. Try it yourself before viewing the solution.
SOLUTION
3 marks · HSC band 4+
Step 1: Identify the transformation.
$$a = 2, \, b = 1, \, d = 0 \quad\Rightarrow\quad \text{Amplitude} = |2| = 2$$
Step 3: Draw a smooth wave. The period is still $2\pi$, but the wave now reaches $y = \pm 2$.
Answer: Amplitude = $2$, Range = $[-2, \, 2]$
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Worked Example, Finding period and amplitude
+5 XP for trying first
For $y = 3\cos(2x)$, find the amplitude, period (in radians), and range.
Your turn first. Try it yourself before viewing the solution.
SOLUTION
3 marks · HSC band 4+
Step 1: Read off the parameters.
$$a = 3, \quad b = 2, \quad d = 0$$
Step 2: Calculate amplitude.
$$\text{Amplitude} = |3| = 3$$
Step 3: Calculate period.
$$\text{Period} = \frac{2\pi}{2} = \pi$$
Step 4: Find range.
$$[-3, \, 3]$$
Answer: Amplitude = $3$, Period = $\pi$, Range = $[-3, \, 3]$
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Worked Example, Graph with vertical shift
+5 XP for trying first
We just saw worked examples for amplitude scaling and period compression. That raises a question: what happens when we add a constant $d$ to the function, how does a vertical shift affect the key points we need to plot? This card answers it → shift every $y$-value up or down by $d$; the shape is identical but the midline moves from $y = 0$ to $y = d$.
Sketch $y = \sin x + 1$ for $0 \leq x \leq 2\pi$ and state the range.
Your turn first. Try it yourself before viewing the solution.
SOLUTION
2 marks · HSC band 4+
Step 1: Identify the shift.
$$d = 1 \quad\Rightarrow\quad \text{The entire sine graph is shifted up by 1 unit.}$$
Step 2: Find new maximum and minimum.
$$\text{Max} = 1 + 1 = 2 \text{ at } x = \frac{\pi}{2}, \quad \text{Min} = -1 + 1 = 0 \text{ at } x = \frac{3\pi}{2}$$
Step 3: Sketch the transformed graph. The wave oscillates between $y = 0$ and $y = 2$ with period $2\pi$.
Answer: Range = $[0, \, 2]$
$y = 2\sin x$: amplitude 2, period $2\pi$, key points $(0,0)$, $(\frac{\pi}{2}, 2)$, $(\pi, 0)$, $(\frac{3\pi}{2}, -2)$, $(2\pi, 0)$; $y = 3\cos(2x)$: amplitude 3, period $\pi$, range $[-3, 3]$
Pause, copy the key points for $y = 2\sin x$ and the parameters for $y = 3\cos(2x)$ (amplitude 3, period $\pi$, range $[-3,3]$) into your book.
Fill in the blank: For $y = 4\sin x - 1$, the range is $[\,\underline{\hspace{40px}},\ \underline{\hspace{40px}}]$.
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Common traps
Meet the mistakes that cost marks, then do it yourself.
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Common traps
We just saw worked examples for amplitude, period, and vertical shift. That raises a question: what mistakes do students make when reading off period or amplitude from the general form under exam conditions? This card answers it → Trap 1: larger $b$ means shorter period (not longer); Trap 2: amplitude is always $|a|$, never negative.
Trap 01
Thinking that $b = 2$ doubles the period
If $b = 2$, the graph is actually compressed, not stretched. The period is halved: $\frac{2\pi}{2} = \pi$. Larger $b$ always means a shorter period.
Trap 02
Writing the amplitude as $-3$ instead of $3$
Amplitude is always a positive quantity representing the distance from the midline to the peak. If $a = -3$, the amplitude is still $3$, but the graph is reflected in the $x$-axis.
Trap 03
Forgetting that the vertical shift affects the range
For $y = a\sin(bx) + d$, the range is $[d - |a|, \, d + |a|]$, not $[-|a|, |a|]$. Always add $d$ to both the minimum and maximum.
Larger $b$ → shorter period (compression); smaller $b$ → longer period (stretch); Amplitude = $|a|$, always positive, never write a negative amplitude
Pause, copy the two period/amplitude traps: larger $b$ compresses the period; amplitude $= |a|$ is always positive (never write $a = -2$ means amplitude $-2$) into your book.
