Periods and graphs

Fundamental period of trigonometric functions and drawing their graphs. 12 problems with step-by-step solutions.

Solve each problem in your notebook first, then open the solution. In all answers \(k, m \in \mathbb{Z}\).

Problem 1 Periodic function

\(f: \mathbb{Z} \to \mathbb{Z}\), \(f(x)\) = the remainder when \(x\) is divided by 3. Describe the graph of \(f\) and find its fundamental period.

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The values repeat: \(f(0) = 0,\ f(1) = 1,\ f(2) = 2,\ f(3) = 0,\ f(4) = 1, \dots\) and for negative numbers \(f(-1) = 2,\ f(-2) = 1,\ f(-3) = 0\).

The graph is a set of points that repeats every 3 units: heights 0, 1, 2, 0, 1, 2, … Since \(f(x + 3) = f(x)\) for all \(x\) and no smaller positive number works, the fundamental period is \(T = 3\).

Problem 2 Periodic function

A function \(f\) has fundamental period 6, and \(f(1) = 2\), \(f(2) = -1\), \(f(4) = 0\). Find \(f(25)\), \(f(38)\) and \(f(-8)\).

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Reduce the argument by multiples of 6.

\(25 = 4\cdot 6 + 1\), so \(f(25) = f(1) = 2\). \(38 = 6\cdot 6 + 2\), so \(f(38) = f(2) = -1\). \(-8 = -2\cdot 6 + 4\), so \(f(-8) = f(4) = 0\).

Problem 3 Fundamental period

Find the fundamental period.

a) \(y = 3\cos 4x\) b) \(y = \sin(2x + 5)\) c) \(y = \cos\frac{x}{3}\) d) \(y = \tan 4x\) e) \(y = \cot\frac{x}{2}\) f) \(y = \sin\frac{\pi x}{4}\) g) \(y = 2\tan\frac{\pi x}{3}\)

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For \(\sin(ax + b)\), \(\cos(ax + b)\): \(T = \frac{2\pi}{|a|}\). For \(\tan(ax + b)\), \(\cot(ax + b)\): \(T = \frac{\pi}{|a|}\). Constants in front or added do not change the period.

a) \(\frac{2\pi}{4} = \frac{\pi}{2}\) b) \(\pi\) c) \(\frac{2\pi}{1/3} = 6\pi\) d) \(\frac{\pi}{4}\) e) \(2\pi\)
f) \(\frac{2\pi}{\pi/4} = 8\) g) \(\frac{\pi}{\pi/3} = 3\)

Problem 4 Period of powers

Find the fundamental period.

a) \(y = \sin^5 6x\) b) \(y = 3\cos^4(2x + 1)\) c) \(y = \sin^2(5x - 2) + 1\) d) \(y = 2\sec^3(3 - 4x)\)

e) \(y = \csc^6(3x + 2) - 1\) f) \(y = \cot^5(4x - 1)\) g) \(y = \tan^2\frac{x}{3}\) h) \(y = 4 - \cos^7(\pi x)\)

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For \(\sin^n\), \(\cos^n\), \(\sec^n\), \(\csc^n\) of \((ax + b)\): \(T = \frac{\pi}{|a|}\) if \(n\) is even, \(T = \frac{2\pi}{|a|}\) if \(n\) is odd. For \(\tan^n\) and \(\cot^n\): always \(T = \frac{\pi}{|a|}\).

a) odd: \(\frac{2\pi}{6} = \frac{\pi}{3}\) b) even: \(\frac{\pi}{2}\) c) even: \(\frac{\pi}{5}\) d) odd: \(\frac{2\pi}{4} = \frac{\pi}{2}\)
e) even: \(\frac{\pi}{3}\) f) \(\frac{\pi}{4}\) g) \(\frac{\pi}{1/3} = 3\pi\) h) odd: \(\frac{2\pi}{\pi} = 2\)

Problem 5 Period of sums

Find the fundamental period.

a) \(y = \sin 2x + \cos 3x\) b) \(y = \sin^3 x + \cos^2 3x\) c) \(y = 2\cos^5 4x - 3\cot^2 6x\)

d) \(y = \cos^2 5x + \tan 3x - \sin^3(2x + 1)\) e) \(y = \sin^4 3x + \cos^3 4x\)

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Find the period of each term and take the least common multiple, using \(\mathrm{LCM}\left(\frac{a}{b}, \frac{c}{d}\right) = \frac{\mathrm{LCM}(a, c)}{\mathrm{GCD}(b, d)}\).

a) \(\pi\) and \(\frac{2\pi}{3}\): \(T = 2\pi\).
b) \(2\pi\) and \(\frac{\pi}{3}\): \(T = 2\pi\).
c) \(\frac{2\pi}{4} = \frac{\pi}{2}\) and \(\frac{\pi}{6}\): \(T = \frac{\pi}{2}\).
d) \(\frac{\pi}{5}\), \(\frac{\pi}{3}\) and \(\pi\): \(T = \pi\).
e) \(\frac{\pi}{3}\) and \(\frac{2\pi}{4} = \frac{\pi}{2}\): \(T = \frac{\mathrm{LCM}(1, 1)}{\mathrm{GCD}(3, 2)}\pi = \pi\).

