Why Trig Graphs Matter#
Trig graphs model repeating patterns — sound waves, tides, heartbeats, seasonal temperatures, alternating current. In exams, trig graphs carry 10–15 marks in Paper 2. You must be able to sketch them, read information from them, and find their equations.
The General Forms#
| Function | General form | Key features |
|---|---|---|
| Sine | $y = a\sin(bx + p) + q$ | Starts at the midline, peaks at $\frac{1}{4}$ period |
| Cosine | $y = a\cos(bx + p) + q$ | Starts at a maximum (if $a > 0$), peaks at $x = 0$ |
| Tangent | $y = a\tan(bx + p) + q$ | Passes through the midline, has vertical asymptotes |
The Four Parameters#
1. Amplitude ($|a|$) — How Tall?#
$$\text{Amplitude} = |a|$$- The distance from the midline ($y = q$) to the peak or trough.
- $|a| > 1$: graph is stretched vertically (taller).
- $|a| < 1$: graph is compressed vertically (shorter).
- $a < 0$: graph is reflected (flipped upside down).
⚠️ Tangent has no amplitude — its range is all real numbers.
For sine and cosine:
- Maximum = $q + |a|$
- Minimum = $q - |a|$
- Range = $[q - |a|;\; q + |a|]$
2. Period ($\frac{360°}{|b|}$) — How Wide?#
$$\text{Period} = \frac{360°}{|b|}$$For tangent: $\text{Period} = \frac{180°}{|b|}$
- $|b| > 1$: graph is compressed horizontally (shorter period, more cycles).
- $|b| < 1$: graph is stretched horizontally (longer period, fewer cycles).
| $b$ value | Sine/Cosine period | Tan period |
|---|---|---|
| $b = 1$ | $360°$ | $180°$ |
| $b = 2$ | $180°$ | $90°$ |
| $b = 3$ | $120°$ | $60°$ |
| $b = \frac{1}{2}$ | $720°$ | $360°$ |
3. Phase Shift ($p$) — Left or Right?#
The phase shift moves the graph horizontally.
$$\text{Shift} = -\frac{p}{b} \text{ (in degrees)}$$- If $p > 0$: graph shifts LEFT.
- If $p < 0$: graph shifts RIGHT.
⚠️ The sign trap: In $y = \sin(2x - 60°)$, we write it as $y = \sin(2(x - 30°))$. The shift is $30°$ to the RIGHT, not $60°$. Always factor out $b$ first!
4. Vertical Shift ($q$) — Up or Down?#
- $q > 0$: graph shifts UP.
- $q < 0$: graph shifts DOWN.
- The midline of the graph is at $y = q$.
- For tangent, the midline and the “centre” of each branch shift by $q$.
The Basic Shapes (Know These Cold)#
$y = \sin x$ (one full cycle: $0°$ to $360°$)#
| $x$ | $0°$ | $90°$ | $180°$ | $270°$ | $360°$ |
|---|---|---|---|---|---|
| $y$ | $0$ | $1$ | $0$ | $-1$ | $0$ |
Pattern: Starts at 0, peaks at $90°$, back to 0 at $180°$, troughs at $270°$, back to 0 at $360°$.
$y = \cos x$ (one full cycle: $0°$ to $360°$)#
| $x$ | $0°$ | $90°$ | $180°$ | $270°$ | $360°$ |
|---|---|---|---|---|---|
| $y$ | $1$ | $0$ | $-1$ | $0$ | $1$ |
Pattern: Starts at maximum (1), drops to 0 at $90°$, troughs at $180°$, back to 0 at $270°$, maximum at $360°$.
💡 $\cos x = \sin(x + 90°)$ — cosine is just sine shifted LEFT by $90°$.
$y = \tan x$ (one full cycle: $0°$ to $180°$)#
| $x$ | $0°$ | $45°$ | $90°$ | $135°$ | $180°$ |
|---|---|---|---|---|---|
| $y$ | $0$ | $1$ | undef | $-1$ | $0$ |
Pattern: Passes through 0, rises to $+\infty$, has asymptote at $90°$, comes from $-\infty$, passes through 0 at $180°$.
Asymptotes at $x = 90° + n \cdot 180°$ (i.e., $90°$, $270°$, $-90°$, …).
How to Sketch a Trig Graph — Step by Step#
Step 1: Identify the parameters#
Read $a$, $b$, $p$, $q$ from the equation.
