Grade by Grade
C. Trigonometric Functions · Sine Functions

Combining Transformations and Finding Equations

C2.6 C2.7About 75 minutes
Learning goals
  • I can sketch y = a sin(x − d) + c by transforming the five key points in the right order.
  • I can state the amplitude, axis, maximum, minimum, domain and range of y = a sin(x − d) + c.
  • I can write an equation for a sinusoidal graph with period 360°.
  • I can check an equation by substituting a maximum or minimum point.

Words to know

combined transformation
Two or more transformations applied to the same graph, such as a stretch followed by translations.
y = a sin(x − d) + c
The transformed sine function: amplitude |a|, horizontal translation d, axis y = c.
equivalent equations
Different equations that produce the same graph, such as y = sin(x − 30°) and y = sin(x + 330°).

Putting it all together

Real periodic situations rarely need only one transformation. A Ferris wheel seat has an amplitude equal to the wheel's radius (that is a), it starts at the bottom rather than level with the centre (that is d), and its centre is above the ground (that is c). So we combine all three:

f(x) = a sin(x − d) + c

Today you will sketch these combined functions, state their properties, and work backwards from a graph to its equation.

Remember from Lessons 6 to 8

  • c: vertical translation, (x, y) → (x, y + c).
  • a: vertical stretch or compression by |a|; reflection in the x-axis if a < 0, (x, y) → (x, ay).
  • d: horizontal translation, right if d > 0, (x, y) → (x + d, y).

Applying the transformations in order

To sketch y = a sin(x − d) + c from y = sin x, apply the transformations to the five key points in this order:

  1. Stretch, compress or reflect with a (multiply each y-coordinate by a).
  2. Translate horizontally by d (add d to each x-coordinate).
  3. Translate vertically by c (add c to each y-coordinate).

All in one rule: (x, y) → (x + d, ay + c).

Stretch before you shift up: if you added c first and then multiplied by a, you would also stretch the shift, and get the wrong graph.

Properties of y = a sin(x − d) + c

  • Amplitude = |a|, axis: y = c, period = 360°.
  • Max = c + |a|, min = c − |a|, range = {y ∈ ℝ | c − |a| ≤ y ≤ c + |a|}, domain = {x ∈ ℝ}.
  • If a > 0, a cycle starts at x = d on the axis going up; the maximum is at x = 90° + d.
y = sin xy = 2 sin xy = 2 sin(x − 30°)y = 2 sin(x − 30°) + 11. a: stretchor reflect2. d: shiftleft/right3. c: shiftup/downCombined mapping: (x, y) → (x + 30°, 2y + 1)
Building y = 2 sin(x − 30°) + 1 from y = sin x in three steps.
Worked example

Example 1: Sketching a combined transformation

Sketch y = 2 sin(x − 30°) + 1 for one cycle. State the amplitude, axis, maximum and minimum points, period, domain and range.

  1. Identify the parameters: a = 2, d = 30°, c = 1.
  2. Apply (x, y) → (x + 30°, 2y + 1) to the key points of y = sin x: (0°, 0) → (30°, 1); (90°, 1) → (120°, 3); (180°, 0) → (210°, 1); (270°, −1) → (300°, −1); (360°, 0) → (390°, 1).
  3. Plot and join with a smooth wave.
  4. Amplitude 2, axis y = 1, max point (120°, 3), min point (300°, −1), period 360°.
  5. Domain {x ∈ ℝ}, range {y ∈ ℝ | −1 ≤ y ≤ 3}.
  6. Check one point in the equation: x = 120° gives 2 sin(90°) + 1 = 2 + 1 = 3 ✓.

Key points (30°, 1), (120°, 3), (210°, 1), (300°, −1), (390°, 1); range −1 ≤ y ≤ 3.

xy0°90°180°270°360°−10123(120°, 3)(300°, −1)(30°, 1)(210°, 1)axis y = 1y = 2 sin(x − 30°) + 1
y = 2 sin(x − 30°) + 1 with its five key points. The amplitude is 2, the axis has moved up to y = 1, and the cycle starts at 30° instead of 0°.
Worked example

Example 2: A reflection in the mix

For y = −0.5 sin(x + 60°) + 2: describe the transformations, list the key points of one cycle, and state the range.

  1. a = −0.5: reflection in the x-axis and vertical compression by 0.5. d = −60°: translation 60° left. c = 2: translation 2 up.
  2. Rule: (x, y) → (x − 60°, −0.5y + 2).
  3. (0°, 0) → (−60°, 2); (90°, 1) → (30°, 1.5); (180°, 0) → (120°, 2); (270°, −1) → (210°, 2.5); (360°, 0) → (300°, 2).
  4. Max = 2 + 0.5 = 2.5 (at 210°); min = 2 − 0.5 = 1.5 (at 30°). Range {y ∈ ℝ | 1.5 ≤ y ≤ 2.5}.

Reflect, compress by 0.5, left 60°, up 2; key points (−60°, 2), (30°, 1.5), (120°, 2), (210°, 2.5), (300°, 2); range 1.5 ≤ y ≤ 2.5.

From a graph to an equation

To write y = a sin(x − d) + c for a given sinusoidal graph with period 360°:

  1. c = (max + min) ÷ 2 (the axis).
  2. |a| = (max − min) ÷ 2 (the amplitude). Choose a > 0 for the simplest equation.
  3. d: with a > 0, the maximum of y = sin x is at 90°, so d = (x-value of a maximum) − 90°. (Or find a point where the graph crosses its axis going up: that x-value is d.)
  4. Check a second point, such as a minimum.
xy−90°0°90°180°270°360°−2−10123456max (60°, 5)min (240°, −1)amplitude 3axis y = 2
A sinusoidal graph with period 360°. Read the maximum (60°, 5) and the minimum (240°, −1) to find the axis, amplitude and shift.
Worked example

Example 3: Equation from a graph

Write an equation for the graph above in the form y = a sin(x − d) + c, with a > 0.

