B. Trigonometric Functions · Radian Measure and Trigonometric Functions

Sketching Transformed Sinusoidal Functions

B2.5About 75 minutes

Learning goals

I can describe, in order, the transformations that map y = sin x or y = cos x onto y = a sin(k(x − d)) + c or y = a cos(k(x − d)) + c.
I can use the mapping rule (x, y) → (x/k + d, ay + c) to find the key points of a transformed graph in radians.
I can sketch one or more cycles of a transformed sinusoidal function by hand.
I can state the period, amplitude, phase shift, domain, range and maximum and minimum points of the sketch.

Words to know

key points
The five points that start, quarter, halve, three-quarter and end one cycle: for sine (0, 0), (π/2, 1), (π, 0), (3π/2, −1), (2π, 0).

mapping rule
A rule that says where each point goes under a transformation. For y = a f(k(x − d)) + c it is (x, y) → (x/k + d, ay + c).

quarter period
One quarter of the period. Consecutive key points are a quarter period apart.

cycle
One complete repetition of the pattern of a periodic function.

From an equation to a sketch

Last lesson you read the amplitude, period, phase shift and axis from an equation. Now you will use them to sketch the graph quickly and accurately. A graphing tool can draw it for you, but sketching by hand shows you understand what each parameter does, and it is expected on tests.

The order of transformations (as in Grade 11):

Stretches, compressions and reflections (a and k) first.
Translations (d and c) last.

The mapping rule does all of this in one step:

(x, y) → (x/k + d, ay + c)

Apply it to the five key points of y = sin x or y = cos x.

	Key point 1	2	3	4	5
y = sin x	(0, 0)	(π/2, 1)	(π, 0)	(3π/2, −1)	(2π, 0)
y = cos x	(0, 1)	(π/2, 0)	(π, −1)	(3π/2, 0)	(2π, 1)

The key points of one cycle of the base functions.

Shortcut for the x-values

When k > 0, the transformed cycle starts at x = d and the key points are a quarter period apart. So you can list the x-values directly: d, d + P/4, d + P/2, d + 3P/4, d + P, where P = 2π/k. Then the y-values follow the pattern: for sine with a > 0 they go axis, max, axis, min, axis.

Worked example

Example 1: Sketch a sine function

Sketch one cycle of y = 3 sin(2(x − π/4)) + 1. State the properties.

a = 3, k = 2, d = π/4, c = 1. Transformations: vertical stretch by 3, horizontal compression by 1/2, translation π/4 right and 1 up.
Period 2π/2 = π; quarter period π/4. Mapping rule: (x, y) → (x/2 + π/4, 3y + 1).
(0, 0) → (π/4, 1); (π/2, 1) → (π/2, 4); (π, 0) → (3π/4, 1); (3π/2, −1) → (π, −2); (2π, 0) → (5π/4, 1).
Draw the axis y = 1 and dashed lines at y = 4 and y = −2. Plot the points and join them smoothly; extend the pattern if needed.

Amplitude 3, period π, phase shift π/4 right, axis y = 1. Domain {x ∈ ℝ}, range {y ∈ ℝ | −2 ≤ y ≤ 4}. Maximum points (π/2 + nπ, 4), minimum points (π + nπ, −2), n ∈ ℤ. In this cycle it increases on [π/4, π/2] and [π, 5π/4] and decreases on [π/2, π].

y = 3 sin(2(x − π/4)) + 1. The cycle from π/4 to 5π/4 is plotted from the five mapped key points, a quarter period (π/4) apart; the curve then repeats.

Worked example

Example 2: A reflected cosine with a phase shift to the left

Sketch y = −2 cos(½(x + π)) − 1 from x = −π to x = 3π.

a = −2, k = 1/2, d = −π, c = −1. Transformations: vertical stretch by 2 with a reflection in the x-axis, horizontal stretch by 2, translation π left and 1 down.
Mapping rule: (x, y) → (2x − π, −2y − 1). (Dividing by k = 1/2 means multiplying by 2.)
(0, 1) → (−π, −3); (π/2, 0) → (0, −1); (π, −1) → (π, 1); (3π/2, 0) → (2π, −1); (2π, 1) → (3π, −3).
Period 2π ÷ (1/2) = 4π, and the points are π apart ✓. Because a < 0, the cycle starts at a minimum instead of a maximum.

Amplitude 2, period 4π, phase shift π left, axis y = −1, range {y ∈ ℝ | −3 ≤ y ≤ 1}. Maximum points (π + 4nπ, 1), minimum points (−π + 4nπ, −3), n ∈ ℤ.

y = −2 cos(½(x + π)) − 1. The reflection makes the cycle start at a minimum, (−π, −3), and the horizontal stretch makes the period 4π.

