Sinusoidal graphs: amplitude, period, midline

Warm-up

Three quick questions from earlier lessons.

Discover

Move the sliders and watch the graph of $y = a\sin(bx) + k$ .

The graph of y = a sin(bx) + k: a wave around the line y = k, a units high, that repeats faster as b grows.

What happens to the waves when $b$ goes from 1 to 2?

They get twice as narrow
They get twice as high
They get twice as wide

Key idea

$y = A\sin(Bx) + D$ and $y = A\cos(Bx) + D$ are waves.

Amplitude

|A|

: the height above the midline. Midline

y = D

. Period

\dfrac{2\pi}{B}

(for

B > 0

): the length of one cycle. Maximum

D + |A|

, minimum

D - |A|

.

Worked example

Describe $y = 3\sin(2x) + 1$ .

Amplitude:

3

The number in front of sin.

Midline:

y = 1

The number added.

Period:

\dfrac{2\pi}{2} = \pi

2π divided by B = 2.

Range: from

-2

to

4

Midline ± amplitude.

Your turn, guided

Your turn: $y = 5\cos(4x) - 2$ .

Amplitude:

5

, midline

y = -2

In front of cos.

Period:

\dfrac{\pi}{2}

2π ÷ 4.

Practice ★

What is the amplitude of $y = 2\sin(5x) + 1$ ?
What is the amplitude of $y = 4\sin(5x) + 3$ ?
What is the amplitude of $y = 4\sin(2x) + 7$ ?
What is the amplitude of $y = 6\sin(3x) + 8$ ?

Watch out!

Practice ★★

What is the period of $y = \sin(6x)$ ?
What is the period of $y = \sin(5x)$ ?
What is the period of $y = \sin(3x)$ ?

What is the midline of $y = 9\cos x - 4$ ?
1. $y = -4$
2. $y = 4$
3. $y = 9$
What is the midline of $y = 8\cos x - 9$ ?
1. $y = -9$
2. $y = 9$
3. $y = 8$

Exit ticket

Three questions to check you have got it.

What is the period of $y = \sin(6x)$ ?
What is the period of $y = \sin(5x)$ ?
What is the period of $y = \sin(3x)$ ?

Summary card

Sinusoidal graphs $y = A\sin(Bx) + D$

Amplitude $|A|$ , midline $y = D$ .
Period $\dfrac{2\pi}{B}$ .
Maximum $D + |A|$ , minimum $D - |A|$ .

Watch out: a bigger $B$ means a shorter period.

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