Phase shift and sinusoidal models

Warm-up

Three quick questions from earlier lessons.

Discover

$y = \sin\left(x - \dfrac{\pi}{2}\right)$ uses the same wave as $y = \sin x$ . Where does it cross its midline going up for the first time with $x \ge 0$ ?

At $x = \dfrac{\pi}{2}$
At $x = -\dfrac{\pi}{2}$

Key idea

A sinusoid can also be shifted sideways.

y = A\sin\big(B(x - h)\big) + D

is shifted

h

to the right (the phase shift). In the form

\sin(Bx - C)

, factor out

B

first: the shift is

\dfrac{C}{B}

.

To model a situation, read $A$ , $D$ and the period from the data, then $B = \dfrac{2\pi}{\text{period}}$ .

Worked example

Find the phase shift of $y = \cos(3x - \pi)$ .

y = \cos\left(3\left(x - \dfrac{\pi}{3}\right)\right)

Factor out B = 3.

Shift:

\dfrac{\pi}{3}

to the right

The form B(x − h).

Your turn, guided

Your turn: $y = \sin(2x - 6)$ .

y = \sin\big(2(x - 3)\big)

Factor out 2.

Shift:

3

to the right

h in B(x − h).

Practice ★

What is the largest value of $y = 5\sin x + 4$ ?
What is the largest value of $y = 2\sin x + 9$ ?
What is the largest value of $y = 6\sin x + 2$ ?

Watch out!

Practice ★★

The graph of $y = \cos(2x - 4)$ is the graph of $y = \cos(2x)$ shifted to the right. By how much?
The graph of $y = \cos(3x - 15)$ is the graph of $y = \cos(3x)$ shifted to the right. By how much?
The graph of $y = \cos(4x - 16)$ is the graph of $y = \cos(4x)$ shifted to the right. By how much?

Write $y = A\sin(Bx) + D$ , with $A, B > 0$ , for a sine curve with amplitude 2, period $\frac{\pi}{2}$ and midline $y = 9$ .
Write $y = A\sin(Bx) + D$ , with $A, B > 0$ , for a sine curve with amplitude 2, period $\frac{2 \pi}{3}$ and midline $y = 6$ .
Write $y = A\sin(Bx) + D$ , with $A, B > 0$ , for a sine curve with amplitude 4, period $\frac{2 \pi}{3}$ and midline $y = 1$ .

Exit ticket

Three questions to check you have got it.

Write $y = A\sin(Bx) + D$ , with $A, B > 0$ , for a sine curve with amplitude 2, period $\frac{\pi}{2}$ and midline $y = 9$ .
Write $y = A\sin(Bx) + D$ , with $A, B > 0$ , for a sine curve with amplitude 2, period $\frac{2 \pi}{3}$ and midline $y = 6$ .
Write $y = A\sin(Bx) + D$ , with $A, B > 0$ , for a sine curve with amplitude 4, period $\frac{2 \pi}{3}$ and midline $y = 1$ .

Summary card

Phase shift and models

$A\sin\big(B(x - h)\big) + D$ : shift $h$ right.
$\sin(Bx - C)$ : shift $\dfrac{C}{B}$ , after factoring out $B$ .
Model: $A$ = half the range, $D$ = middle value, $B = \dfrac{2\pi}{\text{period}}$ .

Watch out: factor out $B$ before reading the shift.

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