Sketch and interpret graphs of $y = a \sin(b x + c) + d$, $y = a \cos(b x + c) + d$ and $y = a \tan(b x + c) + d$, identifying amplitude, period, phase shift and vertical shift

What this dot point is asking

NESA wants you to sketch transformations of $\sin x$, $\cos x$ and $\tan x$ accurately, identify amplitude, period, phase shift and vertical shift from an equation, and read these off a sketch.

The answer

The base graphs

For $y = \sin x$:

• Domain $\mathbb{R}$, range $[-1, 1]$.
• Period $2 \pi$, amplitude $1$.
• Zeros at $x = k \pi$. Maxima at $x = \frac{\pi}{2} + 2 k \pi$. Minima at $x = -\frac{\pi}{2} + 2 k \pi$.
• Odd function: $\sin(-x) = -\sin x$.

For $y = \cos x$:

• Domain $\mathbb{R}$, range $[-1, 1]$.
• Period $2 \pi$, amplitude $1$.
• Zeros at $x = \frac{\pi}{2} + k \pi$. Maxima at $x = 2 k \pi$. Minima at $x = \pi + 2 k \pi$.
• Even function: $\cos(-x) = \cos x$.
• IMATH_22 : cosine is sine shifted left by $\frac{\pi}{2}$.

For $y = \tan x$:

• Domain $\mathbb{R} \setminus \left\{ \frac{\pi}{2} + k \pi \right\}$, range $\mathbb{R}$.
• Period $\pi$, no amplitude (unbounded).
• Zeros at $x = k \pi$. Vertical asymptotes at $x = \frac{\pi}{2} + k \pi$.
• Odd function: $\tan(-x) = -\tan x$.

Transformations of sine and cosine

For $y = a \sin(b(x - h)) + d$ or $y = a \cos(b(x - h)) + d$:

• Amplitude $= |a|$. The graph oscillates between $d - |a|$ and $d + |a|$.
• Period $= \frac{2 \pi}{|b|}$.
• Phase shift $= h$ (right if $h > 0$, left if $h < 0$).
• Vertical shift $= d$. The centre line is $y = d$.
• If $a < 0$, the graph is reflected in the centre line: a sine starts going down from the centre rather than up; a cosine starts at the minimum rather than the maximum.
• If $b < 0$, the graph is reflected in a vertical line. For sine, $\sin(-x) = -\sin x$, which is equivalent to flipping the sign of $a$. For cosine, $\cos(-x) = \cos x$, so a sign on $b$ has no effect.

If the equation is given as $y = a \sin(b x + c) + d$, factor: $b x + c = b\left(x + \frac{c}{b}\right)$. The phase shift is $-\frac{c}{b}$.

Transformations of tangent

For $y = a \tan(b(x - h)) + d$:

• Period $= \frac{\pi}{|b|}$ (note: tangent's period is $\pi$, not $2 \pi$).
• Asymptotes are at $b(x - h) = \frac{\pi}{2} + k \pi$, that is $x = h + \frac{\pi}{2 b} + \frac{k \pi}{b}$.
• Vertical shift by $d$ raises or lowers the graph but does not change the asymptotes.
• IMATH_58 rescales the steepness but does not change the asymptotes or the period.

Reading features off the equation

Given any equation in the standard form, you can extract amplitude, period and shifts in seconds without sketching. Reverse process: given amplitude, period, centre line and a starting point, write the equation.

A sine function with period $T$ has $b = \frac{2 \pi}{T}$. A cosine function with amplitude $A$, centre line $y = c$, period $T$ and starting at the maximum at $x = h$ has equation

Worked examples

Amplitude, period and centre

$y = 4 \sin(\pi x) + 2$: amplitude $4$, period $\frac{2 \pi}{\pi} = 2$, centre line $y = 2$. Max $6$, min $-2$.

Phase shift in the standard form

$y = \sin\left( 2 x + \frac{\pi}{2} \right) = \sin\left( 2\left( x + \frac{\pi}{4} \right) \right)$. Phase shift: left by $\frac{\pi}{4}$. Period $\pi$.

Equivalently, since $\sin\left( 2 x + \frac{\pi}{2} \right) = \cos(2 x)$, the curve is identical to $y = \cos(2 x)$.

Tangent with horizontal compression

$y = \tan(3 x)$: period $\frac{\pi}{3}$. Asymptotes at $x = \frac{\pi}{6} + \frac{k \pi}{3}$, that is at $\frac{\pi}{6}, \frac{\pi}{2}, \frac{5 \pi}{6}, \dots$. Zeros at $x = \frac{k \pi}{3}$.

Writing an equation from features

Find a cosine equation with amplitude $5$, period $\frac{\pi}{2}$, centre line $y = 1$, and the first maximum at $x = \frac{\pi}{8}$.

$b = \frac{2 \pi}{T} = \frac{2 \pi}{\pi / 2} = 4$.

$y = 5 \cos\left( 4\left( x - \frac{\pi}{8} \right) \right) + 1 = 5 \cos\left( 4 x - \frac{\pi}{2} \right) + 1$.

A reflected sine

$y = -3 \sin x + 1$: amplitude $3$, period $2 \pi$, centre line $y = 1$. The negative coefficient reflects: the graph starts at the centre line at $x = 0$ and goes down to the minimum at $y = -2$ first, instead of up to the maximum.

Common traps

Confusing period with $b$. $b$ is not the period; the period is $\frac{2 \pi}{|b|}$ (or $\frac{\pi}{|b|}$ for tan).

Mis-reading the phase shift. In $y = \sin(b x + c)$, the phase shift is $-\frac{c}{b}$, not $-c$. Factor $b$ out of the bracket first.

Confusing amplitude with maximum value. Amplitude is the half-distance from minimum to maximum. The maximum value is $d + |a|$, not just $|a|$.

Forgetting tangent's period is $\pi$. Sine and cosine have period $2 \pi$; tangent has period $\pi$. Use the right formula.

Dropping the absolute value in amplitude. Amplitude is $|a|$, always non-negative. A negative $a$ flips the curve but the amplitude is still $|a|$.

In one sentence

For $y = a \sin(b(x - h)) + d$ (and similarly for cosine), amplitude is $|a|$, period is $\frac{2 \pi}{|b|}$, phase shift is $h$, and vertical shift is $d$; for tangent the period is $\frac{\pi}{|b|}$ and there is no amplitude.