Trigonometric Graphs (Grade 11)

y = a sin(kx + p), y = a cos(kx + p) and y = a tan(kx + p) — amplitude, period, horizontal shift, asymptotes, and reading the graphs to answer questions.

1The three parent graphs

y = sin x · amplitude 1, period 360°
y = cos x · amplitude 1, period 360°
y = tan x · no amplitude, period 180°

2The effect of a — amplitude

y = sin x compared with y = 2 sin x. The amplitude doubles; the period is unchanged.

Amplitude = |a|. It stretches the graph vertically. A negative a flips it in the x-axis.

3The effect of k — period

period of sin and cos = 360°|k| · period of tan = 180°|k|

y = sin x compared with y = sin 2x. Doubling k halves the period to 180°.

⚠️

k does not change the amplitude and a does not change the period. Keep the two effects separate.

4The effect of p — horizontal shift

y = cos x compared with y = cos(x + 60°) — a shift of 60° to the LEFT.

y = cos(x + 60°) moves left by 60°; y = cos(x − 60°) moves right. The sign inside the bracket is the opposite of the direction you might expect.

5The tangent graph

y = tan x on [−360° ; 360°], with asymptotes at −270°, −90°, 90° and 270°.

tan has asymptotes every 180°, starting at 90°, and no maximum or minimum — so it has no amplitude and its range is y ∈ ℝ.

6Reading a trig graph

Worked ExampleFor y = 2 sin x on [−360° ; 360°], state amplitude, period and range

1
Amplitude = |2| = 2.
The graph reaches 2 and −2.
2
Period = 360°1 = 360°.
k = 1.
3
Range: −2 ≤ y ≤ 2.
Between the two extremes, endpoints included.

Worked ExampleFor y = sin 2x, state the period and the number of cycles in [0° ; 360°]

1
Period = 360°2 = 180°.
2
360° ÷ 180° = 2 complete cycles.
Confirmed on the sketch above.

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Always write the period and amplitude down before you draw. Then plot the key points every quarter-period — that is where the maxima, minima and x-intercepts sit.

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
Write down the amplitude of y = 2 sin x.
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2
2
Write down the period of y = sin 2x.
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360°2 = 180°
3
Write down the range of y = 2 sin x.
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−2 ≤ y ≤ 2
4
Describe the transformation from y = cos x to y = cos(x + 60°).
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A shift of 60° to the left
5
Write down the period of y = tan 2x.
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180°2 = 90°
6
Write down the equations of the asymptotes of y = tan x for x ∈ [0° ; 360°].
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x = 90° and x = 270°
7
How many complete cycles does y = sin 2x complete in [0° ; 360°]?
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2
8
Write down the range of y = tan x.
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y ∈ ℝ