Grade by Grade
D. Trigonometric Functions · Sinusoidal Functions

Writing the Equation of a Sinusoidal Function

D2.8About 75 minutes
Learning goals
  • I can find a, k, d and c from a graph or from a list of properties.
  • I can write a sine equation and a cosine equation for the same graph.
  • I can check an equation by substituting points from the graph.

Words to know

maximum point
A highest point of the graph, (x, max).
minimum point
A lowest point of the graph, (x, min).
starting point of a cycle
For y = a sin(k(x − d)) + c with a > 0, the point (d, c) where the graph crosses its axis going up; for y = a cos(k(x − d)) + c with a > 0, the maximum point (d, c + a).

Working backwards

So far you went from an equation to a graph. Now you will go the other way: from a graph, or from a description such as "maximum of 5, period 180°", to an equation. This is the key skill for modelling in the rest of the unit, because real data come as graphs and tables, not as equations.

Four questions, four parameters

  1. Where is the middle? c = (max + min) ÷ 2. The axis is y = c.
  2. How tall? |a| = (max − min) ÷ 2.
  3. How long is one cycle? Find the period (for example, max to the next max, or twice the distance from a max to the next min). Then k = 360° ÷ period.
  4. Where does a cycle start? That gives d, and it depends on which function you choose:
  • cosine with a > 0: d = the x-value of a maximum;
  • cosine with a < 0: d = the x-value of a minimum;
  • sine with a > 0: d = an x-value where the graph crosses the axis going up;
  • sine with a < 0: d = an x-value where the graph crosses the axis going down.

Finally, check by substituting another point from the graph.

−45°0°45°90°135°180°225°−2−10123456axis y = 2max (45°, 5)min (135°, −1)(0°, 2)period = 180°a = 3
Reading the graph for Example 1: the axis is halfway between max and min, the amplitude is the distance from the axis to a max, and the period is max to max.
Worked example

Example 1: One graph, two equations

The graph above has a maximum at (45°, 5) and the next minimum at (135°, −1). Write (a) a cosine equation and (b) a sine equation for it.

  1. c = (5 + (−1)) ÷ 2 = 2. |a| = (5 − (−1)) ÷ 2 = 3.
  2. Max to the next min is half a cycle: 135° − 45° = 90°. Period = 2 × 90° = 180°, so k = 360 ÷ 180 = 2.
  3. (a) Cosine with a > 0 starts at a maximum: d = 45. y = 3 cos(2(x − 45°)) + 2.
  4. (b) Sine with a > 0 starts on the axis going up. That happens a quarter period (45°) before the max: 45° − 45° = 0°. So d = 0 and y = 3 sin 2x + 2.
  5. Check (b) at the max: x = 45°: 3 sin 90° + 2 = 5 ✓. Check (a) at the min: x = 135°: 3 cos(180°) + 2 = −1 ✓.

y = 3 cos(2(x − 45°)) + 2 or y = 3 sin 2x + 2.

Worked example

Example 2: From properties

A sinusoidal function has amplitude 6, period 24, a maximum at x = 15 and axis y = 10. Write three different equations for it.

  1. a = 6 (or −6), c = 10, k = 360 ÷ 24 = 15.
  2. Cosine, starting at the max: y = 6 cos(15(x − 15)) + 10.
  3. Sine, starting on the axis going up, a quarter period (24 ÷ 4 = 6) before the max: d = 15 − 6 = 9. y = 6 sin(15(x − 9)) + 10.
  4. Reflected cosine, starting at a minimum. The minimum is half a period (12) before the max: d = 15 − 12 = 3. y = −6 cos(15(x − 3)) + 10.
  5. Check the third one at x = 15: 15(15 − 3) = 180, cos 180° = −1, so y = −6(−1) + 10 = 16 = 10 + 6 ✓ (the maximum).

y = 6 cos(15(x − 15)) + 10, y = 6 sin(15(x − 9)) + 10 or y = −6 cos(15(x − 3)) + 10.

Worked example

Example 3: A graph that starts low

A graph has a minimum at (20°, −4), the next maximum at (80°, 2) and another minimum at (140°, −4). Write an equation using a reflected cosine and one using a cosine with a > 0.

  1. c = (2 + (−4)) ÷ 2 = −1; |a| = (2 − (−4)) ÷ 2 = 3.
  2. Min to min: 140° − 20° = 120° = period, so k = 360 ÷ 120 = 3.
  3. Reflected cosine starts at a min: y = −3 cos(3(x − 20°)) − 1.
  4. Cosine with a > 0 starts at a max: y = 3 cos(3(x − 80°)) − 1.
  5. Check the first at (80°, 2): 3(80° − 20°) = 180°, so −3 cos 180° − 1 = 3 − 1 = 2 ✓.

y = −3 cos(3(x − 20°)) − 1 or y = 3 cos(3(x − 80°)) − 1.

0°20°40°60°80°100°120°140°160°180°−5−4−3−2−10123axis y = −1(20°, −4)(80°, 2)(140°, −4)
The graph for Example 3. A reflected cosine fits naturally because the cycle shown starts at a minimum.

Which form should I choose?

Choose the form that makes d easiest to read. If the graph shows a clear maximum, use cosine. If it shows a clear minimum, use −cosine. If it crosses the axis going up at a convenient place (often x = 0), use sine. All correct answers have the same a (up to sign), k and c.

Just for fun

Most functions have one equation. A sinusoidal function has infinitely many: shift d by any whole number of periods and you get a new equation for the very same graph. It's the master of disguise of the function world.

Hands-on activity

Equation detective

You need: graph paper, a pencil, a ruler, and a partner (or a free online graphing tool in degree mode).

  1. Each person secretly picks a maximum, a minimum and a period (keep the numbers friendly, such as max 6, min −2, period 120°), and one x-value for a maximum.
  2. Draw two full cycles of the graph carefully, marking the axis, the max and min lines, and the key points. Do not write the equation.
  3. Swap graphs. Each person writes two different equations for the graph they receive: one using sine and one using cosine.
  4. Swap back and check each other's equations by substituting two points from the graph, or graph them with technology.

What to look for: with max 6, min −2, period 120° and a maximum at x = 30°, correct answers include y = 4 cos(3(x − 30°)) + 2 and y = 4 sin(3x) + 2 (the sine starts a quarter period, 30°, before the max, at x = 0°).

Your turn

1

A cosine function has its maximum at x = 0°, a maximum value of 8, a minimum value of 2 and a period of 180°. Write its equation.

2

A sine graph crosses its axis going up at (30°, 0), has maximum 4 and minimum −4, and has period 360°. Write its equation.

3

A graph has a maximum at (60°, 7) and the next minimum at (150°, 1). Write a cosine equation.

4

Write a sine equation for the same graph as the previous question.

5

Write a sine function with amplitude 2, period 90°, phase shift 30° right and axis y = −3.

6

A graph has a minimum at (0°, −2) and the next maximum at (180°, 6). Write an equation using a reflected cosine.

7

Explain why a sinusoidal graph has more than one correct equation.

More practice: warm-up

1

A graph has maximum 10 and minimum 4. Find |a|.

2

For the same graph, find c.

3

The period is 60°. Find k.

4

For a cosine model with a > 0, a maximum is at x = 25°. What is d?

5

A sine model with a > 0 crosses its axis going up at x = −10°. Write the bracket (x − d).

More practice: on your own

1

Write an equation for the data: x = 0°, 30°, 60°, 90°, 120° gives y = 3, 5, 3, 1, 3 (and the pattern repeats).

2

Write an equation of the form y = a sin(k(x − d)) + c with a < 0 (a reflected sine) for the graph of Example 1 (max (45°, 5), min (135°, −1)).

3

Write an equation for a function with range 1 ≤ y ≤ 9, period 30 and a maximum at x = 5.

4

A cosine graph has a maximum at (−30°, 2), a minimum value of −6 and a period of 720°. Write its equation.

5

A sinusoidal graph has a maximum at (10°, 3), and the next point where it crosses its axis is (40°, 0). Write a sine equation.

Challenge

1

A sinusoidal function has maximum points at (40°, 9) and (160°, 9), with no maximum between them, and a minimum value of −3. Write an equation.

2

Show that y = 2 sin(x − 30°) + 1 and y = 2 cos(x − 120°) + 1 have the same graph.

3

A function y = 3 cos(x − d) + 1 passes through (0°, 2.5), where 0° ≤ d < 360°. Find both possible values of d.

Check your understanding

Quiz

1.

A sinusoidal graph has a maximum of 11 and a minimum of 3. What are |a| and c?

2.

A sinusoidal function has a period of 40°. What is k (k > 0)?

3.

For a cosine model y = a cos(k(x − d)) + c with a > 0, the value of d can be the x-value of…

4.

Which function has a maximum at (0°, 5), a minimum of −1 and a period of 360°?

5.

Which function has the same graph as y = cos x?

6.

For a reflected cosine model (a < 0), d is the x-value of…

For parents and teachers

Big idea: D2.8 asks students to represent a sinusoidal function with an equation, given its graph or its properties. The method is four questions: middle (c), height (a), cycle length (k), and starting point (d).

How to help: Draw any wave on paper and ask your teen for the max, the min and the period first; the equation follows. Ask for a second, different equation for the same graph, and have them check both by substituting a point.

Common difficulties:

  • Using max − min for a (it is half of that).
  • Using the max-to-min distance as the period (it is half a period).
  • Choosing d from a maximum while writing a sine equation. Sine with a > 0 starts on the axis going up, a quarter period before the max.
  • Worrying that their answer is different from the back of the book. Different correct equations are normal; substituting points is the way to check.