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Exercises Trigonometry

Example 3 from Algebra and Trigonometry, 8.1 Graphs of the Sine and Cosine Functions

Identifying the Phase Shift of a Function

Determine the direction and magnitude of the phase shift for $f (x) = \sin (x + \frac{π}{6}) - 2.$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
Let’s begin by comparing the equation to the general form $y = A \sin (B x - C) + D .$
In the given equation, notice that $B = 1$ and $C = - \frac{π}{6} .$ So the phase shift is

$\begin{array}{r} \frac{C}{B} = - \frac{\frac{π}{6}}{1} \\ = - \frac{π}{6} \end{array}$
or $\frac{π}{6}$ units to the left.

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Determine the direction and magnitude of the phase shift for $f (x) = 3 \cos (x - \frac{π}{2}) .$

How did it go?

PreviousIdentifying the Amplitude of a Sine or Cosine Function Next Identifying the Vertical Shift of a Function

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 8.1, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.