Problem 6 Period of products

Find the fundamental period.

a) \(y = \sin 3x\cos 3x\) b) \(y = \cos 4x\cos 2x\) c) \(y = \sin 5x\sin x\) d) \(y = 4\sin x\cos x\)

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Write the product as a sum first.

a) \(\frac12\sin 6x\): \(T = \frac{\pi}{3}\).
b) \(\frac12(\cos 6x + \cos 2x)\): periods \(\frac{\pi}{3}\) and \(\pi\), so \(T = \pi\).
c) \(\frac12(\cos 4x - \cos 6x)\): periods \(\frac{\pi}{2}\) and \(\frac{\pi}{3}\), so \(T = \pi\).
d) \(2\sin 2x\): \(T = \pi\).

Problem 7 Graphs: shifting

Draw the graph of \(y = \sin\left(x + \frac{\pi}{4}\right)\).

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This is the graph of \(y = \sin x\) shifted \(\frac{\pi}{4}\) units to the left. Period \(2\pi\), range \([-1, 1]\).

Key points: zeros at \(x = -\frac{\pi}{4}, \frac{3\pi}{4}, \frac{7\pi}{4}\); maximum 1 at \(x = \frac{\pi}{4}\); minimum −1 at \(x = \frac{5\pi}{4}\).

Problem 8 Graphs: amplitude and period

Draw the graph of \(y = 3\sin 2x - 1\) in the interval \([0, \pi]\).

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Period \(\frac{2\pi}{2} = \pi\), so one full wave fits in \([0, \pi]\). Amplitude 3, shifted 1 unit down: range \([-4, 2]\).

\(x\)	0	\(\frac{\pi}{4}\)	\(\frac{\pi}{2}\)	\(\frac{3\pi}{4}\)	\(\pi\)
\(y\)	−1	2	−1	−4	−1

Problem 9 Graphs: a slower wave

Draw the graph of \(y = 2\cos\frac{x}{2} + 1\) in the interval \([-4\pi, 4\pi]\).

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Period \(\frac{2\pi}{1/2} = 4\pi\), so the interval contains two periods. Amplitude 2, shifted 1 unit up: range \([-1, 3]\).

Maximum 3 at \(x = 0, \pm 4\pi\); minimum −1 at \(x = \pm 2\pi\); value 1 at \(x = \pm\pi, \pm 3\pi\).

Problem 10 Graphs: tangent

Draw the graph of \(y = \tan\frac{x}{2}\) in \([-2\pi, 2\pi]\).

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Period \(\frac{\pi}{1/2} = 2\pi\). The function is undefined where \(\frac{x}{2} = \frac{\pi}{2} + k\pi\), i.e. at \(x = \pm\pi\) (vertical asymptotes). It is increasing on each interval.

Points: \(y(0) = 0\), \(y\left(\frac{\pi}{2}\right) = 1\), \(y\left(-\frac{\pi}{2}\right) = -1\).

Problem 11 Graphs: reflection and cotangent

a) Draw \(f: [0, 2\pi] \to \mathbb{R}\), \(f(x) = -\sin x\).

b) Draw \(f: (0, \pi) \to \mathbb{R}\), \(f(x) = \cot 2x\), \(x \ne \frac{\pi}{2}\).

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a) Reflect \(y = \sin x\) in the \(x\)-axis: minimum −1 at \(\frac{\pi}{2}\), maximum 1 at \(\frac{3\pi}{2}\).

b) Period \(\frac{\pi}{2}\); asymptotes where \(2x = k\pi\): \(x = 0, \frac{\pi}{2}, \pi\). Decreasing on each part; \(y = 0\) at \(\frac{\pi}{4}\) and \(\frac{3\pi}{4}\).

Problem 12 Equation from a graph

Each graph is of the form \(y = a + b\cos x\) or \(y = a + b\sin x\) on \([0, 2\pi]\). Find the equations.

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a) The maximum 4 is at \(x = 0\) and \(x = 2\pi\), the minimum 0 at \(x = \pi\): it is a cosine wave. The middle line is \(y = \frac{4 + 0}{2} = 2\) and the amplitude is \(2\). So \(y = 2 + 2\cos x\).

b) At \(x = 0\) the value is the middle value 1 and the graph goes up, with the maximum 4 at \(\frac{\pi}{2}\): it is a sine wave. Middle line \(y = 1\), amplitude \(3\). So \(y = 1 + 3\sin x\).