  1. Max = 5, min = −1. c = (5 + (−1)) ÷ 2 = 2.
  2. a = (5 − (−1)) ÷ 2 = 3.
  3. A maximum is at x = 60°. For y = sin x it would be at 90°, so d = 60° − 90° = −30° (a shift 30° left).
  4. Equation: y = 3 sin(x − (−30°)) + 2 = 3 sin(x + 30°) + 2.
  5. Check the minimum: x = 240° gives 3 sin(270°) + 2 = −3 + 2 = −1 ✓.

y = 3 sin(x + 30°) + 2

More than one right answer

Because the graph repeats every 360°, adding or subtracting 360° to d gives the same graph: y = 3 sin(x − 330°) + 2 is also correct. A reflection with a different shift can also work: y = −3 sin(x − 150°) + 2 gives the same graph. Check any answer by testing a maximum and a minimum.

Just for fun

A sine wave went to get its picture taken.

The photographer said, "Shift a little to the right, now up a bit, and stretch... perfect! Now hold that for 360°."

Hands-on activity

Model a Ferris wheel with a bicycle wheel

You need: a bicycle turned upside down (or a lazy Susan or a round lid on a table), a strip of tape, a measuring tape and a grown-up's help to steady the bike. Keep fingers away from the spokes and chain.

  1. Put a small piece of tape on the tire at its lowest point.
  2. Measure the height of the tape above the floor at the start. Then turn the wheel by hand one-eighth of a turn (45°) at a time, measuring the height each time, for a full turn. Record angle turned and height.
  3. Find the max, min, axis and amplitude from your data.
  4. Write an equation h = a sin(θ − d) + c. Hint: the tape starts at the bottom, which is where sin(θ − d) = −1, so try d = 90°.
  5. Check your equation at θ = 180° (the tape should be at the top).

What to look for: amplitude equal to the wheel's radius (roughly 30 to 35 cm for many adult bikes; use your own measurement), an axis at the height of the hub, and an equation like h = r sin(θ − 90°) + (hub height). At θ = 180° the equation gives r + hub height, the top of the wheel.

Your turn

1

For y = 4 sin(x − 45°) − 1, state a, d and c, and describe each transformation.

2

Write the mapping rule for y = 4 sin(x − 45°) − 1, and give the image of (90°, 1).

3

State the amplitude, axis, maximum, minimum and range of y = 4 sin(x − 45°) − 1.

4

List the five key points of one cycle of y = 3 sin(x + 90°) + 2.

5

A sinusoidal graph with period 360° has max 8 at x = 100° and min 2. Write its equation with a > 0.

6

State the domain and range of y = −2 sin(x − 60°) + 7.

More practice: warm-up

1

What is the axis of y = 5 sin(x − 20°) + 3?

2

What is the amplitude of y = −6 sin(x + 10°) − 2?

3

Which parameter causes a phase shift?

4

What is the maximum of y = 2 sin(x − 30°) + 4?

5

What is the period of y = 7 sin(x − 80°) + 1?

More practice: on your own

1

Sketch y = −2 sin(x − 90°) + 1 by listing its key points, and state the maximum point.

2

A Ferris wheel has radius 15 m and its centre is 17 m above the ground. A seat starts at the bottom. Write h(θ) = a sin(θ − d) + c, where θ is the angle turned, and find the height after 120°.

3

A graph has max 1 at x = 270° and min −7 at x = 90°. Write an equation with a > 0.

4

Evaluate y = 2.5 sin(x + 15°) + 6 at x = 75° and x = 195°.

5

A student sketches y = 3 sin x + 2 by first moving y = sin x up 2 and then multiplying every y-value by 3. What max and min does the student get, and what are the correct ones?

Challenge

1

Find all x from 0° to 360° where y = 2 sin(x − 30°) + 1 equals 2.

2

Write two different equations of the form y = a sin(x − d) + c, one with a > 0 and one with a < 0, for a graph with max 10 at x = 0° and min 4 at x = 180°.

3

For y = a sin(x − d) + c, the range is {y ∈ ℝ | −4 ≤ y ≤ 10} and the graph passes through (50°, 3) going up. Find a (> 0), c and d.

Check your understanding

Quiz

1.

What is the range of y = 3 sin(x − 40°) + 5?

2.

Which mapping rule matches y = 2 sin(x + 30°) − 4?

3.

The point (90°, 1) on y = sin x maps to which point on y = 2 sin(x − 30°) + 1?

4.

A graph with period 360° has max 6 at x = 30° and min −2. With a > 0, the equation is

5.

In which order should the transformations be applied to key points?

6.

What is the maximum value of y = −4 sin(x − 10°) + 1?

For parents and teachers

Big idea: Combining the parameters a, d and c (C2.6) to sketch y = a sin(x − d) + c and state its domain and range (C2.7). This prepares for the modelling problems later in the unit (C3.3).

How to help: Ask your teen to name each parameter's job before sketching: "a stretches, d slides sideways, c lifts." Then ask them to check one point by substituting it into the equation.

Common difficulties:

  • Doing the vertical shift before the stretch.
  • Sign errors in d (x + 30° means left 30°).
  • When finding an equation from a graph, using the whole max-to-min distance as a instead of half of it.