D. Trigonometric Functions · Sinusoidal Functions

Putting It All Together: Sketching y = a sin(k(x − d)) + c

D2.5 D2.6 D2.7About 75 minutes

Learning goals

I can find the amplitude, period, phase shift, equation of the axis, domain and range from y = a sin(k(x − d)) + c or y = a cos(k(x − d)) + c.
I can describe in words the transformations that map y = sin x or y = cos x onto a given function.
I can sketch a sinusoidal function with several transformations using the mapping rule and key points.

Words to know

transformed sinusoidal function
A function of the form y = a sin(k(x − d)) + c or y = a cos(k(x − d)) + c.

amplitude
|a|

period
360° ÷ |k|

phase shift
d (right if d > 0, left if d < 0), read after factoring k(x − d)

vertical translation
c; the axis of the curve is y = c

range
{y ∈ ℝ | c − |a| ≤ y ≤ c + |a|}

All four at once

You now know what each parameter does on its own. Real models usually need all four. Today you will put them together, read every property straight from the equation, and sketch any sinusoidal function by hand.

Parameter	What it does to y = sin x or y = cos x	What it tells you
a	vertical stretch or compression by \|a\|; reflection in the x-axis if a < 0	amplitude = \|a\|
k	horizontal stretch or compression by 1/\|k\|	period = 360° ÷ \|k\|
d	horizontal translation d (right if d > 0, left if d < 0)	phase shift = d
c	vertical translation c (up if c > 0, down if c < 0)	axis: y = c

The four parameters of y = a sin(k(x − d)) + c and y = a cos(k(x − d)) + c.

Domain and range

Every sinusoidal function continues forever left and right, so the domain is {x ∈ ℝ}. The graph swings |a| above and below the axis y = c, so

max = c + |a|, min = c − |a|, range = {y ∈ ℝ | c − |a| ≤ y ≤ c + |a|}

The order of transformations

As in the unit Functions and Equivalent Expressions, apply stretches, compressions and reflections first, then translations. The mapping rule does this for you:

(x, y) → (x/k + d, ay + c)

Five steps to sketch any sinusoidal function. Step 1 matters most: always write the inside as k(x − d) before reading d.

Worked example

Example 1: Sketching y = 2 sin(3(x − 30°)) + 1

State the amplitude, period, phase shift, axis, domain and range, and sketch one cycle.

Already factored. a = 2, k = 3, d = 30, c = 1.
Amplitude 2; period 360° ÷ 3 = 120°; phase shift 30° right; axis y = 1.
Max 1 + 2 = 3, min 1 − 2 = −1. Domain {x ∈ ℝ}, range {y ∈ ℝ | −1 ≤ y ≤ 3}.
Mapping rule (x, y) → (x/3 + 30°, 2y + 1) applied to the sine key points: (0°, 0) → (30°, 1); (90°, 1) → (60°, 3); (180°, 0) → (90°, 1); (270°, −1) → (120°, −1); (360°, 0) → (150°, 1).
Shortcut check: the cycle starts at x = d = 30° and the key points are a quarter period (30°) apart: 30°, 60°, 90°, 120°, 150° ✓.
Draw the axis y = 1 and the lines y = 3 and y = −1, plot the five points, join them smoothly and extend the pattern.

Key points (30°, 1), (60°, 3), (90°, 1), (120°, −1), (150°, 1); range −1 ≤ y ≤ 3.

One and a half cycles of y = 2 sin(3(x − 30°)) + 1. The cycle starting at 30° ends at 150°, one period (120°) later.

Worked example

Example 2: A reflected, stretched cosine

Sketch one cycle of y = −3 cos(0.5(x + 90°)) − 2 and describe the transformations.

a = −3, k = 0.5, d = −90, c = −2.
Transformations of y = cos x: vertical stretch by a factor of 3 with a reflection in the x-axis; horizontal stretch by a factor of 1/0.5 = 2; translation 90° left and 2 units down.
Amplitude 3; period 360° ÷ 0.5 = 720°; axis y = −2; range {y ∈ ℝ | −5 ≤ y ≤ 1}.
Mapping rule (x, y) → (x/0.5 − 90°, −3y − 2) = (2x − 90°, −3y − 2) on the cosine key points: (0°, 1) → (−90°, −5); (90°, 0) → (90°, −2); (180°, −1) → (270°, 1); (270°, 0) → (450°, −2); (360°, 1) → (630°, −5).
Check x = 270°: 0.5(270° + 90°) = 180°, cos 180° = −1, and −3(−1) − 2 = 1 ✓.

Key points (−90°, −5), (90°, −2), (270°, 1), (450°, −2), (630°, −5).

y = −3 cos(0.5(x + 90°)) − 2 (solid) compared with y = cos x (dashed). The reflection makes this cycle start at a minimum, and one cycle now takes 720°.

Worked example

Example 3: Properties from an unfactored equation

State the amplitude, period, phase shift, axis and range of y = 4 sin(2x + 120°) − 1.

