INTRODUCTION
Graph lab
Everything covered in this lesson, in one place - useful for revision or printing.
\( y = a\sin k(x+p) + q \)
Every trig graph in Grade 11 starts as one of three base graphs — sin, cos or tan — and is then changed by four numbers: a, k, p and q.
Learn what each number does on its own and you can sketch any of these graphs with the same five-step recipe.
Have a play with the graph lab below. The faint dashed curve is always the base graph, so you can see exactly what changed. You will meet the sliders one at a time next.
\( y = \sin x \)
Start every sketch from the base graph. For \(y = \sin x\), learn these five key points on one period, from 0° to 360°:
The pattern is 0, up, 0, down, 0. The orange dots in the lab are these key points; the graph repeats them every 360°.
\( y = \cos x \)
Same shape as sine, but cosine starts at its maximum:
The pattern is top, 0, bottom, 0, top. Period 360°, amplitude 1, range \([-1;1]\) — exactly the same as sine.
\( y = \tan x = \dfrac{\sin x}{\cos x} \)
Wherever \(\cos x = 0\) (at 90°, 270°, …) you would divide by zero, so \(\tan x\) is undefined there. The graph has a vertical asymptote: it climbs towards the dashed line but never touches it.
Key points: \((-45^\circ;-1)\), \((0^\circ;0)\), \((45^\circ;1)\). The pattern repeats every 180°, not 360°.
\( \begin{array}{c|c|c} & \text{period} & \text{range} \\ \hline \sin x & 360^\circ & [-1;1] \\ \cos x & 360^\circ & [-1;1] \\ \tan x & 180^\circ & \text{all real} \end{array} \)
Sine and cosine both have amplitude 1. Tangent has no amplitude, because it never stops climbing.
Use the sin / cos / tan buttons in the lab to flip between the three base graphs until you can picture each one with your eyes closed.
\( y = \sin x + 2 \)
q is the number added on at the end. Make your prediction first, then drag q to check.
\( y = \cos x - 1 \)
Moving the graph moves its highest and lowest values too, so the range changes.
\( y = \tan x + 1 \)
Tangent moves up and down in exactly the same way. Think about the dashed asymptote lines before you drag.
\( \text{Range} = [\,q-|a|\,;\;q+|a|\,] \)
• q > 0 moves the graph up; q < 0 moves it down.
• The line \(y = q\) becomes the new “middle” of a sin or cos graph.
• q changes the range and the y-values of the key points. It never changes the shape, the period or the asymptotes.
\( y = 3\sin x \)
a multiplies every y-value of the base graph.
\( y = -2\cos x \)
Now try a negative value of a.
\( y = 2\sin x + 1 \)
A classic exam trap: when \(q \ne 0\), the maximum value and the amplitude are different numbers.
\( y = 2\tan x \)
Tangent has no amplitude, but a still stretches it.
\( \text{Amplitude} = |a| = \dfrac{\text{max} - \text{min}}{2} \)
• |a| > 1 stretches the graph vertically; |a| < 1 squashes it.
• a < 0 reflects the graph in the x-axis.
• sin and cos: amplitude = |a|. tan: no amplitude; a moves the points (45°; 1) to (45°; a).
• Never give the maximum value when the question asks for the amplitude.
\( y = \sin 2x \)
k multiplies x inside the function. It changes the period: how far along the x-axis one full wave takes.
\( y = \cos \tfrac{1}{2}x \)
What if k is smaller than 1?
\( y = \tan 2x \)
Careful — this is the most common period mistake in the exam.
\( \begin{gathered} \sin, \cos: \ \text{period} = \dfrac{360^\circ}{k} \\[4pt] \tan: \ \text{period} = \dfrac{180^\circ}{k} \end{gathered} \)
• k > 1 squashes the graph horizontally (more waves); 0 < k < 1 stretches it (fewer waves).
• k does not change the amplitude or the range.
• The key points are always one quarter of a period apart.
• tan: use 180°, never 360°.
\( y = \sin(x + 30^\circ) \)
p sits inside the bracket with x. To find where the graph moves, ask one question: what x makes the bracket zero?
\( y = \cos(x - 45^\circ) \)
Use the same question for a minus sign.
