Sine/cosine graphs and symmetry | Year 10

01 / LEARN

Understand the idea.

Start with one complete cycle

A basic sine cycle begins at zero, rises to $1$ after a quarter-turn, crosses zero after a half-turn, reaches $- 1$ after three quarters and returns to zero after a full turn. Cosine begins at its maximum.

Both curves repeat every $360^{\circ}$ . Plot key points on labelled axes and join them smoothly; sharp line segments do not represent the continuously changing coordinates of a rotating point on the unit circle.

Gold graph of y equals sine x for x from zero to three hundred and sixty degrees. The y scale runs from minus four to four in unit grid steps, labelled every two units. Degree labels occur every ninety degrees.

Read the parameters

For $y = A sin (B x) + D$ or $y = A cos (B x) + D$ in degrees, with $A, B \neq 0$ , the amplitude is $∣ A ∣$ , period is $\frac{360^{\circ}}{∣ B ∣}$ and midline is $y = D$ . The range is $D - ∣ A ∣ \leq y \leq D + ∣ A ∣$ .

A negative $A$ reflects the curve in its midline. Unit-circle reflection also explains $cos (- x) = cos x$ and $sin (- x) = - sin x$ . If $A$ or $B$ is zero, the function is constant and has no fundamental period.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Read a transformed rule

State amplitude, period, midline and range for y = 2 sin x − 1.

Gold graph of y equals two sine x minus one for x from zero to three hundred and sixty degrees. The y scale runs from minus four to four in unit grid steps, labelled every two units. The curve has no added midline or answer annotations.

Worked solution

1
Identify the coefficients.
$A = 2, B = 1, D = - 1$
Compare the rule with $y = A sin (B x) + D$ .
2
Find amplitude and period.
$amplitude = ∣ 2 ∣ = 2, T = \frac{360^{\circ}}{1} = 360^{\circ}$
Vertical scaling controls amplitude; the inside multiplier controls cycle length.
3
Find the midline and range.
$\begin{array}{c} y = - 1 is the midline, \\ - 1 - 2 \leq y \leq - 1 + 2, \\ - 3 \leq y \leq 1. \end{array}$
The graph extends two units above and below its midline.

Answer

Amplitude 2; period 360°; midline y = −1; −3 ≤ y ≤ 1.

EXAMPLE 02

A compressed cosine graph

For y = 3 cos(2x), state the period and the first maximum and minimum at or after x = 0°.

Graph of y=3 cos(2x) for x from 0 to 360 degrees, on a grid from y=-4 to 4. The horizontal grid intervals are 45 degrees and vertical intervals are one unit.

Worked solution

1
Calculate the compressed period.
$T = \frac{360^{\circ}}{2} = 180^{\circ}$
The inside angle completes one turn while $x$ advances by $180^{\circ}$ .
2
Locate the first maximum.
$2 x = 0^{\circ} \Rightarrow x = 0^{\circ}, y = 3 cos 0^{\circ} = 3$
Cosine starts at its maximum, giving $(0^{\circ}, 3)$ .
3
Locate the first minimum.
$2 x = 180^{\circ} \Rightarrow x = 90^{\circ}, y = 3 cos 180^{\circ} = - 3$
Cosine first reaches $- 1$ after half an inside-angle cycle, giving $(90^{\circ}, - 3)$ .

Answer

Period 180°; maximum (0°,3); minimum (90°,−3).

03 / YOUR TURN

Try it yourself.

Check your understanding

Find the amplitude and period of y = 4 sin(2x) + 1.

Show answer & reasoning

Amplitude 4; period 180°.

For $y = 4 sin (2 x) + 1$ , amplitude is $∣ 4 ∣ = 4$ and period is $\frac{360^{\circ}}{2} = 180^{\circ}$ . The upward shift changes neither quantity.

Keep practising

6 worksheet questions · 17 marks

Open worksheet Worked solutions

Read the questions online

State the amplitude, period and range of $y = 4 sin (2 x) + 1$ .
3 marks
Give $y = cos x$ at $x = 0^{\circ}$ , $90^{\circ}$ , $180^{\circ}$ , $270^{\circ}$ and $360^{\circ}$ .
2 marks
Sketch $y = sin x$ for $0^{\circ} \leq x \leq 360^{\circ}$ on the supplied grid. Label the intercepts and turning points.
4 marks
Describe how $y = - 2 cos x$ differs from $y = 2 cos x$ .
2 marks
Explain why $sin (360^{\circ} - x) = - sin x$ using the unit circle.
3 marks
A sine graph has maximum 3, minimum $- 1$ and period $360^{\circ}$ . It crosses its midline upwards at $x = 0^{\circ}$ . Give a rule.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

State the amplitude, period and range of $y = 4 sin (2 x) + 1$ .
3 marks
Use $∣ A ∣ = 4$ and $\frac{360^{\circ}}{∣ B ∣} = 180^{\circ}$ .
Add and subtract 4 from the midline 1.
Amplitude 4; period $180^{\circ}$ ; $- 3 \leq y \leq 5$ .
Give $y = cos x$ at $x = 0^{\circ}$ , $90^{\circ}$ , $180^{\circ}$ , $270^{\circ}$ and $360^{\circ}$ .
2 marks
Use the $x$ -coordinates of the unit-circle points.
1, 0, $- 1$ , 0, 1.
Sketch $y = sin x$ for $0^{\circ} \leq x \leq 360^{\circ}$ on the supplied grid. Label the intercepts and turning points.
4 marks
Evaluate sine at $0^{\circ}, 90^{\circ}, 180^{\circ}, 270^{\circ}, 360^{\circ}$ .
Plot the five key points and join with a smooth periodic curve.
Intercepts $(0^{\circ}, 0)$ , $(180^{\circ}, 0)$ , $(360^{\circ}, 0)$ ; maximum $(90^{\circ}, 1)$ ; minimum $(270^{\circ}, - 1)$ .
Describe how $y = - 2 cos x$ differs from $y = 2 cos x$ .
2 marks
Every $y$ -value changes sign.
Both amplitudes are 2 and periods $360^{\circ}$ .
It is reflected in the $x$ -axis; amplitude and period stay the same.
Explain why $sin (360^{\circ} - x) = - sin x$ using the unit circle.
3 marks
The two points have opposite vertical coordinates.
Sine is the vertical coordinate.
Reflection across the $x$ -axis keeps $x$ -coordinate but reverses $y$ -coordinate.
A sine graph has maximum 3, minimum $- 1$ and period $360^{\circ}$ . It crosses its midline upwards at $x = 0^{\circ}$ . Give a rule.
3 marks
Midline $= \frac{3 + (- 1)}{2} = 1$ .
Amplitude $= \frac{3 - (- 1)}{2} = 2$ .
An upward midline crossing at $0^{\circ}$ matches sine.
$y = 2 sin x + 1$ .

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Curriculum reference & lesson scope

Australian Curriculum v9 and South Australian school course scope

Right-triangle applications include AC9M10M03. The approved chapter also develops non-right-triangle methods, exact values and trigonometric functions within one normal Year 10 course.

Official curriculum source ↗

In this lesson

Understand the idea.

Start with one complete cycle

Read the parameters

See the method in action.

Read a transformed rule

Identify the coefficients.

Find amplitude and period.

Find the midline and range.

A compressed cosine graph

Calculate the compressed period.

Locate the first maximum.

Locate the first minimum.

Try it yourself.

Check your understanding

Keep practising