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Graph of Tan
y=tan x has vertical asymptotes where cos x=0 (x=pi/2 + k pi). In each open interval between asymptotes it runs over all reals, through 0 at k pi. Period pi. y=tan(cx) period pi/|c|, asymptotes when cx=pi/2+k pi. Meaning: tan is a repeating family of branches, not a bounded wave. Do not draw it like a sine.
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Graph of Tan
What is it?
y=tan x has vertical asymptotes where cos x=0 (x=pi/2 + k pi). In each open interval between asymptotes it runs over all reals, through 0 at k pi. Period pi. y=tan(cx) period pi/|c|, asymptotes when cx=pi/2+k pi. Meaning: tan is a repeating family of branches, not a bounded wave. Do not draw it like a sine.
In simple words
- y=tan x has vertical asymptotes where cos x=0 (x=pi/2 + k pi).
- In each open interval between asymptotes it runs over all reals, through 0 at k pi.
- Period pi.
- tan is unbounded. Do not give it amplitude 1.
Important terms
GRAPH OF TAN: y=tan x has vertical asymptotes where cos x=0 (x=pi/2 + k pi). In each open interval between asymptotes it runs over all reals, through 0 at k pi. Period pi. y=tan(cx) period pi/|c|, asymptotes when cx=pi/2+k pi. Meaning: tan is a repeating family of branches, not a bounded wave. Do not draw it like a sine.
1. PARENT TAN: Through origin. Up to +inf as x->(pi/2)-. From -inf just after -pi/2. Next copy shifted by pi. Never a horizontal amplitude.
2. tan(cx+d): Find asymptotes by setting inside = pi/2 + k pi. Period pi/|c|. Sketch one branch, then repeat.
What you should learn
- State the main ideas and notation used in graph of tan.
- Apply the rules of graph of tan in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Examples
- Asymptote pi/2
- Zero at 0, pi, ...
- Period pi
PARENT TAN
Through origin. Up to +inf as x->(pi/2)-. From -inf just after -pi/2. Next copy shifted by pi. Never a horizontal amplitude.
tan(cx+d)
Find asymptotes by setting inside = pi/2 + k pi. Period pi/|c|. Sketch one branch, then repeat.
Recommended for Class 11
Full school-level explanation
y=tan x has vertical asymptotes where cos x=0 (x=pi/2 + k pi). In each open interval between asymptotes it runs over all reals, through 0 at k pi. Period pi. y=tan(cx) period pi/|c|, asymptotes when cx=pi/2+k pi. Meaning: tan is a repeating family of branches, not a bounded wave. Do not draw it like a sine.
Branches, then a stretch:
WHAT — GRAPH OF TAN
y=tan x has vertical asymptotes where cos x=0 (x=pi/2 + k pi). In each open interval between asymptotes it runs over all reals, through 0 at k pi. Period pi. y=tan(cx) period pi/|c|, asymptotes when cx=pi/2+k pi. Meaning: tan is a repeating family of branches, not a bounded wave. Do not draw it like a sine.
RULE — 1. PARENT TAN
Through origin. Up to +inf as x->(pi/2)-. From -inf just after -pi/2. Next copy shifted by pi. Never a horizontal amplitude.
METHOD — 2. tan(cx+d)
Find asymptotes by setting inside = pi/2 + k pi. Period pi/|c|. Sketch one branch, then repeat.
WORKED EXAMPLE
Worked: period and first positive asymptote of tan(2x). T = pi / 2 2x = pi/2 -> x = pi/4 asymptote x=pi/4 (and pi/4 + k pi/2)
PARENT TAN
Through origin. Up to +inf as x->(pi/2)-. From -inf just after -pi/2. Next copy shifted by pi. Never a horizontal amplitude.
Remember
- unbounded
Examples
- Asymptote pi/2
- Zero at 0, pi, ...
- Period pi
tan(cx+d)
Find asymptotes by setting inside = pi/2 + k pi. Period pi/|c|. Sketch one branch, then repeat.
Remember
- solve for holes
Examples
- x = pi/2 + k pi - dc
- One branch then copy
- Not between -1 and 1
How to answer a graph of tan question
How it works
- Identify the given quantities, notation, and required result.
- Choose the matching rule and write it before substitution.
- Work one algebraic step at a time, keeping brackets and signs clear.
- Check restrictions, simplify the result, and verify it where possible.
Remember
- asymptotes at odd pi/2
- period pi
- T=pi/2, x=pi/4
- no amplitude for tan
- unbounded
- solve for holes
- asymptotes at odd pi/2
- period pi
How the ideas connect
Worked: period and first positive asymptote of tan(2x). T = pi / 2 2x = pi/2 -> x = pi/4 asymptote x=pi/4 (and pi/4 + k pi/2)
Deeper understanding
Go beyond memorising Graph of Tan. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked: period and first positive asymptote of tan(2x). T = pi / 2 2x = pi/2 -> x = pi/4 asymptote x=pi/4 (and pi/4 + k pi/2)
Common mistakes
Exam focus
Core Idea
1. PARENT TAN
2. tan(cx+d)
Worked Example
Exam focus
Quick revision
- asymptotes at odd pi/2
- period pi
- T=pi/2, x=pi/4
- no amplitude for tan
- GRAPH OF TAN: y=tan x has vertical asymptotes where cos x=0 (x=pi/2 + k pi). In each open interval between asympto
- 1. PARENT TAN: Through origin. Up to +inf as x->(pi/2)-. From -inf just after -pi/2. Next copy shifted by pi. Never
tan is unbounded. Do not give it amplitude 1.
For long answers on Graph of Tan: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- asymptotes at odd pi/2
- period pi
- T=pi/2, x=pi/4
- no amplitude for tan
Important formulas
- asymptotes at odd pi/2
- period pi
- T=pi/2, x=pi/4
Common exam traps
- tan is unbounded. Do not give it amplitude 1.
- Do not use a remembered formula before checking that its conditions fit the question.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
tan is unbounded. Do not give it amplitude 1.
- Mistake
Do not use a remembered formula before checking that its conditions fit the question.
- Mistake
Do not hide sign changes or restrictions; write them on the line where they are used.
- Mistake
tan is unbounded. Do not give it amplitude 1.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
asymptotes at odd pi/2
period pi
T=pi/2, x=pi/4
no amplitude for tan
y=tan x has vertical asymptotes where cos x=0 (x=pi/2 + k pi). In each open interval between asymptotes it runs over all reals, through 0 at k pi. Period pi. y=
T=pi/2, x=pi/4
- Definition
- y=tan x has vertical asymptotes where cos x=0 (x=pi/2 + k pi). In each open interval between asymptotes it runs over all reals, through 0 at k pi. Period pi. y=
- Formula
- T=pi/2, x=pi/4
- Example
- Asymptote pi/2
- Critical warning
- tan is unbounded. Do not give it amplitude 1.
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Exam Practice
Define
Q1. Define Graph of Tan.
Explain
Q2. Explain: PARENT TAN.
Example
Q3. Give one clear example of Graph of Tan and say why it fits.
Mistake
Q4. State one common mistake students make about Graph of Tan.
Exam note
Q5. Write a short exam note on Graph of Tan: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.