Find the period and the phase shift: y = 4cot (x + 3π/4)

Period π, phase shift 3π/4 left

Find the period and the phase shift: y = -3tan (x - π/4)

Period π, phase shift π/4 right

Find the period and the phase shift: y = -3cot (3x - 3π/2)

Period π/3, phase shift π/2 right

Write a sine or cosine equation for the graph.

Sample answer: y = 3sin (x) - 1

Worksheets · Precalculus

Amplitude, period and phase shift worksheet for Precalculus

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.

Also taught in Algebra 2, at its own level and with its own examples.

The preview can look cramped on a phone. The PDF and printed sheet come out normal.



Sigma Prep · sigmaprep.io/worksheets Name Amplitude, Period and Phase Shift Date Period Find each value from the equation, or write the equation. Find the amplitude Find the amplitude: $y = - 3 \cos (4 x)$ Find the amplitude: $y = - 4 \sin (4 x)$ Find the amplitude: $y = 2 \sin (4 x)$ Find the amplitude: $y = \sin (3 x)$ Find the period Find the period: $y = 7 \sin ⁣ (x + \frac{π}{6}) - 3$ Find the period: $y = - 6 \cos (2 x)$ Find the period: $y = - 6 \sin ⁣ (5 x - \frac{π}{3}) - 4$ Find the period: $y = - 6 \cos (3 x) + 2$ Find the phase shift Find the phase shift: $y = - 9 \sin ⁣ (5 x - \frac{2 π}{3}) - 6$ Find the phase shift: $y = 10 \sin ⁣ (2 x + \frac{3 π}{4}) - 7$ Find the phase shift: $y = - 10 \sin ⁣ (4 x + \frac{π}{2}) + 4$ Write the function from its features Write a cosine function with amplitude $5$ and period $π$ . Write a sine function with amplitude $4$ and period $2 π$ . Write a sine function with amplitude $5$ and period $π$ . Find the midline, maximum and minimum Find the midline, the maximum and the minimum: $y = - 2 \sin (x) + 5$ Find the midline, the maximum and the minimum: $y = - ⁣ \sin (3 x) + 3$ Find the midline, the maximum and the minimum: $y = \cos (x) + 2$ Find the period or phase shift of $\tan$ , $\cot$ , $\sec$ , $\csc$ Find the period and the phase shift: $y = 4 \cot ⁣ (x + \frac{3 π}{4})$ Find the period and the phase shift: $y = - 3 \tan ⁣ (x - \frac{π}{4})$ Find the period and the phase shift: $y = - 3 \cot ⁣ (3 x - \frac{3 π}{2})$ 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, or write the equation.

Find the amplitude
Find the amplitude: $y = - 3 \cos (4 x)$
Find the amplitude: $y = - 4 \sin (4 x)$
Find the amplitude: $y = 2 \sin (4 x)$
Find the amplitude: $y = \sin (3 x)$

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

Find the phase shift
Find the phase shift:
$y = - 9 \sin ⁣ (5 x - \frac{2 π}{3}) - 6$
Find the phase shift:
$y = 10 \sin ⁣ (2 x + \frac{3 π}{4}) - 7$
Find the phase shift:
$y = - 10 \sin ⁣ (4 x + \frac{π}{2}) + 4$

Write the function from its features
Write a cosine function with amplitude $5$ and period $π$ .
Write a sine function with amplitude $4$ and period $2 π$ .
Write a sine function with amplitude $5$ and period $π$ .

Find the midline, maximum and minimum
Find the midline, the maximum and the minimum: $y = - 2 \sin (x) + 5$
Find the midline, the maximum and the minimum: $y = - ⁣ \sin (3 x) + 3$
Find the midline, the maximum and the minimum: $y = \cos (x) + 2$

Find the period or phase shift of $\tan$ , $\cot$ , $\sec$ , $\csc$
Find the period and the phase shift:
$y = 4 \cot ⁣ (x + \frac{3 π}{4})$
Find the period and the phase shift:
$y = - 3 \tan ⁣ (x - \frac{π}{4})$
Find the period and the phase shift:
$y = - 3 \cot ⁣ (3 x - \frac{3 π}{2})$



Sigma Prep · sigmaprep.io/worksheets Name Amplitude, Period and Phase ShiftAnswer keyVersion 1 Date Period Find each value from the equation, or write the equation. Find the amplitude Find the amplitude: $y = - 3 \cos (4 x)$ $3$ Find the amplitude: $y = - 4 \sin (4 x)$ $4$ Find the amplitude: $y = 2 \sin (4 x)$ $2$ Find the amplitude: $y = \sin (3 x)$ $1$ Find the period Find the period: $y = 7 \sin ⁣ (x + \frac{π}{6}) - 3$ $2 π$ Find the period: $y = - 6 \cos (2 x)$ $π$ Find the period: $y = - 6 \sin ⁣ (5 x - \frac{π}{3}) - 4$ $\frac{2 π}{5}$ Find the period: $y = - 6 \cos (3 x) + 2$ $\frac{2 π}{3}$ Find the phase shift Find the phase shift: $y = - 9 \sin ⁣ (5 x - \frac{2 π}{3}) - 6$ $\frac{2 π}{15}$ right Find the phase shift: $y = 10 \sin ⁣ (2 x + \frac{3 π}{4}) - 7$ $\frac{3 π}{8}$ left Find the phase shift: $y = - 10 \sin ⁣ (4 x + \frac{π}{2}) + 4$ $\frac{π}{8}$ left Write the function from its features Write a cosine function with amplitude $5$ and period $π$ .Sample answer: $y = 5 \cos (2 x)$ Write a sine function with amplitude $4$ and period $2 π$ .Sample answer: $y = 4 \sin (x)$ Write a sine function with amplitude $5$ and period $π$ .Sample answer: $y = 5 \sin (2 x)$ Find the midline, maximum and minimum Find the midline, the maximum and the minimum: $y = - 2 \sin (x) + 5$ Midline $y = 5$ , max $7$ , min $3$ Find the midline, the maximum and the minimum: $y = - ⁣ \sin (3 x) + 3$ Midline $y = 3$ , max $4$ , min $2$ Find the midline, the maximum and the minimum: $y = \cos (x) + 2$ Midline $y = 2$ , max $3$ , min $1$ Find the period or phase shift of $\tan$ , $\cot$ , $\sec$ , $\csc$ Find the period and the phase shift: $y = 4 \cot ⁣ (x + \frac{3 π}{4})$ Period $π$ , phase shift $\frac{3 π}{4}$ left Find the period and the phase shift: $y = - 3 \tan ⁣ (x - \frac{π}{4})$ Period $π$ , phase shift $\frac{π}{4}$ right Find the period and the phase shift: $y = - 3 \cot ⁣ (3 x - \frac{3 π}{2})$ Period $\frac{π}{3}$ , phase shift $\frac{π}{2}$ right 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, or write the equation.

Find the amplitude
Find the amplitude: $y = - 3 \cos (4 x)$ $3$
Find the amplitude: $y = - 4 \sin (4 x)$ $4$
Find the amplitude: $y = 2 \sin (4 x)$ $2$
Find the amplitude: $y = \sin (3 x)$ $1$

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

Find the phase shift
Find the phase shift:
$y = - 9 \sin ⁣ (5 x - \frac{2 π}{3}) - 6$ $\frac{2 π}{15}$ right
Find the phase shift:
$y = 10 \sin ⁣ (2 x + \frac{3 π}{4}) - 7$ $\frac{3 π}{8}$ left
Find the phase shift:
$y = - 10 \sin ⁣ (4 x + \frac{π}{2}) + 4$ $\frac{π}{8}$ left

Write the function from its features
Write a cosine function with amplitude $5$ and period $π$ .Sample answer: $y = 5 \cos (2 x)$
Write a sine function with amplitude $4$ and period $2 π$ .Sample answer: $y = 4 \sin (x)$
Write a sine function with amplitude $5$ and period $π$ .Sample answer: $y = 5 \sin (2 x)$

Find the midline, maximum and minimum
Find the midline, the maximum and the minimum: $y = - 2 \sin (x) + 5$ Midline $y = 5$ , max $7$ , min $3$
Find the midline, the maximum and the minimum: $y = - ⁣ \sin (3 x) + 3$ Midline $y = 3$ , max $4$ , min $2$
Find the midline, the maximum and the minimum: $y = \cos (x) + 2$ Midline $y = 2$ , max $3$ , min $1$

Find the period or phase shift of $\tan$ , $\cot$ , $\sec$ , $\csc$
Find the period and the phase shift:
$y = 4 \cot ⁣ (x + \frac{3 π}{4})$ Period $π$ , phase shift $\frac{3 π}{4}$ left
Find the period and the phase shift:
$y = - 3 \tan ⁣ (x - \frac{π}{4})$ Period $π$ , phase shift $\frac{π}{4}$ right
Find the period and the phase shift:
$y = - 3 \cot ⁣ (3 x - \frac{3 π}{2})$ Period $\frac{π}{3}$ , phase shift $\frac{π}{2}$ right

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 $π$ , phase 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: $y = 4 \cos (x)$ → $[- 4, 4]$
Write the function from its features: Write a sine function with amplitude $4$ and period $\frac{2 π}{3}$ . → Sample answer: $y = 4 \sin (3 x)$
Write the equation from the graph: Write a sine or cosine equation for the graph. → Sample answer: $y = 3 \sin (x) - 1$
The midline, maximum and minimum: Find the midline, the maximum and the minimum: $y = 4 \cos (3 x) - 4$ → Midline $y = - 4$ , max $0$ , min $- 8$
Tangent, cotangent, secant and cosecant: Find the period: $y = \sec (3 x)$ → Period $\frac{2 π}{3}$

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 8: amplitude, period, phase shift, range, write the function from its features, write the equation from the graph, the midline, maximum and minimum and tangent, cotangent, secant and cosecant. 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 for Precalculus

Find the amplitude

Find the period

Find the phase shift

Write the function from its features

Find the midline, maximum and minimum

Find the period or phase shift of tan⁡\tantan, cot⁡\cotcot, sec⁡\secsec, csc⁡\csccsc

Find the amplitude

Find the period

Find the phase shift

Write the function from its features

Find the midline, maximum and minimum

Find the period or phase shift of tan⁡\tantan, cot⁡\cotcot, sec⁡\secsec, csc⁡\csccsc

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these amplitude, period and phase shift worksheets free?+

Do the amplitude, period and phase shift worksheets come with an answer key?+

Are these amplitude, period and phase shift worksheets printable?+

What grade are these amplitude, period and phase shift worksheets for?+

Can I get a different version of the same amplitude, period and phase shift worksheet?+

Is this a free alternative to Kuta Software for amplitude, period and phase shift?+

Can I choose how hard the amplitude, period and phase shift questions are?+

What kinds of question are on a amplitude, period and phase shift worksheet?+

What do students get wrong with amplitude, period and phase shift?+

Find the period or phase shift of $\tan$ , $\cot$ , $\sec$ , $\csc$

Find the period or phase shift of $\tan$ , $\cot$ , $\sec$ , $\csc$