Translations of Trigonometric Graphs

Q: What causes a vertical shift?

A constant added outside the function, $+d$, which moves the whole wave (and its midline) up or down.

Adding constants shifts a trig wave without changing its shape. A vertical shift raises or lowers the midline; a phase shift slides it left or right.

In $y = a\sin(x) + d$ , the $+d$ moves the midline to $y = d$ . Reading these shifts is the focus here.

Vertical shift and midline

The midline is the horizontal line the wave oscillates around. A plain sine wave has midline $y = 0$ ; adding $d$ moves it to $y = d$ .

For $y = 2\sin\theta + 5$ , the $+5$ raises the midline to $y = 5$ . The wave still rises and falls by the amplitude around that line.

Phase shift

A horizontal shift, or phase shift, comes from a constant added inside the function, like $\sin(x - c)$ . It slides the wave left or right by $c$ .

Vertical and phase shifts are independent — one moves the midline, the other moves the wave sideways.

Worked examples

Example 1: reading the midline

What is the midline of $y = 2\sin\theta + 5$ ?

The added constant sets the midline

+5

Midline is y = d

y = 5

Answer: $y = 5$

Example 2: a lowered midline

What is the midline of $y = \sin\theta - 3$ ?

The constant is -3

d = -3

Answer: $y = -3$

Example 3: reading a phase shift

What is the phase shift of $y = \sin(\theta - 45^\circ)$ ?

The constant sits inside the function

\theta - 45^\circ

Subtracting inside slides the wave right

45^\circ \text{ right}

Answer: $45^\circ$ to the right

Try one yourself

Common questions

What is the midline?

The horizontal line a trig wave oscillates around. For $y = a\sin(x) + d$ , it is $y = d$ .

What causes a vertical shift?

A constant added outside the function, $+d$ , which moves the whole wave (and its midline) up or down.

What is a phase shift?

A horizontal slide caused by a constant inside the function, like $\sin(x - c)$ moving the wave right by $c$ .

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Translations of Trigonometric Graphs

Vertical shift and midline

Phase shift

Worked examples

Example 1: reading the midline

Example 2: a lowered midline

Example 3: reading a phase shift

Try one yourself

Common questions

Keep learning

Graphing Sine & Cosine Functions

Angles & Angle Measure

Trigonometric Identities