C. Trigonometric Functions · Sine Functions

Vertical Stretches, Compressions and Reflections: f(x) = a sin x

C2.6 C2.7About 65 minutes

Learning goals

I can describe the effect of a on the graph of f(x) = a sin x.
I can tell a stretch from a compression and recognize a reflection in the x-axis.
I can sketch f(x) = a sin x and state its amplitude, maximum, minimum, domain and range.
I can find a from a graph.

Words to know

vertical stretch
A transformation that multiplies every y-value by a factor greater than 1, making the graph taller.

vertical compression
A transformation that multiplies every y-value by a factor between 0 and 1, making the graph flatter.

reflection in the x-axis
A flip over the x-axis: (x, y) → (x, −y). Peaks become troughs.

absolute value |a|
The distance of a from 0, always non-negative. |−3| = 3.

Taller waves, flatter waves, upside-down waves

Compare a small Ferris wheel at a school fun fair with a huge observation wheel. Both seats go up and down once per turn, but the big wheel's seat swings much higher above its centre and much lower below it. The amplitude is bigger. In the equation, the number that controls the amplitude is the multiplier a in f(x) = a sin x.

Remember from Grade 10

For parabolas, in y = ax² the value of a stretches the graph vertically if |a| > 1, compresses it if 0 < |a| < 1, and reflects it in the x-axis if a < 0. The sine function behaves the same way.

The role of a in f(x) = a sin x

Every output of sin x is multiplied by a, so every point moves by the rule

(x, y) → (x, ay)

If |a| > 1: a vertical stretch by a factor of |a|. The wave gets taller.
If 0 < |a| < 1: a vertical compression by a factor of |a|. The wave gets flatter.
If a < 0: also a reflection in the x-axis. The wave flips upside down: peaks become troughs.

What changes and what stays the same?

Amplitude = |a|. (It is a distance, so use the absolute value.)
Max = |a|, min = −|a|, and range = {y ∈ ℝ | −|a| ≤ y ≤ |a|}.
If a < 0, the maximum moves to where sin x had its minimum (270°) and vice versa.
The axis (y = 0), period (360°), domain and zeros (0°, 180°, 360°, ...) do not change, because 0 × a = 0.

x	90°	270°
sin x	1	−1
2 sin x	2	−2
0.5 sin x	0.5	−0.5
−2 sin x	−2	2

Multiplying each output by a. The zeros never move.

y = 2 sin x (solid) is y = sin x (dashed) stretched vertically by a factor of 2: amplitude 2. The dotted curve, y = 0.5 sin x, is compressed by a factor of 0.5: amplitude 0.5. All three share the same zeros.

Worked example

Example 1: A vertical stretch

Sketch y = 3 sin x for 0° ≤ x ≤ 360°. State the amplitude, maximum and minimum points, zeros and range.

Multiply the y-coordinates of the five key points by 3: (0°, 0), (90°, 3), (180°, 0), (270°, −3), (360°, 0).
Join them with a smooth wave that is three times as tall as y = sin x.
Amplitude = |3| = 3. Maximum point (90°, 3), minimum point (270°, −3).
Zeros: 0°, 180°, 360° (unchanged). Range: {y ∈ ℝ | −3 ≤ y ≤ 3}.

Amplitude 3; max (90°, 3); min (270°, −3); zeros 0°, 180°, 360°; range −3 ≤ y ≤ 3.

Worked example

Example 2: A reflection and a stretch

Describe the transformations in y = −2 sin x. Give its key points, maximum and minimum points, and range.

a = −2. The negative sign means a reflection in the x-axis. |a| = 2 means a vertical stretch by a factor of 2.
Mapping: (x, y) → (x, −2y). Key points: (0°, 0), (90°, −2), (180°, 0), (270°, 2), (360°, 0).
The maximum is now at (270°, 2) and the minimum at (90°, −2).
Range: {y ∈ ℝ | −2 ≤ y ≤ 2}. Amplitude: 2 (not −2).

Reflected in the x-axis and stretched vertically by 2; max (270°, 2), min (90°, −2); range −2 ≤ y ≤ 2.

y = −2 sin x: the dashed y = sin x is flipped over the x-axis and stretched by 2. The highest point is now at 270° and the lowest at 90°.

Worked example

Example 3: Finding a from a graph

A sine graph passes through the origin, has zeros at 0°, 180° and 360°, a minimum point at (90°, −0.8) and a maximum point at (270°, 0.8). Write its equation.

Zeros at 0°, 180°, 360° and axis y = 0, so there is no translation: y = a sin x.
Amplitude = 0.8, so |a| = 0.8.
At 90°, sin x has its maximum, but this graph has its minimum there, so it is reflected: a < 0.

y = −0.8 sin x

Investigate with technology

Graph Y1 = sin(X) and Y2 = A·sin(X) for A = 2, 3, 0.5, 0.25, −1 and −3 (degree mode; window 0 ≤ x ≤ 360, −4 ≤ y ≤ 4). Record the amplitude and the maximum point each time. Then try a slider for a, and notice what happens as a passes through 0.

Just for fun

Why does y = −sin x see the world so differently?

It has spent a lot of time reflecting.

Hands-on activity

Stretching and flipping waves

You need: a sheet of graph paper, a pencil, a ruler and three coloured pencils (or three pencil styles: solid, dashed, dotted).

Draw axes with x from 0° to 360° (one square = 30°) and y from −4 to 4 (two squares = 1 unit).
Graph y = sin x using the five key points.
On the same axes, graph y = 3 sin x and y = −1.5 sin x. Use a different colour or line style for each.
For each graph, write the amplitude, the maximum point and the minimum point beside it.
Hold the paper up to a light with the back facing you. Look at y = sin x through the paper upside down (rotate it about the x-axis). Which graph does the flipped y = sin x look like (ignoring size)?

What to look for: y = 3 sin x has max (90°, 3) and min (270°, −3); y = −1.5 sin x has max (270°, 1.5) and min (90°, −1.5). All three cross the x-axis at 0°, 180° and 360°. The flipped y = sin x has the same shape as y = −1.5 sin x (ignoring size), because both are reflections in the x-axis.

Your turn

Describe how y = 4 sin x is related to y = sin x.

Describe how y = −0.5 sin x is related to y = sin x.

State the amplitude and range of y = −5 sin x.

Give the key points of y = 1.5 sin x from 0° to 360°.

Where is the maximum of y = −3 sin x between 0° and 360°?

A graph has amplitude 6, axis y = 0, and a maximum at (90°, 6). Write its equation.

More practice: warm-up

What is the amplitude of y = 7 sin x?

What is the amplitude of y = −7 sin x?

What are the zeros of y = 2.5 sin x from 0° to 360°?

Write the mapping rule for y = 3 sin x.

Is y = 0.2 sin x a stretch or a compression?

More practice: on your own

Find the value of y = −4 sin x at x = 30°, 150° and 210°.

The point (240°, −0.866) is on y = sin x. Find the image of this point on y = 3 sin x and on y = −2 sin x.

A sine graph has zeros at 0°, 180°, 360°, a maximum point (270°, 0.4) and a minimum point (90°, −0.4). Write its equation and describe the transformations.

For which values of a is the range of y = a sin x equal to {y ∈ ℝ | −9 ≤ y ≤ 9}?

Explain why the zeros of y = a sin x are the same as the zeros of y = sin x for any a ≠ 0.

Challenge

For 0° ≤ x ≤ 360°, where is y = −2 sin x increasing?

Solve 3 sin x = 1.5 for 0° ≤ x ≤ 360°.

The graphs of y = 2 sin x and y = −2 sin x meet at certain points between 0° and 360°. Find them and explain.

Check your understanding

Quiz

What is the amplitude of y = −3 sin x?

Which transformation maps y = sin x onto y = 0.25 sin x?

Where is the maximum point of y = −2 sin x for 0° ≤ x ≤ 360°?

Which property of y = sin x is unchanged in y = 5 sin x?

What is the range of y = 4.5 sin x?

A graph has axis y = 0, amplitude 2.5 and a minimum at (90°, −2.5). Its equation is

For parents and teachers

Big idea: The role of a in f(x) = a sin x: vertical stretches, compressions and reflections in the x-axis (C2.6), and sketching the result with its domain and range (C2.7).

How to help: Ask, "Is the wave taller or flatter? Is it upside down?" before your teen sketches. Check the maximum point: is it at 90° or at 270°?

Common difficulties:

Writing a negative amplitude. Amplitude = |a|.
Thinking a stretch changes the zeros. It does not.
Forgetting that a < 0 swaps the positions of the maximum and minimum.