MathematicsChapter 9: Trigonometric Functions
Graph of Tan
y=tan x has vertical asymptotes where cos x=0 (x=pi/2 + k pi).
Learning objectives
- 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.
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=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.
Mathematical form
T = pi / 2
asymptote x=pi/4 (and pi/4 + k pi/2)
y=tan x has vertical asymptotes where cos
x=0 (x
Where / symbols
- 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.
Overview
Branches, then a stretch:
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.
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.
Examples x = (pi/2 + k pi - d)/c · One branch then copy · Not between -1 and 1
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)
How to answer a graph of tan question
- 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.
Worked example
- 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)
In simple words
tan is unbounded. Do not give it amplitude 1.
Key points
- asymptotes at odd pi/2
- period pi
- unbounded
- solve for holes
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Graph of Tan
Sticky Notes · Quick Revision
Core Idea
- 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.
1. PARENT TAN
- Through origin.
- Up to +inf as x->(pi/2)-.
- From -inf just after -pi/2.
- Next copy shifted by pi.
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)
Common Misconception
- tan is unbounded. Do not give it amplitude 1.
FBISE Exam Check
- State the rule for graph of tan before substituting values.
- Show the main algebra steps, keep exact values where possible, and check every condition.
60-second revision
- 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.
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