D. Trigonometric Functions · Sinusoidal Functions

Transformations I: Vertical Stretches, Reflections and Translations (a and c)

D2.5 D2.6 D2.7About 70 minutes

Learning goals

I can explain how the parameter a in y = a sin x and y = a cos x changes the graph.
I can explain how the parameter c in y = sin x + c and y = cos x + c changes the graph.
I can find the amplitude, equation of the axis, maximum, minimum and range from an equation.
I can sketch y = a sin x + c and y = a cos x + c using key points.

Words to know

vertical stretch / compression
Multiplying every y-value by a number. If |a| > 1 the graph is stretched (taller); if 0 < |a| < 1 it is compressed (flatter).

reflection in the x-axis
Flipping the graph upside down; it happens when a is negative.

vertical translation
Sliding the graph up (c > 0) or down (c < 0) by |c| units.

mapping rule
A rule that shows where each point goes, for example (x, y) → (x, ay + c).

Turning up the volume

In Grade 10 you saw that y = a(x − h)² + k is the parabola y = x² stretched by a and moved by h and k. Sinusoidal functions work the same way. In the next four lessons we study

y = a sin(k(x − d)) + c and y = a cos(k(x − d)) + c

one parameter at a time. Today: the two vertical parameters, a and c. They change the height of the wave, but not its timing.

Investigate with technology

Open a free online graphing tool (such as Desmos) and type y = a sin(x) + c. Make sure it is in degree mode, and add sliders for a and c. Drag each slider and watch what happens. Questions to answer: Which parameter changes the height of the wave? Which one moves it? What happens when a is negative? Does either one change the period or the zeros of y = a sin x?

The parameter a: vertical stretch and reflection

In y = a sin x, every y-value of y = sin x is multiplied by a.

If |a| > 1, the graph is stretched vertically by a factor of |a|.
If 0 < |a| < 1, the graph is compressed vertically by a factor of |a|.
If a < 0, the graph is also reflected in the x-axis (turned upside down).

Since the maximum of sin x is 1, the maximum of a sin x is |a|. So the amplitude is |a|.

The x-values do not change, so the period stays 360° and the zeros stay where they were (0 × a = 0).

Multiplying by a stretches or compresses the wave vertically. y = 3 sin x has amplitude 3; y = 0.5 sin x has amplitude 0.5. The zeros and the period do not change.

The parameter c: vertical translation

In y = sin x + c, every y-value is increased by c. The whole graph slides up c units (or down if c < 0).

The axis moves from y = 0 to y = c. The amplitude and period do not change.

Putting a and c together

For y = a sin x + c or y = a cos x + c:

amplitude = |a|
equation of the axis: y = c
maximum = c + |a|, minimum = c − |a|
range: {y ∈ ℝ | c − |a| ≤ y ≤ c + |a|}

Each point moves by the mapping rule (x, y) → (x, ay + c): multiply by a first, then add c (the same order as the order of operations).

Worked example

Example 1: Sketching y = 3 sin x + 2

Sketch one cycle of y = 3 sin x + 2 and state the amplitude, axis, maximum, minimum and range.

Start with the key points of y = sin x: (0°, 0), (90°, 1), (180°, 0), (270°, −1), (360°, 0).
Apply (x, y) → (x, 3y + 2): (0°, 2), (90°, 5), (180°, 2), (270°, −1), (360°, 2).
Draw the axis y = 2 as a dashed line, plot the five points, and join them with a smooth curve.
Amplitude |3| = 3. Axis y = 2. Max 2 + 3 = 5; min 2 − 3 = −1. Range {y ∈ ℝ | −1 ≤ y ≤ 5}.

Key points (0°, 2), (90°, 5), (180°, 2), (270°, −1), (360°, 2); range −1 ≤ y ≤ 5.

y = 3 sin x + 2 (solid) is y = sin x (dashed) stretched vertically by 3 and then moved up 2. The new axis is y = 2.

Worked example

Example 2: A reflected cosine

Describe the transformations that map y = cos x onto y = −2 cos x − 1, list the key points of one cycle, and state the range.

a = −2: a vertical stretch by a factor of 2 and a reflection in the x-axis. c = −1: a translation 1 unit down.
Key points of y = cos x: (0°, 1), (90°, 0), (180°, −1), (270°, 0), (360°, 1).
Apply (x, y) → (x, −2y − 1): (0°, −3), (90°, −1), (180°, 1), (270°, −1), (360°, −3).
The reflection made the cosine start at a minimum. Max = −1 + 2 = 1; min = −1 − 2 = −3. Range {y ∈ ℝ | −3 ≤ y ≤ 1}.

Stretch by 2, reflect in the x-axis, translate 1 down; range −3 ≤ y ≤ 1.

The reflection (a = −2) turns the cosine upside down, so it starts at its minimum, (0°, −3). The axis is y = −1.

Worked example

Example 3: Finding a and c from the maximum and minimum

A sinusoidal function of the form y = a sin x + c has a maximum value of 7 and a minimum value of −1. Its graph is increasing at x = 0°. Write its equation.

Amplitude: |a| = (7 − (−1)) ÷ 2 = 4.
Axis: c = (7 + (−1)) ÷ 2 = 3.
y = sin x increases at x = 0°. A negative a would flip it so it decreases there. So a is positive: a = 4.
Check: max = 3 + 4 = 7 ✓, min = 3 − 4 = −1 ✓.

y = 4 sin x + 3

Just for fun

When someone asks you to "turn it up", they are really asking for a bigger a. The loudness of a sound depends on the amplitude of its pressure wave, so a volume knob is basically an amplitude slider.

Hands-on activity

Make waves with a skipping rope

You need: a skipping rope (or a long piece of rope or string) and a partner. Do this outside or in a room with plenty of space.

Your partner holds one end of the rope still. You hold the other end, with the rope gently slack.
Move your hand up and down a little, at a steady rhythm. Watch the wave. Now move your hand up and down a lot at the same rhythm. What changed? What stayed the same?
Keep the same size and rhythm of motion, but do it with your hand held at waist height, then at chest height. What changed?
Match each change to a parameter: a or c.

What to look for: a bigger hand motion makes taller waves: that is a bigger amplitude (|a|). Raising your hand moves the whole wave up: that is like changing c. Neither change makes the waves come faster, so the period does not change. (If you have a computer, repeat the investigation with sliders for a and c in a free graphing tool, as described in the tip above.)

Your turn

For y = 4 sin x, state the amplitude, maximum, minimum and range.

For y = sin x − 3, state the equation of the axis and the range.

For y = 2 cos x + 5, state the amplitude, axis and range, and where the maximum and minimum occur for 0° ≤ x ≤ 360°.

List the five key points of one cycle (0° to 360°) of y = −3 sin x.

Describe the transformations that map y = cos x onto y = −0.5 cos x + 4.

Write the equation of a cosine function (no reflection) with amplitude 5 and axis y = −2.

Under the transformation y = 3 sin x − 1, where does the point (90°, 1) of y = sin x go?

More practice: warm-up

State the amplitude of y = 7 cos x.

State the amplitude of y = −2 sin x.

State the equation of the axis of y = sin x + 6.

What is the maximum value of y = sin x − 4?

In y = a sin x + c, which parameter can reflect the graph in the x-axis?

More practice: on your own

For y = 1.5 sin x + 2.5, list the key points of one cycle and state the range.

For y = −4 cos x + 1 on 0° ≤ x ≤ 360°, find the maximum and minimum values and where they occur.

A function y = a sin x + c has a maximum of 10, a minimum of 2, and is increasing at x = 0°. Write its equation.

A function y = a cos x + c has axis y = 3 and passes through (0°, −1). Find a and write the equation.

The range of y = a sin x + c is {y ∈ ℝ | −6 ≤ y ≤ 2}. Find |a| and c.

Challenge

Use the graph of y = 2 sin x + 1 to find all x between 0° and 360° where the graph crosses the x-axis.

A sinusoidal function y = a sin x + c has a maximum of 6, and its graph just touches the x-axis at each minimum. Find two possible equations.

Explain why the graph of y = −2 sin x is the same as the graph of y = 2 sin x translated 180° to the right.

Check your understanding

Quiz

What is the amplitude of y = −5 sin x + 2?

What is the range of y = 3 cos x − 1?

Which function has a maximum of 6 and a minimum of 2?

In y = a sin x, what does a negative value of a do?

Changing c in y = sin x + c changes…

Where does the point (90°, 1) on y = sin x go under y = 2 sin x − 3?

For parents and teachers

Big idea: D2.5 asks students to investigate, with technology, how each parameter transforms the graphs; D2.6 to find amplitude, domain and range from an equation; D2.7 to sketch transformed graphs. This lesson covers a (vertical stretch or compression, reflection) and c (vertical translation).

How to help: Ask your teen to explain y = 3 sin x + 2 in plain words ("three times as tall, moved up 2"), then to predict the highest and lowest values before sketching. A free graphing website with sliders is a quick way to check predictions.

Common difficulties:

Saying the amplitude is negative when a is negative. Amplitude is |a|.
Adding c before multiplying by a. The mapping rule is (x, y) → (x, ay + c): multiply first.
Thinking a or c changes the period. Neither touches the x-values.
Forgetting that a reflected cosine starts at a minimum and a reflected sine starts by going down.