Trigonometric graphs and modelling

Amplitude, period, phase shift and mean value of y = a sin n(x − h) + k, sketching transformed trigonometric graphs, and using trigonometric functions to model periodic situations such as tides.

The general wave

Transforming y = sin x

The dashed line is the centre line y = k.

Try this

Which slider changes the period? Which one moves the centre line?

Example 1●●

For $y = 3\sin 2x + 1$, state the amplitude, period, mean value and range.

Amplitude $3$, period $\dfrac{2\pi}{2} = \pi$, mean value $1$.

Range: $1 - 3 \le y \le 1 + 3$, that is $-2 \le y \le 4$.

Modelling

Example 2: tides●●

The depth of water at a jetty is $h = 1.5 + 1.2\cos\dfrac{\pi t}{6}$ metres, $t$ hours after midnight. Find the greatest and least depths, the period, and the first time the depth is $2.1$ m.

Greatest $1.5 + 1.2 = 2.7$ m, least $1.5 - 1.2 = 0.3$ m.

Extremes

$\dfrac{2\pi}{\pi / 6} = 12$ hours.

Period

$1.2\cos\dfrac{\pi t}{6} = 0.6$ gives $\cos\dfrac{\pi t}{6} = \tfrac12$, so $\dfrac{\pi t}{6} = \dfrac{\pi}{3}$ and $t = 2$. That is $2$ am.

Solve

Flashcards

Period of $sin nx$ and $cos nx$

$\dfrac{2\pi}{n}$

Period of $ an nx$

$\dfrac{\pi}{n}$

Range of $asin nx + k$

$k - |a| \le y \le k + |a|$

Practice

Find the amplitude of $y = 3\sin 2x + 1$.
The period of $y = 3\sin 2x + 1$ is:
Find the mean value of $y = 3\sin 2x + 1$.
The range of $y = 3\sin 2x + 1$ is:
$y = \sin\left(x - \tfrac{\pi}{3}\right)$ is $y = \sin x$:
The phase shift of $y = \sin(2x - \pi)$ is:
For $h = 1.5 + 1.2\cos\dfrac{\pi t}{6}$, find the greatest depth.
Find the period of that tide, in hours.
The period of $y = \tan 3x$ is: