Precalculus Graphs of Trigonometric Functions Worksheet (PDF)

Precalculus math worksheet: Graphs of Trigonometric Functions – Maths PDF

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This precalculus trig graphs worksheet for Precalculus covers graphs of trigonometric functions (Common Core F-TF.B.5, F-IF.C.7e, F-TF.B.7), with a free PDF to print.

These 20 problems practice graphs of trigonometric functions (Common Core F-TF.B.5, F-IF.C.7e and F-TF.B.7). You will find amplitude, period, phase shift and midline, graph sine, cosine and tangent, and build periodic models.

Precalculus · Common Core State Standards: F-TF.B.5, F-IF.C.7e, F-TF.B.7

Precalculus trig graphs worksheet: exercises 1 to 10

Exercise 1 – Amplitude and period

Find the amplitude and period of \(y=3\sin2x\).

Exercise 2 – Reflected cosine

Find the amplitude and period of \(y=-\frac12\cos\frac{x}{3}\), and tell whether it is reflected.

Exercise 3 – Period with pi in b

Find the period of \(y=4\cos(\pi x)\).

Exercise 4 – Phase shifts

Find the phase shift of (a) \(y=\sin\left(x-\frac{\pi}{3}\right)\); (b) \(y=\cos(2x+\pi)\).

Exercise 5 – Midline and range

Find the midline and range of \(y=2\sin x-3\).

Exercise 6 – Write an equation

Write a sine function with amplitude \(2\), period \(\pi\), midline \(y=1\) and no phase shift.

Exercise 7 – Evaluate

Evaluate \(y=3\sin2x\) at \(x=\frac{\pi}{4}\) and \(x=\frac{\pi}{6}\).

Exercise 8 – Table of values

Complete the table.

Complete the table for \(y=2\cos x\).

Exercise 9 – Read a graph

Graph of a sine curve.

The graph of \(y=3\sin2x\) is shown. State the amplitude, the period and the coordinates of the first maximum.

Exercise 10 – Tangent with a stretch

Find the period and the asymptotes of \(y=\tan\frac{x}{2}\) for \(-2\pi< x< 2\pi\).

Exercises 11 and up

Exercise 11 – Tangent with compression

Find the period of \(y=\tan2x\) and the asymptote with the smallest positive \(x\).

Exercise 12 – Equation from a graph (cosine)

A cosine-shaped graph.

The graph has maximum \(5\) at \(x=0\) and minimum \(-1\) at \(x=\pi\). Write a cosine equation.

Exercise 13 – Equation from a graph (shifted sine)

A shifted sine curve.

The sinusoid has midline \(y=1\), amplitude \(2\), period \(2\pi\), and crosses the midline going up at \(x=\frac{\pi}{4}\). Write a sine equation.

Exercise 14 – Range

Find the range of \(y=-4\cos x+1\).

Exercise 15 – Phase shift identity

Show that \(\sin\left(x+\frac{\pi}{2}\right)\) and \(\cos x\) give the same value at \(x=0\), \(\frac{\pi}{3}\) and \(\pi\).

Exercise 16 – Alternating current

A circuit voltage is \(V(t)=170\sin(120\pi t)\) volts with \(t\) in seconds. Find the amplitude, the period and the number of cycles per second.

Exercise 17 – Tide

Depth of water at Priya’s pier.

The water depth at a pier is \(d(t)=5+2\cos\left(\frac{\pi}{6}t\right)\) feet, with \(t\) in hours after high tide. Find the high and low depths, the period and the depth at \(t=2\).

Exercise 18 – Daylight hours

The hours of daylight at a city are \(D(t)=12+2.5\sin\left(\frac{2\pi}{365}(t-80)\right)\), with \(t\) the day of the year. Find the amplitude, the longest day length, and \(D(80)\).

Exercise 19 – Ferris wheel

Height of a rider on a Ferris wheel.

A rider’s height is \(h(t)=30-25\cos\left(\frac{\pi}{15}t\right)\) feet. Find the lowest and highest heights, the time for one rotation, and the height at \(t=10\) s.

Exercise 20 – A model from data

A sinusoid has maximum \(7\) at \(x=0\), minimum \(-1\) at \(x=4\pi\). Write a cosine model.

Key facts behind this precalculus trig graphs worksheet

Key facts

Amplitude = |a| = (max – min)/2; midline y = k = (max + min)/2.
Period of sine and cosine = 2 pi / |b|; period of tangent = pi / |b|.
Phase shift h moves the graph right when h is positive.
Tangent has vertical asymptotes at x = pi/2 + k pi.
Use cosine for a model that starts at a maximum and sine for one that starts at the midline.

How to use this precalculus trig graphs worksheet

Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.

Download the precalculus trig graphs worksheet as a PDF with the button below and keep it with your notes.

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