State the amplitude, period, phase shift and midline of $y = 3\sin(2(x - 45^\circ)) + 1$, and its maximum and minimum.

Amplitude $3$, period $180^\circ$, $45^\circ$ right, midline $y = 1$; maximum $4$, minimum $-2$.

For $y = -2\cos x + 5$ with $0^\circ \le x \le 360^\circ$, find the maximum and minimum and where they happen.

Maximum $7$ at $180^\circ$; minimum $3$ at $0^\circ$ and $360^\circ$.

Use the mapping rule to find the images of the five key points of $y = \sin x$ under $y = 2\sin(3(x + 30^\circ)) - 1$.

$(-30^\circ, -1)$, $(0^\circ, 1)$, $(30^\circ, -1)$, $(60^\circ, -3)$, $(90^\circ, -1)$

Write $y = 4\cos(2x - 120^\circ)$ so you can read the phase shift, then state the period and the phase shift.

$y = 4\cos(2(x - 60^\circ))$: period $180^\circ$, phase shift $60^\circ$ right.

A sinusoidal graph has a maximum at $(30^\circ, 7)$ and the next minimum at $(120^\circ, 1)$. Write a cosine equation and a sine equation for it.

$y = 3\cos(2(x - 30^\circ)) + 4$ or $y = 3\sin(2(x + 15^\circ)) + 4$

Grade 11 · MCR3U · Functions

MCR3U: Transformations of Sine and Cosine

Every sinusoidal graph in MCR3U is $y = \sin x$ or $y = \cos x$ stretched, shifted or reflected. Read four numbers from $y = a\sin(k(x - d)) + c$ and you can sketch the graph, or read them from a graph and write its equation.

How It Works

The parent graphs. $y = \sin x$ starts at $(0, 0)$ , peaks at $(90^\circ, 1)$ , crosses the axis at $(180^\circ, 0)$ , bottoms out at $(270^\circ, -1)$ and is back to $(360^\circ, 0)$ . $y = \cos x$ has the same shape but starts at its maximum, $(0, 1)$ . Both have period $360^\circ$ , amplitude $1$ and midline $y = 0$ .

Amplitude. $|a|$ stretches the graph vertically. If $a < 0$ , the graph is also reflected in the midline, so a sine curve goes down first.

Period. $k$ compresses the graph horizontally by a factor of $\frac{1}{|k|}$ , so the period is $\frac{360^\circ}{|k|}$ . Factor $k$ out of the brackets before you read the shift: $y = \sin(2x - 60^\circ)$ is $y = \sin(2(x - 30^\circ))$ , a shift of $30^\circ$ , not $60^\circ$ .

Phase shift. $d$ moves the graph right when you see $x - d$ and left when you see $x + d$ .

Midline. $c$ moves the graph up or down, so the midline is $y = c$ . The maximum is $c + |a|$ and the minimum is $c - |a|$ .

The mapping rule. Each point $(x, y)$ on the parent graph goes to $\left(\frac{x}{k} + d, ay + c\right)$ . Map the five key points, then join them with a smooth wave. The stretches come before the shifts, which is why you divide by $k$ before adding $d$ .