\( y = \tan(x - 45^\circ) \)
A horizontal shift moves everything sideways — including the asymptotes.
\( \sin(2x + 60^\circ) = \sin 2(x + 30^\circ) \)
When k and p appear together, take k out of the bracket before you read the shift. The bracket-zero question still works on either form.
\( x + p = 0 \;\Rightarrow\; x = -p \)
• Ask “what x makes the bracket zero?” That x-value is where the base pattern now starts.
• (x + 30°) moves the graph 30° left; (x − 30°) moves it 30° right.
• p moves the asymptotes of tan as well.
• If k is in the bracket, factor it out first.
\( y = 2\cos 2x - 1 \)
Sketch \(y = 2\cos 2x - 1\) for \(x \in [-180^\circ; 180^\circ]\). Every sketch follows the same five steps:
\( \text{Period} = \dfrac{360^\circ}{2} = 180^\circ \)
Step 1. k = 2, so the period is 180°.
Step 2. There is no p, so the bracket is zero at x = 0°. A quarter-period is 180° ÷ 4 = 45°, so the key x-values are:
\( y = 2 \times (\text{base}) - 1 \)
Step 3. Write the cosine pattern (1, 0, −1, 0, 1) under the x-values, multiply by a = 2, then add q = −1:
| x | 0° | 45° | 90° | 135° | 180° |
|---|---|---|---|---|---|
| cos 2x | 1 | 0 | −1 | 0 | 1 |
| y | 1 | −1 | −3 | −1 | 1 |
\( y = 2\cos 2x - 1, \ \ x \in [-180^\circ; 180^\circ] \)
Step 4. The interval is two periods long, so repeat the pattern to the left: (−45°; −1), (−90°; −3), (−135°; −1), (−180°; 1). Both end points are (−180°; 1) and (180°; 1).
Step 5. Join with a smooth curve — rounded at the turning points, never pointed. Label the graph and the end points.
\( y = \tan(x - 30^\circ) \)
For tan, sketch the asymptotes before anything else. On \([-180^\circ;180^\circ]\):
• Bracket zero: x = 30°, so the centre point is (30°; 0).
• Asymptotes are half a period (90°) either side: x = −60° and x = 120°.
• Key points a quarter-period (45°) either side: (−15°; −1) and (75°; 1).
• End points: (−180°; tan(−210°)) and (180°; tan 150°), both about −0,58 (tan repeats every 180°).
\( \begin{gathered} \text{period} \to \text{key } x \to \text{key } y \\ \to \text{plot} \to \text{join} \end{gathered} \)
• Key x-values are a quarter-period apart, starting where the bracket is zero.
• Key y-values: base pattern × a, then + q.
• Stay inside the given interval and plot both end points.
• tan: dashed asymptotes first, then the three key points.
Practise in the lab: pick a graph from your textbook, set the sliders, and check your hand sketch against it.
\( f(x) = \sin x \qquad g(x) = \cos 2x \)
Both graphs are drawn for \(x \in [-90^\circ; 270^\circ]\). Every “for which values of x” answer is built from the x-values where the graphs cross, so always find them first.
\( f(x) > g(x) \)
\(f(x) > g(x)\) means: where is the graph of f above the graph of g? Read it between the crossings.
\( f(x) \cdot g(x) \le 0 \)
A product is negative when one factor is positive and the other is negative. On the graph: one curve above the x-axis and the other below it. Because of the ≤, the product may also be 0, so the x-intercepts of either graph are included.
\( \tan x \ge 1, \quad x \in [0^\circ; 360^\circ] \)
Inequalities with tan add one more rule about brackets. Here the second “graph” is the horizontal line \(y = 1\).
\( \begin{gathered} \;\to\; (\ ) \qquad \le,\ \ge \;\to\; [\ ] \\[4pt] \text{asymptote} \;\to\; (\ ) \text{ always} \end{gathered} \)
\[ \begin{gathered} y = a\sin k(x+p) + q \\[4pt] \text{period } \tfrac{360^\circ}{k} \ \ (\tan\!: \tfrac{180^\circ}{k}) \\[4pt] \text{amplitude } |a| \\[4pt] \text{range } [\,q-|a|\,;\,q+|a|\,] \end{gathered} \]
The four mistakes to avoid: