State the amplitude, period, maximum, minimum and axis of $y = 3 \sin x + 1$.

Amplitude $3$, period $2\pi$, axis $y = 1$, maximum $4$, minimum $-2$

Find the period of $y = \cos 2x$ and the five key points of its first cycle from $x = 0$.

Period $\pi$; key points $(0, 1)$, $\left(\frac{\pi}{4}, 0\right)$, $\left(\frac{\pi}{2}, -1\right)$, $\left(\frac{3\pi}{4}, 0\right)$, $(\pi, 1)$

Describe the transformations in $y = -2 \sin\left(x - \frac{\pi}{3}\right) + 1$ and give the key points of one cycle.

Amplitude $2$ with a reflection, period $2\pi$, phase shift $\frac{\pi}{3}$ right, axis $y = 1$; the cycle runs from $\left(\frac{\pi}{3}, 1\right)$ down to $\left(\frac{5\pi}{6}, -1\right)$, up through $\left(\frac{4\pi}{3}, 1\right)$ to $\left(\frac{11\pi}{6}, 3\right)$ and back to the axis at $x = \frac{7\pi}{3}$

For $y = 4 \cos(3x + \pi) - 2$, find the amplitude, period, phase shift, axis, maximum and minimum.

Amplitude $4$, period $\frac{2\pi}{3}$, phase shift $\frac{\pi}{3}$ left, axis $y = -2$, maximum $2$, minimum $-6$

A sinusoidal function has a maximum of $7$ at $x = \frac{\pi}{4}$ and the next minimum, $-1$, at $x = \frac{3\pi}{4}$. Write its equation as a cosine function and as a sine function.

$y = 4 \cos\left(2\left(x - \frac{\pi}{4}\right)\right) + 3$, or equivalently $y = 4 \sin 2x + 3$

Grade 12 · MHF4U · Advanced Functions

MHF4U: Graphing Sine and Cosine with Transformations

Every sinusoidal graph is the basic sine or cosine wave, stretched and slid. Four numbers in $y = a \sin(k(x - d)) + c$ say how, and reading them off correctly is most of the work.

How It Works

Amplitude is $|a|$ : half the distance from the maximum to the minimum. A negative $a$ reflects the wave in its axis, so sine starts by going down instead of up.

Period is $\frac{2\pi}{k}$ . A bigger $k$ squeezes more cycles into the same width. The number in front of $x$ has to be factored out before you can read the phase shift: $\sin(3x + \pi)$ is $\sin\left(3\left(x + \frac{\pi}{3}\right)\right)$ , a shift of $\frac{\pi}{3}$ left, not $\pi$ .

Phase shift is $d$ : right for $x - d$ , left for $x + d$ . Vertical shift is $c$ , the equation of the axis of the wave. The maximum is $c + |a|$ and the minimum is $c - |a|$ .

Five key points draw one cycle. For sine they are start, max, middle, min, end; for cosine, max, middle, min, middle, max. Space them a quarter period apart, beginning at $x = d$ , and read their heights from $c$ and $a$ . Then repeat the cycle in both directions.

Sine and cosine are the same wave: $\cos x = \sin\left(x + \frac{\pi}{2}\right)$ . A graph that has a maximum on the $y$ -axis is simplest as a cosine; one that crosses its axis there going up is simplest as a sine. Either can describe any sinusoid, with a different $d$ .

Going from a graph to an equation, read the numbers in order: $c$ is halfway between the max and the min, $|a|$ is half the gap, the period is the distance between two consecutive maxima (or twice the gap between a max and the next min), $k = \frac{2\pi}{\text{period}}$ , and $d$ is wherever you choose to start the cycle, matched to sine or cosine.

5 Worked Examples

Easy first, test level last. Try each one on paper, then open the solution.