5 Worked Examples

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

Easy
State the amplitude, period, phase shift and midline of $y = 3\sin(2(x - 45^\circ)) + 1$ , and its maximum and minimum.
Show solution
1. $a = 3$ , $k = 2$ , $d = 45^\circ$ and $c = 1$ .
2. Period: $\frac{360^\circ}{2} = 180^\circ$ .
3. Maximum: $1 + 3 = 4$ . Minimum: $1 - 3 = -2$ .
Answer: Amplitude $3$ , period $180^\circ$ , $45^\circ$ right, midline $y = 1$ ; maximum $4$ , minimum $-2$ .
Easy
For $y = -2\cos x + 5$ with $0^\circ \le x \le 360^\circ$ , find the maximum and minimum and where they happen.
Show solution
1. $a = -2$ : amplitude $2$ , reflected in the midline $y = 5$ . The graph starts at a minimum instead of a maximum.
2. Minimum $5 - 2 = 3$ where $\cos x = 1$ , at $x = 0^\circ$ and $x = 360^\circ$ .
3. Maximum $5 + 2 = 7$ where $\cos x = -1$ , at $x = 180^\circ$ .
Answer: Maximum $7$ at $180^\circ$ ; minimum $3$ at $0^\circ$ and $360^\circ$ .
Medium
Use the mapping rule to find the images of the five key points of $y = \sin x$ under $y = 2\sin(3(x + 30^\circ)) - 1$ .
Show solution
1. $a = 2$ , $k = 3$ , $d = -30^\circ$ , $c = -1$ , so $(x, y) \to \left(\frac{x}{3} - 30^\circ, 2y - 1\right)$ .
2. $(0, 0) \to (-30^\circ, -1)$ and $(90^\circ, 1) \to (0^\circ, 1)$ .
3. $(180^\circ, 0) \to (30^\circ, -1)$ , $(270^\circ, -1) \to (60^\circ, -3)$ and $(360^\circ, 0) \to (90^\circ, -1)$ .
4. Check: one cycle runs from $-30^\circ$ to $90^\circ$ , which is $120^\circ = \frac{360^\circ}{3}$ .
Answer: $(-30^\circ, -1)$ , $(0^\circ, 1)$ , $(30^\circ, -1)$ , $(60^\circ, -3)$ , $(90^\circ, -1)$
Medium
Write $y = 4\cos(2x - 120^\circ)$ so you can read the phase shift, then state the period and the phase shift.
Show solution
1. Factor $2$ out of the brackets: $2x - 120^\circ = 2(x - 60^\circ)$ .
2. $y = 4\cos(2(x - 60^\circ))$ , so $d = 60^\circ$ .
3. Period: $\frac{360^\circ}{2} = 180^\circ$ .
Answer: $y = 4\cos(2(x - 60^\circ))$ : period $180^\circ$ , phase shift $60^\circ$ right.
Test level
A sinusoidal graph has a maximum at $(30^\circ, 7)$ and the next minimum at $(120^\circ, 1)$ . Write a cosine equation and a sine equation for it.
Show solution
1. Midline: $c = \frac{7 + 1}{2} = 4$ . Amplitude: $a = \frac{7 - 1}{2} = 3$ .
2. Maximum to minimum is half a cycle: $120^\circ - 30^\circ = 90^\circ$ , so the period is $180^\circ$ and $k = \frac{360^\circ}{180^\circ} = 2$ .
3. Cosine starts at a maximum, so $d = 30^\circ$ : $y = 3\cos(2(x - 30^\circ)) + 4$ .
4. Sine reaches its maximum a quarter cycle, $45^\circ$ , after it starts on the midline, so it starts at $30^\circ - 45^\circ = -15^\circ$ : $y = 3\sin(2(x + 15^\circ)) + 4$ .
5. Check the sine equation at $x = 30^\circ$ : $3\sin(90^\circ) + 4 = 7$ .
Answer: $y = 3\cos(2(x - 30^\circ)) + 4$ or $y = 3\sin(2(x + 15^\circ)) + 4$

Learn It on Martian Lab

Martian Lab teaches this in the Graphing Trignometric Functions lesson of its MCR3U Trigonometric Functions unit. Watch it, then take the Trigonometric Functions practice tests to see if it stuck.

Watch the Graphing Trignometric Functions lesson free

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Free Trigonometric Functions practice test, with solutions

More Trigonometric Functions topics: Sinusoidal Word Problems

More MCR3U topics: Inverse Functions · Transformations of Functions · Sine Law and the Ambiguous Case · Domain, Range and Function Notation · Simplifying Rational Expressions · Operations with Rational Expressions · Max/Min and Vertex Form · Linear-Quadratic Systems · Quadratic Word Problems · Simplifying Radicals · Exponent Laws and Rational Exponents · Graphing Exponential Functions · Exponential Growth and Decay · Special Angles and the CAST Rule · Cosine Law and 3D Problems · Arithmetic Sequences and Series · Geometric Sequences and Series · Pascal's Triangle and the Binomial Theorem · Simple and Compound Interest · Annuities: Future and Present Value

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