Step 2: Calculate key values#
| Property | Formula |
|---|---|
| Amplitude | $\|a\|$ |
| Period | $\frac{360°}{\|b\|}$ (or $\frac{180°}{\|b\|}$ for tan) |
| Phase shift | $-\frac{p}{b}$ degrees |
| Vertical shift | $q$ |
| Maximum (sin/cos) | $q + \|a\|$ |
| Minimum (sin/cos) | $q - \|a\|$ |
Step 3: Find the five key points (for sin/cos)#
Divide one period into 4 equal intervals. The key points are at:
- Start, $\frac{1}{4}$ period, $\frac{1}{2}$ period, $\frac{3}{4}$ period, End
Apply the phase shift to each $x$-value.
Step 4: Draw#
- Draw the midline $y = q$ as a light dashed line.
- Plot the five key points.
- Draw asymptotes (for tangent) as dashed vertical lines.
- Connect with a smooth curve.
- Label all intercepts, turning points, and asymptotes.
Worked Example 1: Sine with Amplitude and Vertical Shift#
Sketch $y = 2\sin x + 1$ for $x \in [0°;\, 360°]$
Parameters: $a = 2$, $b = 1$, $p = 0$, $q = 1$
| Property | Value |
|---|---|
| Amplitude | $2$ |
| Period | $360°$ |
| Phase shift | $0°$ |
| Midline | $y = 1$ |
| Maximum | $1 + 2 = 3$ |
| Minimum | $1 - 2 = -1$ |
Key points:
| $x$ | $0°$ | $90°$ | $180°$ | $270°$ | $360°$ |
|---|---|---|---|---|---|
| $y$ | $1$ | $3$ | $1$ | $-1$ | $1$ |
Range: $y \in [-1;\, 3]$
Worked Example 2: Cosine with Period Change#
Sketch $y = \cos 2x$ for $x \in [0°;\, 360°]$
Parameters: $a = 1$, $b = 2$, $p = 0$, $q = 0$
| Property | Value |
|---|---|
| Amplitude | $1$ |
| Period | $\frac{360°}{2} = 180°$ |
| Phase shift | $0°$ |
| Midline | $y = 0$ |
This means two full cycles fit in $[0°;\, 360°]$.
Key points for first cycle ($0°$ to $180°$):
| $x$ | $0°$ | $45°$ | $90°$ | $135°$ | $180°$ |
|---|---|---|---|---|---|
| $y$ | $1$ | $0$ | $-1$ | $0$ | $1$ |
Second cycle repeats from $180°$ to $360°$.
Worked Example 3: Sine with Phase Shift#
Sketch $y = \sin(x - 30°)$ for $x \in [0°;\, 360°]$
Parameters: $a = 1$, $b = 1$, $p = -30°$, $q = 0$
| Property | Value |
|---|---|
| Amplitude | $1$ |
| Period | $360°$ |
| Phase shift | $-\frac{-30°}{1} = 30°$ to the RIGHT |
| Midline | $y = 0$ |
Key points: Take the standard sine points and shift each $x$-value RIGHT by $30°$:
| $x$ | $30°$ | $120°$ | $210°$ | $300°$ | $390°$ |
|---|---|---|---|---|---|
| $y$ | $0$ | $1$ | $0$ | $-1$ | $0$ |
Since we only plot to $360°$, the curve doesn’t quite finish its cycle — it ends between the trough and the final zero.
Worked Example 4: Reflected Cosine with All Parameters#
Sketch $y = -3\cos(2x + 60°) + 1$ for $x \in [-90°;\, 270°]$
Step 1 — Factor out $b$: $y = -3\cos(2(x + 30°)) + 1$
Parameters: $a = -3$, $b = 2$, $p = 60°$, $q = 1$
| Property | Value |
|---|---|
| Amplitude | $3$ |
| Period | $\frac{360°}{2} = 180°$ |
| Phase shift | $-\frac{60°}{2} = 30°$ LEFT |
| Midline | $y = 1$ |
| Maximum | $1 + 3 = 4$ |
| Minimum | $1 - 3 = -2$ |
Since $a < 0$: the cosine is reflected — it starts at a MINIMUM instead of a maximum.
