Trigonometric Graphs (Grade 10)

The shapes of y = sin x, y = cos x and y = tan x on [0° ; 360°], and what a and q do in y = a sin x + q — amplitude, vertical shift, range and asymptotes.

1The two wave graphs

y = sin x and y = cos x on [0° ; 360°]. Both have amplitude 1 and period 360°.

y = sin x starts at 0, peaks at 90°, back to 0 at 180°, minimum at 270°.
y = cos x starts at 1, crosses at 90°, minimum at 180°, back to 1 at 360°.
Both have amplitude 1, period 360° and range −1 ≤ y ≤ 1.

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cos is just sin shifted 90° to the left. If you can draw one you can draw the other.

2The effect of a — amplitude

y = 2 sin x reaches 2 and −2: the amplitude doubles, the period does not change.

In y = a sin x + q amplitude = |a|

A negative a flips the graph in the x-axis. y = −sin x starts by going down instead of up.

3The effect of q — vertical shift

y = cos x + 1 is the whole cosine graph moved up 1 unit.

y = a sin x + q ⇒ range: q − |a| ≤ y ≤ q + |a|

Worked ExampleFor y = 2 sin x + 1, state the amplitude, period and range

1
Amplitude = |2| = 2.
2
Period = 360°.
Grade 10 does not change the period — that is Grade 11.
3
Range: 1 − 2 ≤ y ≤ 1 + 2.
Centre line is y = q = 1, reaching 2 either side.
4
−1 ≤ y ≤ 3.

⚠️

The maximum of y = 2 sin x + 1 is 3, not 2. Work out the amplitude and the shift separately, then combine.

4The tangent graph

y = tan x on [0° ; 360°], with asymptotes at 90° and 270°.

tan has no amplitude (it has no maximum or minimum), a period of 180°, and vertical asymptotes wherever cos x = 0 — at 90° and 270° in this interval.

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Draw the asymptotes as dashed lines before you draw the curve. Marks are awarded for them, and the shape becomes obvious once they are there.

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 x.
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360°
3
Write down the range of y = 2 sin x + 1.
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−1 ≤ y ≤ 3
4
Describe the transformation from y = cos x to y = cos x + 1.
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A shift of 1 unit upwards
5
Write down the equations of the asymptotes of y = tan x for x ∈ [0° ; 360°].
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x = 90° and x = 270°
6
Write down the maximum value of y = 3 cos x − 1.
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3 − 1 = 2
7
For what value of x in [0° ; 360°] does y = sin x reach its minimum?
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x = 270°
8
What does a negative value of a do to y = a sin x?
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It reflects the graph in the x-axis.