5.2: Phase Shift | Precalculus | Code Scholars Tutoring

How it works

Factor the coefficient of x before reading a horizontal shift. A positive h in the form x minus h shifts the graph right. The order matters: reading the constant before factoring usually gives the wrong shift.

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

Worked examples

Example 1

Find the shift in $\displaystyle y=\sin(2x-\pi)$ .

Factor as $\displaystyle \sin(2(x-\pi/2))$ . The shift is right $\displaystyle \pi/2$ .

Example 2

Find the shift in $\displaystyle y=3\cos(x+\pi/4)$ .

Write $\displaystyle x-(-\pi/4)$ : the shift is left $\displaystyle \pi/4$ .

Problems

Problem 5.2.1

For $\displaystyle y=2\sin(x-\pi/2)+1$ , find the shift and the value at $\displaystyle x=\pi/2$ .

Sine curve shifted right pi/2, with amplitude 2 and midline 1. — The graph crosses its midline upward at x = pi/2.

Show worked solution

The shift is right $\displaystyle \pi/2$ . The value is $\displaystyle 2\sin0+1=1$ .

Sine curve shifted right pi/2, with amplitude 2 and midline 1. The solution adds labeled relationships. — The highlighted midline crossing is (pi/2, 1).

Problem 5.2.2

Factor the argument and find the shift of $\displaystyle y=\cos(3x+\pi)$ .

Show worked solution

It is $\displaystyle \cos(3(x+\pi/3))$ , shifted left $\displaystyle \pi/3$ .

Problem 5.2.3

Write a sine function of amplitude $\displaystyle 1$ , period $\displaystyle 2\pi$ , shifted right $\displaystyle \pi/3$ .

Show worked solution

One answer is $\displaystyle y=\sin(x-\pi/3)$ .

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 5.2.1

Problem 5.2.2

Problem 5.2.3

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