Amplitude, period and phase shift worksheet

Every number in y = a sin(bx − c) + d moves the graph in one particular way. Read them off one at a time and you have the whole picture without plotting a single point.



Sigma Prep · sigmaprep.io/worksheets Name Amplitude, Period and Phase Shift Date Period Find each value from the equation. Find the amplitude: $y = - 7 \cos (5 x + \frac{5 π}{3}) + 1$ Find the period: $y = - 6 \sin (3 x + \frac{π}{3}) - 1$ Find the phase shift: $y = 6 \cos (4 x + \frac{2 π}{3}) + 3$ Find the amplitude: $y = - 8 \cos (2 x) - 3$ Find the period: $y = - 6 \cos (4 x) - 3$ Find the phase shift: $y = - 8 \sin (5 x + \frac{5 π}{2}) + 5$ Find the amplitude: $y = - 8 \cos (5 x) - 4$ Find the period: $y = 6 \sin (x) - 2$ Find the phase shift: $y = - 8 \cos (2 x + \frac{π}{2}) - 2$ Find the amplitude: $y = - 6 \cos (x - \frac{3 π}{2}) + 2$ Find the period: $y = - 8 \cos (5 x) - 3$ Find the phase shift: $y = 8 \sin (2 x + π) + 3$ Find the amplitude: $y = 6 \sin (2 x) + 4$ Find the period: $y = - 6 \sin (2 x + \frac{π}{6}) + 5$ Find the phase shift: $y = - 6 \sin (2 x + 2 π)$ Find the amplitude: $y = 8 \cos (2 x - \frac{3 π}{4}) + 2$ Find the period: $y = - 7 \sin (5 x - \frac{π}{2}) + 1$ Find the phase shift: $y = - 7 \cos (3 x - \frac{π}{2}) - 5$ Find the amplitude: $y = - 8 \cos (x) + 5$ Find the period: $y = - 7 \sin (2 x) - 4$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Amplitude, Period and Phase Shift

Date Period

Find each value from the equation.

Find the amplitude: $y = - 7 \cos (5 x + \frac{5 π}{3}) + 1$
Find the period: $y = - 6 \sin (3 x + \frac{π}{3}) - 1$
Find the phase shift: $y = 6 \cos (4 x + \frac{2 π}{3}) + 3$
Find the amplitude: $y = - 8 \cos (2 x) - 3$
Find the period: $y = - 6 \cos (4 x) - 3$
Find the phase shift: $y = - 8 \sin (5 x + \frac{5 π}{2}) + 5$
Find the amplitude: $y = - 8 \cos (5 x) - 4$
Find the period: $y = 6 \sin (x) - 2$
Find the phase shift: $y = - 8 \cos (2 x + \frac{π}{2}) - 2$
Find the amplitude: $y = - 6 \cos (x - \frac{3 π}{2}) + 2$
Find the period: $y = - 8 \cos (5 x) - 3$
Find the phase shift: $y = 8 \sin (2 x + π) + 3$
Find the amplitude: $y = 6 \sin (2 x) + 4$
Find the period: $y = - 6 \sin (2 x + \frac{π}{6}) + 5$
Find the phase shift: $y = - 6 \sin (2 x + 2 π)$
Find the amplitude: $y = 8 \cos (2 x - \frac{3 π}{4}) + 2$
Find the period: $y = - 7 \sin (5 x - \frac{π}{2}) + 1$
Find the phase shift: $y = - 7 \cos (3 x - \frac{π}{2}) - 5$
Find the amplitude: $y = - 8 \cos (x) + 5$
Find the period: $y = - 7 \sin (2 x) - 4$



Sigma Prep · sigmaprep.io/worksheets Name Amplitude, Period and Phase ShiftAnswer keyVersion 1 Date Period Find each value from the equation. Find the amplitude: $y = - 7 \cos (5 x + \frac{5 π}{3}) + 1$ $7$ Find the period: $y = - 6 \sin (3 x + \frac{π}{3}) - 1$ $\frac{2 π}{3}$ Find the phase shift: $y = 6 \cos (4 x + \frac{2 π}{3}) + 3$ $\frac{π}{6}$ left Find the amplitude: $y = - 8 \cos (2 x) - 3$ $8$ Find the period: $y = - 6 \cos (4 x) - 3$ $\frac{π}{2}$ Find the phase shift: $y = - 8 \sin (5 x + \frac{5 π}{2}) + 5$ $\frac{π}{2}$ left Find the amplitude: $y = - 8 \cos (5 x) - 4$ $8$ Find the period: $y = 6 \sin (x) - 2$ $2 π$ Find the phase shift: $y = - 8 \cos (2 x + \frac{π}{2}) - 2$ $\frac{π}{4}$ left Find the amplitude: $y = - 6 \cos (x - \frac{3 π}{2}) + 2$ $6$ Find the period: $y = - 8 \cos (5 x) - 3$ $\frac{2 π}{5}$ Find the phase shift: $y = 8 \sin (2 x + π) + 3$ $\frac{π}{2}$ left Find the amplitude: $y = 6 \sin (2 x) + 4$ $6$ Find the period: $y = - 6 \sin (2 x + \frac{π}{6}) + 5$ $π$ Find the phase shift: $y = - 6 \sin (2 x + 2 π)$ $π$ left Find the amplitude: $y = 8 \cos (2 x - \frac{3 π}{4}) + 2$ $8$ Find the period: $y = - 7 \sin (5 x - \frac{π}{2}) + 1$ $\frac{2 π}{5}$ Find the phase shift: $y = - 7 \cos (3 x - \frac{π}{2}) - 5$ $\frac{π}{6}$ right Find the amplitude: $y = - 8 \cos (x) + 5$ $8$ Find the period: $y = - 7 \sin (2 x) - 4$ $π$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Amplitude, Period and Phase ShiftAnswer keyVersion 1