Easy
State the amplitude, period, maximum, minimum and axis of $y = 3 \sin x + 1$ .
Show solution
1. $a = 3$ , so the amplitude is $3$ . $k = 1$ , so the period is $2\pi$ .
2. $c = 1$ : the axis is $y = 1$ . Maximum $1 + 3 = 4$ , minimum $1 - 3 = -2$ .
Answer: Amplitude $3$ , period $2\pi$ , axis $y = 1$ , maximum $4$ , minimum $-2$
Easy
Find the period of $y = \cos 2x$ and the five key points of its first cycle from $x = 0$ .
Show solution
1. $k = 2$ , so the period is $\frac{2\pi}{2} = \pi$ . A quarter period is $\frac{\pi}{4}$ .
2. Cosine starts at its maximum: $(0, 1)$ , then $\left(\frac{\pi}{4}, 0\right)$ , $\left(\frac{\pi}{2}, -1\right)$ , $\left(\frac{3\pi}{4}, 0\right)$ , $(\pi, 1)$ .
Answer: Period $\pi$ ; key points $(0, 1)$ , $\left(\frac{\pi}{4}, 0\right)$ , $\left(\frac{\pi}{2}, -1\right)$ , $\left(\frac{3\pi}{4}, 0\right)$ , $(\pi, 1)$
Medium
Describe the transformations in $y = -2 \sin\left(x - \frac{\pi}{3}\right) + 1$ and give the key points of one cycle.
Show solution
1. $a = -2$ : amplitude $2$ , reflected in the axis. $k = 1$ : period $2\pi$ . $d = \frac{\pi}{3}$ : shifted right. $c = 1$ : axis $y = 1$ , maximum $3$ , minimum $-1$ .
2. Start the cycle at $x = \frac{\pi}{3}$ on the axis. Because of the reflection the wave goes down first: $\left(\frac{\pi}{3}, 1\right)$ , $\left(\frac{5\pi}{6}, -1\right)$ , $\left(\frac{4\pi}{3}, 1\right)$ , $\left(\frac{11\pi}{6}, 3\right)$ , $\left(\frac{7\pi}{3}, 1\right)$ .
Answer: Amplitude $2$ with a reflection, period $2\pi$ , phase shift $\frac{\pi}{3}$ right, axis $y = 1$ ; the cycle runs from $\left(\frac{\pi}{3}, 1\right)$ down to $\left(\frac{5\pi}{6}, -1\right)$ , up through $\left(\frac{4\pi}{3}, 1\right)$ to $\left(\frac{11\pi}{6}, 3\right)$ and back to the axis at $x = \frac{7\pi}{3}$
Medium
For $y = 4 \cos(3x + \pi) - 2$ , find the amplitude, period, phase shift, axis, maximum and minimum.
Show solution
1. Factor out $k$ first: $3x + \pi = 3\left(x + \frac{\pi}{3}\right)$ , so $y = 4 \cos\left(3\left(x + \frac{\pi}{3}\right)\right) - 2$ .
2. Amplitude $4$ . Period $\frac{2\pi}{3}$ . Phase shift $\frac{\pi}{3}$ to the left, not $\pi$ .
3. Axis $y = -2$ ; maximum $-2 + 4 = 2$ , minimum $-2 - 4 = -6$ .
4. Check one point: at $x = -\frac{\pi}{3}$ the cosine's argument is $0$ , so $y = 4 - 2 = 2$ , a maximum, which is where a cosine shifted $\frac{\pi}{3}$ left should peak.
Answer: Amplitude $4$ , period $\frac{2\pi}{3}$ , phase shift $\frac{\pi}{3}$ left, axis $y = -2$ , maximum $2$ , minimum $-6$
Test level
A sinusoidal function has a maximum of $7$ at $x = \frac{\pi}{4}$ and the next minimum, $-1$ , at $x = \frac{3\pi}{4}$ . Write its equation as a cosine function and as a sine function.
Show solution
1. Axis: $c = \frac{7 + (-1)}{2} = 3$ . Amplitude: $a = \frac{7 - (-1)}{2} = 4$ .
2. A max to the next min is half a period: $\frac{3\pi}{4} - \frac{\pi}{4} = \frac{\pi}{2}$ , so the period is $\pi$ and $k = \frac{2\pi}{\pi} = 2$ .
3. Cosine starts at a maximum, so $d = \frac{\pi}{4}$ : $y = 4 \cos\left(2\left(x - \frac{\pi}{4}\right)\right) + 3$ .
4. Sine starts on the axis going up, a quarter period before the maximum: $\frac{\pi}{4} - \frac{\pi}{4} = 0$ , so $y = 4 \sin 2x + 3$ .
5. Check: $4 \sin\left(2 \cdot \frac{\pi}{4}\right) + 3 = 4 \sin \frac{\pi}{2} + 3 = 7$ and $4 \sin \frac{3\pi}{2} + 3 = -1$ .
Answer: $y = 4 \cos\left(2\left(x - \frac{\pi}{4}\right)\right) + 3$ , or equivalently $y = 4 \sin 2x + 3$

Learn It on Martian Lab

Martian Lab teaches this in the Sinusoidal Functions lesson of its MHF4U Trigonometric Functions unit. Watch it, then take the Trigonometric Functions practice tests to see if it stuck.

Watch the Sinusoidal Functions lesson free

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Free Trigonometric Functions practice test, with solutions

More Trigonometric Functions topics: Solving Trig Equations · Graphs of Tan, Sec, Csc and Cot · Sinusoidal Word Problems

More MHF4U topics: Long and Synthetic Division · Vertical and Horizontal Asymptotes · Proving Trig Identities · Laws of Logarithms · Solving Logarithmic Equations · End Behaviour and Degree · Finding the Equation of a Polynomial · Even and Odd Functions · Remainder and Factor Theorem · Solving Polynomial Equations · Polynomial Inequalities · Sum and Difference of Cubes · Holes in Rational Functions · Graphing Rational Functions · Rational Equations and Inequalities · Radian Measure · Unit Circle and Special Angles · Compound Angle Formulas · Double Angle Formulas · Solving Exponential Equations · Graphing Logarithmic Functions · Exponential and Log Word Problems · Sums, Products and Quotients of Functions · Composition of Functions · Average vs Instantaneous Rate of Change

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