Sketch tangent graphs and label their asymptotes
Sketch translated tangent curves over the stated domain, using their period, centre points and vertical asymptotes.
Section 10 · Trigonometry
Cambridge IGCSE Additional Mathematics 0606, 2028–2030 syllabus, Version 1. Paper 1 is non-calculator; Paper 2 requires a scientific calculator. These original lessons provide self-checked topic practice, not a Cambridge grade or a timed paper. Full-paper assessment and human specialist review are still outstanding.
Before you begin
- Cambridge assumes knowledge of IGCSE Mathematics or an equivalent course. Check the domain before using an inverse or logarithm, and retain exact values where the question asks for them.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
At which input values is tan x undefined?
What you will learn
- Find the period and vertical translation of a tangent graph
- Locate and label every vertical asymptote within a stated domain
- Sketch the increasing or decreasing branches between consecutive asymptotes
Tangent repeats more often and has no amplitude
For y = a tan(bx) + c, the period is 180°/|b| in degrees or π/|b| in radians. The vertical translation is c, and the graph passes through its centre point (where bx is a multiple of π). A tangent graph has no maximum or minimum, so amplitude is not defined.
The vertical asymptotes occur when bx = 90° + 180°k or bx = π/2 + πk, where k is any integer. Solve for x, then retain only asymptotes inside the stated domain. When a 0606 question gives b as a fraction, its denominator is 2, 3, 4, 6 or 8.
Sketch each branch only on its domain
The 0606 form uses a positive integer a, so each branch increases between consecutive asymptotes. The coefficient a sets vertical stretch, while c moves the centre line to y = c.
Mark each asymptote with a dashed vertical line, place the centre and useful quarter-period points, and draw a smooth branch approaching the asymptotes without touching them.
Pause and explain
Does y = 3 tan(2x) + 5 have an amplitude of 3?
Worked example
Sketch y = 2 tan(2x) − 1 for −π/2 < x < π/2. Give its period, asymptotes and centre.
Show the worked solution
- The period is π/2. Asymptotes satisfy 2x = ±π/2 within this domain, so x = −π/4 and x = π/4.
- At x = 0, y = −1, so the centre point is (0,−1). At x = ±π/8, tan(2x) = ±1, giving y = 1 and −3.
- Draw the increasing branch through these points, approaching −∞ at the left asymptote and +∞ at the right.
Answer Period π/2; vertical asymptotes x = −π/4 and x = π/4; centre (0,−1).
Try a topic test
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Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Sketch y = 2 tan(3x) − 1 on −π/6 < x < π/6. Find its period, asymptotes, centre and points where 3x = ±π/4.
Give me a hint
Solve 3x = ±π/2 for the boundary asymptotes; for the interior points, use tan(±π/4) = ±1.
Compare my reasoning
- The period is π/3 and the asymptotes in the stated interval are x = −π/6 and x = π/6.
- The centre is (0,−1). At x = ±π/12, the tangent values are ±1.
- Therefore the two points are (−π/12,−3) and (π/12,1); join them in an increasing branch.
Period π/3; asymptotes x = ±π/6; centre (0,−1); points (−π/12,−3) and (π/12,1).
Look for these in your work
- Period computed
- Asymptotes lie at domain boundaries
- Vertical stretch and shift applied
For y = 2 tan(2x) + 3 on −π/2 < x < π/2, find all asymptotes and sketch the branch through its centre.
Give me a hint
The period is π/2; locate every asymptote and centre in the open interval.
Compare my reasoning
- Asymptotes satisfy 2x = π/2 + kπ, giving x = π/4 + kπ/2.
- Inside the domain these are x = −π/4 and x = π/4; the only centre x = kπ/2 inside the open domain is x = 0.
- The centre is (0,3). At x = ±π/8, tan(2x) = ±1, giving y = 1 and 5; draw the increasing branch between the asymptotes.
Period π/2; asymptotes x = −π/4 and π/4; centre (0,3), with points (−π/8,1) and (π/8,5).
Look for these in your work
- Asymptote family solved
- Open domain handled correctly
- Centre and vertical scale applied
A controller follows y = 3 tan(πx/4) + 2 for 0 < x < 8. Find the period, all asymptotes in the interval and the centre between them.
Give me a hint
Solve πx/4 = π/2 + kπ, and check that the asymptote is strictly inside the domain.
Compare my reasoning
- The period is π/(π/4) = 4.
- Asymptotes occur at x = 2 + 4k; in 0 < x < 8 they are x = 2 and x = 6.
- The branch between them is centred halfway at x = 4, where y = 2.
Period 4; asymptotes x = 2 and x = 6; centre (4,2).
Look for these in your work
- Period interpreted in x-units
- Only in-domain asymptotes included
- Centre and translation identified
Common mistakes
- Assigning a tangent graph an amplitude
- Finding asymptotes from bx = kπ instead of bx = π/2 + kπ
- Including an asymptote outside the stated domain
- Drawing a branch through or across a vertical asymptote
What can you explain now?
What is the period of y = tan(4x) for x in radians?
Compare with the explanation
π/4
Tangent has period π/|b| in radians.
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Make it stick
How do you find a tangent graph's period, asymptotes and centre before sketching its branches?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.