Date Period

Find each value from the equation.

Find the amplitude: $y = - 7 \cos (5 x + \frac{5 π}{3}) + 1$ $7$
Find the period: $y = - 6 \sin (3 x + \frac{π}{3}) - 1$ $\frac{2 π}{3}$
Find the phase shift: $y = 6 \cos (4 x + \frac{2 π}{3}) + 3$ $\frac{π}{6}$ left
Find the amplitude: $y = - 8 \cos (2 x) - 3$ $8$
Find the period: $y = - 6 \cos (4 x) - 3$ $\frac{π}{2}$
Find the phase shift: $y = - 8 \sin (5 x + \frac{5 π}{2}) + 5$ $\frac{π}{2}$ left
Find the amplitude: $y = - 8 \cos (5 x) - 4$ $8$
Find the period: $y = 6 \sin (x) - 2$ $2 π$
Find the phase shift: $y = - 8 \cos (2 x + \frac{π}{2}) - 2$ $\frac{π}{4}$ left
Find the amplitude: $y = - 6 \cos (x - \frac{3 π}{2}) + 2$ $6$
Find the period: $y = - 8 \cos (5 x) - 3$ $\frac{2 π}{5}$
Find the phase shift: $y = 8 \sin (2 x + π) + 3$ $\frac{π}{2}$ left
Find the amplitude: $y = 6 \sin (2 x) + 4$ $6$
Find the period: $y = - 6 \sin (2 x + \frac{π}{6}) + 5$ $π$
Find the phase shift: $y = - 6 \sin (2 x + 2 π)$ $π$ left
Find the amplitude: $y = 8 \cos (2 x - \frac{3 π}{4}) + 2$ $8$
Find the period: $y = - 7 \sin (5 x - \frac{π}{2}) + 1$ $\frac{2 π}{5}$
Find the phase shift: $y = - 7 \cos (3 x - \frac{π}{2}) - 5$ $\frac{π}{6}$ right
Find the amplitude: $y = - 8 \cos (x) + 5$ $8$
Find the period: $y = - 7 \sin (2 x) - 4$ $π$

How to do these

$y = 3 \sin (2 x - \frac{π}{3}) + 1$

The amplitude is 3, the size of the number in front.
The period is 2 pi over 2, which is pi.
The shift is pi over 3 divided by 2.

The answer is Period $π$ , shift $\frac{π}{6}$ right.

Where students go wrong

Reading the phase shift straight off the parenthesis. In sin(2x − π/3) the shift is not π/3: you have to divide by the 2 as well, because the whole parenthesis is what has to be zero.

Also called amplitude and period, phase shift, sine and cosine graphs, transformations of sine or properties of trig graphs.

What you can put on this worksheet

Amplitude: Find the amplitude: $y = 4 \cos (x)$ → $4$
Period: Find the period: $y = 4 \cos (x)$ → $2 π$
Phase shift: Find the phase shift: $y = 4 \cos (x - \frac{π}{2})$ → $\frac{π}{2}$ right
Range: Find the range: $y = 4 \cos (x)$ → $[- 4, 4]$

Questions about these worksheets

Yes. Every amplitude, period and phase shift sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Precalculus is usually taken around 11th or 12th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the amplitude, period and phase shift questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 4: amplitude, period, phase shift and range. Tick as many as you want and set how many of each, or let it spread them evenly.

Reading the phase shift straight off the parenthesis. In sin(2x − π/3) the shift is not π/3: you have to divide by the 2 as well, because the whole parenthesis is what has to be zero.

Amplitude, period and phase shift worksheet

Amplitude, Period and Phase Shift

Amplitude, Period and Phase ShiftAnswer keyVersion 1

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

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