Worked example

Example 3: A period with no π

Sketch one cycle of y = 4 sin(πx/6) + 5 for 0 ≤ x ≤ 12, on a decimal x-axis.

k = π/6, so period = 2π ÷ (π/6) = 12 and the quarter period is 3.
No phase shift: the cycle starts at x = 0. x-values: 0, 3, 6, 9, 12.
Sine pattern with a = 4 and c = 5: y-values 5, 9, 5, 1, 5.

Key points (0, 5), (3, 9), (6, 5), (9, 1), (12, 5). Models of things like temperature over 12 months look exactly like this: in models, x is usually time, not an angle.

Just for fun

A tuning fork vibrates in a motion very close to a pure sine wave, which is one reason musicians use it to tune. Your sketch of y = 4 sin(πx/6) + 5 is quieter, but much easier to carry around.

Hands-on activity

Roll-the-dice sinusoids

You need: two standard dice, a coin, graph paper (or lined paper with axes drawn), a pencil, and if possible a free online graphing calculator to check.

Roll the first die: that number is a. If the coin shows tails, make a negative.
Roll the second die: c is that number minus 3 (so c is between −2 and 3).
Use k = 2 and d = π/4 for every round.
Write the equation y = a sin(2(x − π/4)) + c. Write the mapping rule and the five key points. Sketch one cycle and label the axis, maximum and minimum.
Check by graphing the equation with technology, or by substituting one of your key points into the equation.
Play three rounds. Then challenge a family member: you roll, they sketch, you check.

What to look for: the cycle always runs from x = π/4 to x = 5π/4 with key points π/4 apart (only a and c change). When a is negative, the second key point (x = π/2) is a minimum instead of a maximum. The maximum value is always c + |a|.

Your turn

List the five key points of one cycle of y = 2 sin(x − π/3).

List the five key points of one cycle of y = cos 3x + 2, starting at x = 0.

Write the mapping rule for y = −4 sin(2(x + π/6)) − 1.

Use your rule from the last question to map the point (π/2, 1).

For y = 5 cos(x − π/4) + 2, state the range and the maximum points.

Describe the transformations that map y = sin x onto y = ½ sin 4x − 3.

Give the key points of y = −sin x + 1 for 0 ≤ x ≤ 2π.

More practice: warm-up

Where does one cycle of y = sin(x − π/2) start?

What is the quarter period of y = cos 2x?

What is the maximum value of y = 3 sin x + 1?

Find the image of (π, 0) under (x, y) → (x/2, 3y).

State the range of y = cos x + 2.

More practice: on your own

List the key points of one cycle of y = 3 cos(½(x − π/2)).

Sketch one cycle of y = −2 sin 3x + 4 from x = 0: give the key points and the maximum points.

Give the key points of one cycle of y = 2 sin(πx) − 1 from x = 0, on a decimal axis.

In the cycle π/4 ≤ x ≤ 5π/4 of y = 3 sin(2(x − π/4)) + 1, where is the function decreasing? Explain.

Describe, in order, the transformations that map y = cos x onto y = 2 cos(3x + π) − 4.

Challenge

Find the key points of one cycle of y = 2 sin(2x − π/2) and show that the graph is the same as y = −2 cos 2x.

Find the x-coordinates of all the maximum points of y = 4 sin(2(x − π/6)) + 1.

How many complete cycles of y = sin 6x lie in 0 ≤ x ≤ 2π, and how many maximum points are there in that interval? Explain.

Check your understanding

Quiz

What is the mapping rule for y = 2 sin(3(x − π/4)) + 5?

One cycle of y = sin(2(x − π/3)) starts at x = π/3. Where does it end?

What is the maximum value of y = −3 cos x + 2?

For 0 ≤ x ≤ 2π, where does y = −cos x reach its maximum?

What is the quarter period of y = sin(πx/2)?

The key point (π/2, 1) of y = sin x maps to which point on y = 4 sin 2x − 1?

For parents and teachers

Big idea (B2.5): to sketch y = a sin(k(x − d)) + c, map the five key points of y = sin x with (x, y) → (x/k + d, ay + c). The cycle starts at x = d and the key points are a quarter period apart.

How to help: Play the dice game in the activity. Ask your teen to explain each number in the mapping rule. If you have a free online graphing calculator, have them sketch first and graph second.

Common difficulties:

Multiplying x by k instead of dividing.
Forgetting to factor k out of the bracket, which gives the wrong phase shift.
Starting a reflected graph at a maximum.
Unequal spacing of key points on the sketch.