Mathematics Advanced • Year 11 • Module 2 • Lesson 10
Graphs of Sine and Cosine
Build fluency in reading amplitude, period, vertical shift, and range from $y = a \sin(bx) + d$ and $y = a \cos(bx) + d$, and in sketching key features.
1. Quick recall
Answer each question in the space provided. 1 mark each
Q1.1 For $y = a \sin(bx) + d$, fill in the formulas:
Amplitude = ____________ Period (in radians) = ____________ Range = [____________, ____________]
Q1.2 Five key $x$-values of one cycle of $y = \sin x$ on $[0, 2\pi]$ are 0, π/2, π, 3π/2, 2π. State the $y$-value at each.
x = 0: y = ____ x = π/2: y = ____ x = π: y = ____ x = 3π/2: y = ____ x = 2π: y = ____
Q1.3 Why is the amplitude of $y = -3 \sin x$ equal to 3, not $-3$? Answer in one sentence.
2. Worked example — sketching $y = 2 \sin x$
Every step annotated. Use as the template for the faded version.
Problem. Sketch one cycle of $y = 2 \sin x$ for $0 \leq x \leq 2\pi$ and state its amplitude and range.
Step 1 — Identify the parameters.
a = 2, b = 1, d = 0.
Reason: read directly from the form $y = a \sin(bx) + d$.
Step 2 — Compute amplitude and period.
Amplitude = |a| = 2. Period = 2π/|b| = 2π/1 = 2π.
Reason: amplitude is always positive (Trap 2); period unchanged because $b = 1$.
Step 3 — Plot the five key points.
(0, 0), (π/2, 2), (π, 0), (3π/2, −2), (2π, 0).
Reason: same x-pattern as $\sin x$, but y multiplied by 2.
Step 4 — State the range and conclude.
Range = [−2, 2]; one cycle complete from 0 to 2π.
Conclusion. Amplitude = $\mathbf{2}$, Period = $\mathbf{2\pi}$, Range = $\mathbf{[-2, 2]}$.
3. Faded example — fill in the missing steps
For $y = 3 \cos(2x) + 1$, find amplitude, period, and range. Fill in each blank line. 4 marks
Step 1 — Read off parameters.
a = ____, b = ____, d = ____
Step 2 — Amplitude.
Amplitude = |a| = ________
Step 3 — Period.
Period = 2π / |b| = 2π / ____ = ________
Step 4 — Range.
Range = [d − |a|, d + |a|] = [____, ____]
Conclusion. Amplitude = ____, Period = ____, Range = ____.
4. Graduated practice
State amplitude, period, and (where there is a vertical shift) range. For one question, sketch.
Foundation — basic transforms (4 questions)
| Q | Function | Amplitude | Period |
|---|---|---|---|
| 4.1 1 | $y = \sin x$ | ||
| 4.2 1 | $y = 2 \cos x$ | ||
| 4.3 1 | $y = \sin(2x)$ | ||
| 4.4 1 | $y = \cos\left(\frac{x}{2}\right)$ |
Standard — transformations + ranges (6 questions)
4.5 Find the amplitude, period, and range of $y = 4 \sin(3x)$. 2 marks
4.6 Find the amplitude, period, and range of $y = \cos x + 2$. 2 marks
4.7 Find the amplitude, period, and range of $y = -2 \cos(4x) + 3$. (Mind the negative amplitude coefficient.) 2 marks
4.8 Find the amplitude, period, and range of $y = 5 \sin\left(\frac{x}{3}\right) - 1$. 2 marks
4.9 A student claims the period of $y = \sin(2x)$ is $4\pi$. Identify which trap from the lesson this commits and give the correct period. 2 marks
4.10 Sketch one cycle of $y = 3 \sin x$ for $0 \leq x \leq 2\pi$. Label the five key points $(x, y)$, the amplitude, and the period. 3 marks
Extension — combine ideas (2 questions)
4.11 Find amplitude, period, and range of $y = -3 \cos\left(\frac{\pi x}{2}\right) + 4$. State the period in the same units as $x$. 3 marks
4.12 The graph of $y = a \sin(bx) + d$ has maximum value 7, minimum value $-3$, and period $\pi$. Find $a$ (positive), $b$ (positive), and $d$. Show every step. 3 marks
5. Self-check the easy 3
Tick the first three once you've verified your method works.
