Trigonometric functions and graphs (QCE Math Methods Unit 2)

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 $y = \cos x$

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 two are the same wave with a quarter-period offset. Both are bounded between $-1$ and $1$ and repeat every $2\pi$, which is why amplitude and period are the first two features to read from any transformed version.

The transformed form $y = a \sin(b(x - h)) + k$

Each parameter has a distinct effect.

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

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

Why each parameter does what it does

The four parameters map onto the four standard transformations of a graph. The amplitude $a$ is a vertical dilation, stretching the wave away from its centre line and flipping it when negative. The factor $b$ is a horizontal dilation, and because it acts inside the function it compresses the period to $\dfrac{2\pi}{b}$ (a larger $b$ means more cycles in the same span). The phase shift $h$ is a horizontal translation, and $k$ is a vertical translation that sets the centre line. Reading these straight off the equation is the fastest route to both a sketch and a description.

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, because $\cos$ starts at its peak. Plotting these five quarter-period points and joining them with a smooth curve produces an accurate sketch over one period, which then repeats.

Periodic modelling

Periodic functions are the natural model for any quantity that repeats: tides rising and falling, daily temperature, hours of daylight through the year, and alternating current. The skill is to read the four parameters from the described behaviour rather than from an equation. Worded scenarios (tides, temperature, alternating current) give the maximum, minimum and the time of one extremum. From those:

• $k$ = (max + min) / 2 (centre line).
• $|a|$ = (max - min) / 2 (amplitude).
• $b = 2\pi /$ period.
• $h$ = time at which the sine starts at zero rising, or the cosine starts at maximum.