Sketch and analyse graphs of $y = a \sin(b(x - h)) + k$ and $y = a \cos(b(x - h)) + k$, identifying amplitude, period, phase shift and vertical translation

What this dot point is asking

QCAA wants you to sketch and analyse transformed sine and cosine graphs, identifying the four key parameters (amplitude, period, phase shift, vertical translation) and using them to model periodic phenomena like tides, temperatures or pendulum displacement.

The parent functions $y = \sin x$ and IMATH_1

For $\sin x$:

• Domain: all real $x$.
• Range: $[-1, 1]$.
• Period: $2\pi$.
• Amplitude: $1$.
• Zeros at $x = n\pi$ for integer $n$.
• Maximum $1$ at $x = \pi/2 + 2n\pi$. Minimum $-1$ at $x = 3\pi/2 + 2n\pi$.

For $\cos x$:

• Same domain, range, period, amplitude.
• Zeros at $x = \pi/2 + n\pi$.
• Maximum $1$ at $x = 2n\pi$. Minimum $-1$ at $x = (2n+1)\pi$.

$\cos x = \sin(x + \pi/2)$, so the cosine graph is the sine graph shifted left by $\pi/2$.

The transformed form IMATH_21

Each parameter has a distinct effect.

• IMATH_22 : amplitude. Vertical dilation. Range becomes $[k - |a|, k + |a|]$. If $a < 0$, the graph is reflected vertically.
• IMATH_25 : angular frequency. Period $= 2\pi / |b|$. Larger $|b|$ compresses the graph horizontally.
• IMATH_28 : horizontal phase shift. Graph moves $h$ units right (if $h > 0$).
• IMATH_31 : vertical translation. Centre line moves to $y = k$.

The same parameters apply to $y = a \cos(b(x - h)) + k$.

Key features for sketching

For $y = a \sin(b(x-h)) + k$:

• Centre line: $y = k$.
• Maximum: $y = k + |a|$.
• Minimum: $y = k - |a|$.
• Period: $2\pi / |b|$.
• A complete cycle goes: centre, max, centre, min, centre.
• First zero of the parent sine occurs at $x = h$.

For cosine, replace "centre" first with "maximum" first.

Worked example

Sketch $y = -3 \cos(2(x - \pi/4)) + 5$.

Parameters: $a = -3$, $b = 2$, $h = \pi/4$, $k = 5$.

• Amplitude $3$, range $[2, 8]$.
• Period $2\pi / 2 = \pi$.
• Phase shift: $\pi/4$ right.
• Reflection (because $a < 0$): the cosine starts at a minimum at $x = \pi/4$ instead of a maximum.

Key points within one period:

• IMATH_51 minimum.
• IMATH_52 centre line crossing (rising).
• IMATH_53 maximum.
• IMATH_54 centre line crossing (falling).
• IMATH_55 next minimum.

Periodic modelling

Worded scenarios (tides, temperature, alternating current) give the maximum, minimum and the time of one extremum. From those:

• IMATH_56 = (max + min) / 2 (centre line).
• IMATH_57 = (max - min) / 2 (amplitude).
• IMATH_58 period.
• IMATH_59 = time at which the sine starts at zero rising, or the cosine starts at maximum.

Common traps

Confusing $b$ with the period. Period is $2\pi / b$, not $b$ itself.

Forgetting that $h$ shifts in the same direction as its sign. $\sin(2(x - \pi/4))$ shifts right by $\pi/4$, not left.

Missing the reflection when $a < 0$. Negative $a$ flips the graph; max becomes min and vice versa.

Reading the model in degrees by accident. QCAA Math Methods always uses radians.

In one sentence

For $y = a \sin(b(x-h)) + k$ and $y = a \cos(b(x-h)) + k$, the amplitude is $|a|$, the period is $2\pi/|b|$, the phase shift is $h$ (right if positive), and the centre line is $y = k$, with the range $[k - |a|, k + |a|]$.