How did this worksheet feel?
What I'll revisit before next class:
Q1.1 — Formulas
Amplitude = $|a|$. Period = $\frac{2\pi}{|b|}$. Range = $[d - |a|, d + |a|]$.
Q1.2 — Five key points of $y = \sin x$
$x = 0$: $y = 0$. $x = \pi/2$: $y = 1$. $x = \pi$: $y = 0$. $x = 3\pi/2$: $y = -1$. $x = 2\pi$: $y = 0$.
Q1.3 — Why amplitude is positive
Amplitude is a distance from the midline to the peak; distance is always positive. The negative sign on $a = -3$ instead reflects the graph in the $x$-axis (Trap 2 in the lesson).
Q3 — Faded example: $3 \cos(2x) + 1$
Step 1: $a = 3$, $b = 2$, $d = 1$.
Step 2: Amplitude $= |3| = 3$.
Step 3: Period $= 2\pi / 2 = \pi$.
Step 4: Range $= [1 - 3, 1 + 3] = [-2, 4]$.
Conclusion: Amplitude $= \mathbf{3}$, Period $= \mathbf{\pi}$, Range $= \mathbf{[-2, 4]}$.
Q4.1 — $\sin x$
Amplitude $= 1$, Period $= 2\pi$.
Q4.2 — $2 \cos x$
Amplitude $= 2$, Period $= 2\pi$.
Q4.3 — $\sin(2x)$
Amplitude $= 1$, Period $= \frac{2\pi}{2} = \pi$.
Q4.4 — $\cos\left(\frac{x}{2}\right)$
$b = \frac{1}{2}$. Amplitude $= 1$, Period $= \frac{2\pi}{1/2} = 4\pi$.
Q4.5 — $4 \sin(3x)$
Amplitude $= 4$, Period $= \frac{2\pi}{3}$, Range $= [-4, 4]$.
Q4.6 — $\cos x + 2$
Amplitude $= 1$, Period $= 2\pi$, Range $= [2 - 1, 2 + 1] = [1, 3]$.
Q4.7 — $-2 \cos(4x) + 3$
Amplitude $= |-2| = 2$, Period $= \frac{2\pi}{4} = \frac{\pi}{2}$, Range $= [3 - 2, 3 + 2] = [1, 5]$.
Q4.8 — $5 \sin\left(\frac{x}{3}\right) - 1$
Amplitude $= 5$, Period $= \frac{2\pi}{1/3} = 6\pi$, Range $= [-1 - 5, -1 + 5] = [-6, 4]$.
Q4.9 — Critique of "period of $\sin(2x)$ is $4\pi$"
This commits Trap 1 in the lesson: $b = 2$ compresses the graph (halves the period) rather than stretching. Correct period: $\frac{2\pi}{2} = \mathbf{\pi}$.
Q4.10 — Sketch $y = 3 \sin x$
Amplitude = 3, period = $2\pi$. Five key points: $(0, 0)$, $\left(\frac{\pi}{2}, 3\right)$, $(\pi, 0)$, $\left(\frac{3\pi}{2}, -3\right)$, $(2\pi, 0)$. Sketch is a smooth sine wave oscillating between $y = -3$ and $y = 3$ across one cycle. Range $= [-3, 3]$.
Q4.11 — $-3 \cos\left(\frac{\pi x}{2}\right) + 4$
$a = -3$, $b = \frac{\pi}{2}$, $d = 4$. Amplitude $= |-3| = 3$. Period $= \frac{2\pi}{\pi/2} = \frac{2\pi \cdot 2}{\pi} = 4$ (in the same units as $x$). Range $= [4 - 3, 4 + 3] = [1, 7]$.
Q4.12 — Find $a$, $b$, $d$
$d = \frac{\max + \min}{2} = \frac{7 + (-3)}{2} = \mathbf{2}$.
Amplitude $= \frac{\max - \min}{2} = \frac{7 - (-3)}{2} = 5$, so $a = \mathbf{5}$ (taking positive).
$b = \frac{2\pi}{\text{period}} = \frac{2\pi}{\pi} = \mathbf{2}$.
So $y = 5 \sin(2x) + 2$. Check: max $= 5 + 2 = 7$ ✓, min $= -5 + 2 = -3$ ✓, period $= \frac{2\pi}{2} = \pi$ ✓.