Odd one out: Which of these does NOT change the amplitude of $y = \sin x$?
Quick-fire practice · 5 reps
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Drill it, then lock it in
Run the quick drill and copy the summary into your book.
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$y = \sin x$, state amplitude and period.
Show answer
Amplitude: $1$. Period: $2\pi$.
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$y = 2\cos x$, state amplitude and period.
Show answer
Amplitude: $2$. Period: $2\pi$.
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$y = \sin(2x)$, state amplitude and period.
Show answer
Amplitude: $1$. Period: $\pi$.
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$y = \cos x + 2$, state amplitude, period, and range.
Teach it back: Explain in 1–2 sentences why increasing $b$ in $y = \sin(bx)$ shortens the period rather than lengthening it.
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Revisit, sine and cosine shapes
+5 XP for checking
The points $(0, 0)$, $(\frac{\pi}{2}, 1)$, $(\pi, 0)$, $(\frac{3\pi}{2}, -1)$, $(2\pi, 0)$ form a smooth wave, the sine curve. The cosine curve uses $(0, 1)$, $(\frac{\pi}{2}, 0)$, $(\pi, -1)$, $(\frac{3\pi}{2}, 0)$, $(2\pi, 1)$, giving the same wave shape shifted left by $\frac{\pi}{2}$.
Return to your original answer from Section 01. What did you get right? What has changed in your thinking?
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Multiple choice
Answer the drill bank and rate your confidence.
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Multiple choice
+5 XP per correct · +25 XP all-correct
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Short answer
Write full responses, then check them against the model answers.
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Short answer
ApplyBand 44 marks
Q1. Sketch $y = 3\cos(2x)$ for $0 \leq x \leq 2\pi$. Label the amplitude, period, and the coordinates of all maximum and minimum points. (4 marks)
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View comprehensive answer
Amplitude = 3 [0.5], Period = $\pi$ [0.5].
Max points at $(0, 3)$, $(\pi, 3)$, $(2\pi, 3)$ [1.5].
Min points at $(\frac{\pi}{2}, -3)$, $(\frac{3\pi}{2}, -3)$ [1.5].
ApplyBand 43 marks
Q2. Find the exact period and range of $y = 2\sin\left(\frac{x}{2}\right) + 1$. (3 marks)
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View comprehensive answer
Period = $\frac{2\pi}{\frac{1}{2}} = 4\pi$ [1].
Amplitude = 2 [0.5].
Range = $[1 - 2, \, 1 + 2] = [-1, \, 3]$ [1.5].
AnalyseBand 53 marks
Q3. Explain why $y = \cos x$ can be obtained from $y = \sin x$ by a horizontal translation. State the exact size and direction of this translation. (3 marks)
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View comprehensive answer
$\cos x = \sin\left(x + \frac{\pi}{2}\right)$ [1].
This means replacing $x$ with $x + \frac{\pi}{2}$ in $y = \sin x$ [1].
This corresponds to a horizontal translation of $\frac{\pi}{2}$ units to the left [1].
Comprehensive answers (click to reveal)
Drill 1: Amplitude $1$, Period $2\pi$ · 2: Amplitude $2$, Period $2\pi$ · 3: Amplitude $1$, Period $\pi$ · 4: Amplitude $1$, Period $2\pi$, Range $[1, 3]$ · 5: Amplitude $3$, Period $2\pi$, Range $[-3, 3]$
Q1 (4 marks): Amplitude = 3 [0.5], Period = $\pi$ [0.5], max at $(0,3)$, $(\pi,3)$, $(2\pi,3)$ [1.5], min at $(\frac{\pi}{2},-3)$, $(\frac{3\pi}{2},-3)$ [1.5].
Q2 (3 marks): Period = $4\pi$ [1], Amplitude = 2 [0.5], Range = $[-1, 3]$ [1.5].
Q3 (3 marks): $\cos x = \sin(x + \frac{\pi}{2})$ [1]. Replacing $x$ with $x + \frac{\pi}{2}$ [1]. Translation $\frac{\pi}{2}$ to the left [1].
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Review and finish
Take the module quiz if you are ready, then mark the lesson complete.
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quiz
A full module quiz covering every lesson in this module, not just this one. Set aside a decent block of time and treat it like a real assessment.