Key points (standard cosine shifted left $30°$, reflected):
| $x$ | $-30°$ | $15°$ | $60°$ | $105°$ | $150°$ |
|---|---|---|---|---|---|
| $y$ (before reflection + shift) | $\cos(0°) = 1$ | $\cos(90°) = 0$ | $\cos(180°) = -1$ | $\cos(270°) = 0$ | $\cos(360°) = 1$ |
| $y$ (final: $-3 \times \text{above} + 1$) | $-2$ | $1$ | $4$ | $1$ | $-2$ |
Worked Example 5: Tangent with Period Change#
Sketch $y = \tan 2x$ for $x \in [0°;\, 180°]$
Parameters: $a = 1$, $b = 2$, $p = 0$, $q = 0$
| Property | Value |
|---|---|
| Period | $\frac{180°}{2} = 90°$ |
| Asymptotes | At $x = 45°$, $x = 135°$ (every $\frac{90°}{2} = 45°$ from $0°$, offset by half a period) |
Key points:
| $x$ | $0°$ | $22.5°$ | $45°$ | $67.5°$ | $90°$ |
|---|---|---|---|---|---|
| $y$ | $0$ | $1$ | undef | $-1$ | $0$ |
Two complete cycles in $[0°;\, 180°]$.
Finding the Equation from a Graph#
Strategy#
- Read $q$ from the midline: $q = \frac{\text{max} + \text{min}}{2}$
- Read $|a|$ from the amplitude: $|a| = \frac{\text{max} - \text{min}}{2}$. Check if reflected ($a < 0$).
- Read the period and calculate $b$: $b = \frac{360°}{\text{period}}$
- Read the phase shift by comparing to the standard shape.
Worked Example#
A graph has maximum $y = 5$, minimum $y = -1$, and completes one full cycle from $0°$ to $180°$.
$q = \frac{5 + (-1)}{2} = 2$
$|a| = \frac{5 - (-1)}{2} = 3$
Period = $180°$, so $b = \frac{360°}{180°} = 2$
If the graph starts at a maximum at $x = 0°$: it’s cosine with $a = 3$.
Equation: $y = 3\cos 2x + 2$
Reading Information from Trig Graphs#
Exam questions frequently ask:
| Question | How to answer |
|---|---|
| Amplitude | $\frac{\text{max} - \text{min}}{2}$ |
| Period | Horizontal distance for one full cycle |
| Range | $[q - \|a\|;\; q + \|a\|]$ |
| $f(x) = g(x)$ | Read the $x$-values where the graphs intersect |
| $f(x) > g(x)$ | Where $f$ is ABOVE $g$ — read the $x$-interval |
| $f(x) \cdot g(x) \leq 0$ | Where one graph is above and the other below the $x$-axis |
| Maximum of $f(x) - g(x)$ | Find where the vertical distance between $f$ and $g$ is greatest |
🚨 Common Mistakes#
- Phase shift calculation: In $y = \sin(2x - 60°)$, the shift is NOT $60°$. Factor out $b$: $y = \sin(2(x - 30°))$. The shift is $30°$ to the right.
- Period of tangent: Tan period is $\frac{180°}{|b|}$, NOT $\frac{360°}{|b|}$.
- Reflected graphs: If $a < 0$, the graph is FLIPPED. A reflected sine starts by going DOWN from the midline, not up. A reflected cosine starts at a MINIMUM.
- Not labelling turning points: In exams, you must label all maxima, minima, intercepts, and asymptotes with coordinates/equations.
- Drawing tan through asymptotes: The tangent curve approaches the asymptote but NEVER crosses it. Leave a clear gap.
- Confusing domain restriction with period: If you’re asked to sketch on $[0°;\, 360°]$ but the period is $180°$, you must draw TWO full cycles.
- Range for reflected graphs: The range is still $[q - |a|;\; q + |a|]$ regardless of reflection. The $a < 0$ only flips the shape, not the range boundaries.
💡 Pro Tip: The “5-Point” Method#
For sine and cosine, you only need 5 key points per cycle (start, quarter, half, three-quarter, end). Calculate these 5 points, plot them, and connect with a smooth curve. This guarantees an accurate sketch every time.
For tangent, plot the asymptotes first, then the zeros (midway between asymptotes), then one point between each zero and asymptote.
🔗 Related Grade 11 topics:
- Reduction Formulas & CAST — you need CAST to understand why sin/cos/tan behave differently in each quadrant
- Trig Identities & Equations — solving $f(x) = g(x)$ where $f$ and $g$ are trig graphs requires equation-solving skills
- Functions: The Parabola — trig graphs use the same domain/range/transformation language
📌 Grade 10 foundation: Trig Ratios & Special Angles — the special angle values power your key points
📌 Grade 12 extension: Trigonometry — compound and double angle identities change the shape of trig expressions