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Exercises Trigonometry

Example 5 from Algebra and Trigonometry, 8.1 Graphs of the Sine and Cosine Functions

Identifying the Variations of a Sinusoidal Function from an Equation

Determine the midline, amplitude, period, and phase shift of the function $y = 3 \sin (2 x) + 1.$

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 $y = A \sin (B x - C) + D .$
$A = 3,$ so the amplitude is $| A | = 3.$
Next, $B = 2,$ so the period is $P = \frac{2 π}{| B |} = \frac{2 π}{2} = π .$
There is no added constant inside the parentheses, so $C = 0$ and the phase shift is $\frac{C}{B} = \frac{0}{2} = 0.$

Finally, $D = 1,$ so the midline is $y = 1.$

Your turn

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

Determine the midline, amplitude, period, and phase shift of the function $y = \frac{1}{2} \cos (\frac{x}{3} - \frac{π}{3}) .$

How did it go?

PreviousIdentifying the Vertical Shift of a Function Next Identifying the Equation for a Sinusoidal Function from a Graph

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.