Trigonometric Graphs — Grade 10 Maths (CAPS)

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.

5Working backwards: finding the equation of a given graph

Sketching a graph from an equation is half the skill. The ATP also asks you to find the equation of a given graph, and papers ask it constantly. Read off three things, in this order.

Which curve is it? A sine graph starts at its middle value and rises; a cosine graph starts at a maximum or minimum; a tangent graph has asymptotes.
q — the vertical shift. q = (maximum + minimum) ÷ 2. That is the value the curve oscillates about.
a — the amplitude. a = (maximum − minimum) ÷ 2. If the curve is flipped (a sine graph that goes down first), a is negative.

For y = a sinθ + q: a = (max − min)/2 (for a > 0) q = (max + min)/2

Worked ExampleA cosine graph starts at its maximum of 5 and has a minimum of −1. Determine its equation.

1
q = (5 + (−1))/2 = 2
The centre line of the curve.
2
a = (5 − (−1))/2 = 3
Half the total vertical spread.
3
The graph starts at its maximum, so a is positive.
A negative a would flip it to start at the minimum.
4
y = 3 cos θ + 2

Worked ExampleA sine graph has a range of [−4 ; 2] and falls from θ = 0°. Determine its equation.

1
max = 2 and min = −4
Read straight off the range.
2
q = (2 + (−4))/2 = −1
3
a = (2 − (−4))/2 = 3, but the graph FALLS first, so a = −3
A normal sine graph rises from its centre line. Falling means it is reflected.
4
y = −3 sin θ − 1

⚠️

Reading the amplitude as the maximum. For y = 3 cosθ + 2 the maximum is 5, not 3. Amplitude is the distance from the centre line to the maximum — and the centre line is only the θ-axis when q = 0.

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.
9
The graph of y = a sinθ + q, where a > 0, has a maximum value of 7 and a minimum value of 1. Determine the values of a and q, and write down the equation of the graph.
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q = (7 + 1)/2 = 4 ✓; a = (7 − 1)/2 = 3 ✓; y = 3 sinθ + 4 ✓
10
The graph of y = a cosθ + q, where a > 0, has a range of [−5 ; 3]. Determine a and q.
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q = (3 + (−5))/2 = −1 ✓; a = (3 − (−5))/2 = 4 ✓; so y = 4 cosθ − 1 ✓
11
A cosine graph passes through a MINIMUM at θ = 0° and has amplitude 2 with no vertical shift. Write down its equation and explain the sign of a.
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y = −2 cosθ ✓✓; a is negative because an ordinary cosine graph starts at a maximum — starting at a minimum means it has been reflected in the θ-axis ✓