Trigonometric Graphs and Transformations
Trigonometric graphs and transformations covers the shapes, periods and key points of y = sin(x), y = cos(x) and y = tan(x), and how transformations such as y = a sin(x) + d, y = sin(x + c) or y = sin(bx) stretch, translate or reflect these graphs. It builds on function transformations and trig identities, and Level 2 Further Maths questions can combine graph reasoning with solving equations for up to 7 marks.
Before you start
Make sure you're comfortable with these topics first:
Method
- Learn the key features of y = sin(x), y = cos(x) and y = tan(x): amplitude, period, and the coordinates of the maximum, minimum and intercepts.
- For y = a sin(x) + d, apply a vertical stretch by scale factor a, then a vertical translation by vector (0, d): the amplitude becomes |a| and the graph oscillates about y = d.
- For y = sin(x - c), apply a horizontal translation by vector (c, 0): every key point shifts c degrees to the right (or left if c is negative).
- For y = sin(bx), apply a horizontal stretch by scale factor 1/b parallel to the x-axis: the period becomes 360/b degrees.
- To find where a transformed graph meets a horizontal line or another curve, form and solve the resulting trig equation over the given range.
- Track a transformed maximum or minimum point by applying each transformation, in order, to the known key point of the original graph.
Worked example
The graph of y = sin(x) has a maximum point at (90, 1). Find the coordinates of the maximum point of y = 4sin(x) - 3.
- The transformation is a vertical stretch of scale factor 4, followed by a translation of -3 in the y-direction.
- The x-coordinate of the maximum is unaffected by these vertical transformations, so it stays at 90.
- Apply the stretch to the y-value: 4 x 1 = 4.
- Apply the translation: 4 - 3 = 1.
- So the maximum point of y = 4sin(x) - 3 is (90, 1).
Practice questions
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Q1State the period, in degrees, of y = cos(x).Show answer
Answer: 360 degrees
Q2State the amplitude of y = 6sin(x).Show answer
Answer: 6
Q3Describe fully the single transformation that maps y = cos(x) onto y = cos(x) + 4.Show answer
Answer: translation by vector (0, 4)
Q4State the period, in degrees, of y = sin(5x).Show answer
Answer: 72 degrees (360 divided by 5)
Q5Describe fully the single transformation that maps y = cos(x) onto y = 7cos(x).Show answer
Answer: stretch, scale factor 7, parallel to the y-axis
Q6The graph of y = sin(x) has a minimum point at (270, -1). Find the coordinates of the minimum point of y = 2sin(x) + 5.Show answer
Answer: (270, 3) (2 x -1 + 5 = 3)
Exam-style questions
Written in the style of a GCSE Further Maths exam paper, with a full mark scheme.
The graph of y = sin(x) is translated by vector (30, 0) and then stretched by scale factor 3 parallel to the y-axis, in that order. Write down the equation of the resulting graph.
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Solve cos(3x) = 0.5 for 0 <= x <= 360, giving all solutions.
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The graph of y = a sin(x) + b, where a > 0, has a maximum point at (90, 9) and a minimum point at (270, 1). Find the values of a and b.
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See real GCSE Further Maths past-paper questions, with official mark schemes →
Free printable worksheet
Want more practice on paper? Download the trigonometric graphs and transformations worksheet pack - 13 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
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