Factor out k = 2: 2x + 120° = 2(x + 60°). So y = 4 sin(2(x + 60°)) − 1.
Amplitude 4; period 360° ÷ 2 = 180°; phase shift 60° left; axis y = −1.
Max −1 + 4 = 3, min −1 − 4 = −5. Range {y ∈ ℝ | −5 ≤ y ≤ 3}; domain {x ∈ ℝ}.

Amplitude 4, period 180°, phase shift 60° left, axis y = −1, range −5 ≤ y ≤ 3.

Check one point in the original equation

After you find your key points, substitute one of them into the original equation. If it does not work, look for a sign error in d or a missing factor of k. This 20-second check catches most mistakes.

Just for fun

Riddle: What do you get when you stretch, squeeze, flip, slide and lift a sine wave?

Still a sine wave. However much you transform a sinusoidal function, it never loses its shape. That is more than can be said for a sweater in the dryer.

Hands-on activity

Transformation card challenge

You need: 16 small pieces of paper, a pencil, graph paper, and a partner (or a free online graphing tool in degree mode to check your work).

Make four piles of four cards each:

a pile: 2, −1, 0.5, −3
k pile: 1, 2, 3, 0.5
d pile: 0°, 30°, −45°, 90°
c pile: 0, 1, −2, 3

Shuffle each pile and draw one card from each. Write the function y = a sin(k(x − d)) + c.
State the amplitude, period, phase shift, axis and range.
Use the mapping rule to find the five key points, and sketch one cycle.
Swap with your partner to check, or graph it with technology. Then draw again, this time using cosine.

What to look for: for example, drawing a = −3, k = 2, d = 30°, c = 1 gives y = −3 sin(2(x − 30°)) + 1, with amplitude 3, period 180°, phase shift 30° right, axis y = 1, range −2 ≤ y ≤ 4, and key points (30°, 1), (75°, −2), (120°, 1), (165°, 4), (210°, 1).

Your turn

For y = 3 sin(2(x − 45°)) + 1, state the amplitude, period, phase shift, axis and range.

List the key points of one cycle of y = 3 sin(2(x − 45°)) + 1.

List the key points of one cycle of y = −2 cos(x + 60°) + 4.

State the amplitude, period, phase shift and range of y = 5 sin(0.5x − 30°).

Describe the transformations that map y = cos x onto y = 2 cos(3(x + 20°)) − 5.

State the domain and range of y = −4 sin 6x + 7.

Find the y-intercept of y = 2 sin(3(x − 30°)) + 1.

More practice: warm-up

State the amplitude of y = 6 cos(2(x − 10°)) − 1.

State the period of y = 6 cos(2(x − 10°)) − 1.

State the phase shift of y = 6 cos(2(x − 10°)) − 1.

State the equation of the axis of y = 6 cos(2(x − 10°)) − 1.

State the range of y = 6 cos(2(x − 10°)) − 1.

More practice: on your own

List the key points of one cycle of y = 4 cos(2(x − 30°)) − 1.

Find the amplitude, period and phase shift of y = −sin(3x + 90°) + 2, and list the key points of one cycle.

Find the maximum value of y = 3 sin(2(x − 15°)) + 4 and the first x > 0° where it occurs.

Does the graph of y = 2 sin(x − 30°) − 3 ever cross the x-axis? Explain.

Explain why the mapping rule divides x by k before adding d.

Challenge

Find every x between 0° and 360° where y = 2 sin(3(x − 30°)) + 1 has a maximum.

The point (90°, 1) on y = sin x is mapped to (40°, 7) by y = 3 sin(k(x − 10°)) + c. Find k and c.

How many times does the graph of y = 2 sin(3(x − 30°)) + 1 cross the line y = 2 for 0° ≤ x ≤ 360°? Explain.

Check your understanding

Quiz

What is the period of y = 2 sin(4(x − 10°)) + 3?

What is the phase shift of y = cos(2x + 40°)?

What is the range of y = −3 sin(x − 45°) + 2?

Which is the mapping rule for y = a sin(k(x − d)) + c?

Which describes y = 0.5 cos 2x − 4 compared with y = cos x?

What is the value of y = 2 sin(3(x − 30°)) + 1 at x = 60°?

For parents and teachers

Big idea: D2.6 asks students to determine amplitude, period, phase shift, domain and range from an equation, and D2.7 to sketch graphs with one or more transformations. This lesson combines all four parameters.

How to help: Give your teen an equation and ask for the five properties out loud (amplitude, period, phase shift, axis, range) before they sketch anything. Then ask them to check one key point in the original equation.

Common difficulties:

Skipping the factoring step (2x + 120° must become 2(x + 60°)).
Getting the period backwards (360° ÷ k, not 360° × k).
Sign errors in d: (x + 60°) means 60° left.
Plotting key points but forgetting the reflection (a < 0) when choosing max or min.
Sharp, pointy sketches. Remind your teen to draw smooth, rounded peaks and to label the axis and the max and min lines.