Trig Graphs

Video lesson4 worked examples

Theory

You must be familiar with the basic shapes of $y = \sin x$ , $y = \cos x$ , and $y = \tan x$ graphs between $0^\circ$ and $360^\circ$ .

The General Equation of a Trigonometric Function is $y = a\sin(bx + c) + d$ .

a: Amplitude (half the distance between the maximum and minimum values).
b: Frequency (the number of complete waves within $360^\circ$ ).
c: Phase Angle (translation left or right).
d: Vertical Shift (translation up or down).

The Golden Rule: read the four numbers off a graph like this — $a = \frac{\text{max} - \text{min}}{2}$ (amplitude), $d = \frac{\text{max} + \text{min}}{2}$ (the midline), $b$ is the number of complete waves in $360^\circ$ , and $c$ is the horizontal shift.

⚠️ Common Examiner Traps

b is cycles, not period: if the wave repeats every $120^\circ$ then $b = 360 \div 120 = 3$ .
Shift direction: $(x + c)$ shifts the graph left, $(x - c)$ shifts it right — the opposite of the sign.
Amplitude is half: the amplitude is half the total height between max and min, not the whole height.
Max/min of the shape: a plain $\cos$ graph has its maximum at $0^\circ$ and minimum at $180^\circ$ ; scale and shift from there.

Worked examples

Example 1

State the amplitude and vertical shift of the graph $y = 3\cos x + 4$ .

The amplitude ( $a$ ) is 3. The graph is shifted up vertically by 4 units ( $d$ ).

Example 2

A sine graph has a maximum value of 5 and a minimum of -5. It completes two full cycles between $0^\circ$ and $360^\circ$ . State its equation.

Step 1: The amplitude is 5 ( $a = 5$ ).

Step 2: The frequency is 2 ( $b = 2$ ).

Answer: $y = 5\sin(2x)$ .

Example 3

The graph of $y = \cos(x - a)^\circ$ is shifted to the right by $45^\circ$ . What is the value of $a$ ?

A shift to the right means the phase angle is subtracted from $x$ .

Answer: $a = 45$ .

Example 4

🎯 Exam-style (turning point)

The graph of $y = 2\cos(x - 30)^\circ$ has a minimum turning point at A. State the coordinates of A.

Step 1: A cosine graph reaches its minimum when the angle inside equals $180^\circ$ . So set $x - 30 = 180$ :

x = 210

Step 2: The amplitude is 2, so the minimum value of $y$ is $-2$ .

Answer: A is $(210, -2)$ .

Ready to practise?

8 Trig Graphs questions with answers and video solutions