Graph of Tangent

The tangent graph models slope, cyclic ratios, and asymptotic behaviour. Understanding its domain is essential for equations and inverse tangent.

Intuition and core definition

The tangent function is $\tan x=\sin x/\cos x$. Its graph has period $\pi$, zeros where $\sin x=0$, and vertical asymptotes where $\cos x=0$. On each interval between asymptotes, $y=\tan x$ increases from negative to positive infinity.

Notation, language, and conditions

For $y=a\tan(b(x-h))+k$ with $b\ne0$, period is $\pi/|b|$, phase shift is $h$, midline is $y=k$, and vertical scale or reflection comes from $a$. Asymptotes occur when $b(x-h)=\pi/2+n\pi$. The range is all real numbers when $a\ne0$; if $a=0$, the graph is the constant $y=k$ on the tangent expression's domain.

Why this idea matters

The tangent graph records repeating slopes, with zeros at horizontal directions and vertical asymptotes where cosine vanishes.

A dependable method

1. Identify $a,b,h,k$ from a grouped form.
2. Compute period $\pi/|b|$ and locate a central crossing at $(h,k)$.
3. Place nearest asymptotes half a period on either side of $h$.
4. Use quarter-period points where base tangent is $\pm1$, scaled by $a$.
5. Repeat periodically and check asymptote/domain behaviour.

Worked example

Representations and interpretation

On the unit circle, tangent is slope $y/x$ and becomes undefined when $x$-coordinate (cosine) is zero. The graph’s asymptotes record those missing angles; periodic branches record repeated slopes after half-turns.

Reasoning about variations

Unlike sine and cosine, tangent has no maximum or minimum and no amplitude. Its period is half the $2\pi$ period of sine/cosine because a terminal ray reversed by $\pi$ has the same slope.

Common mistakes

How to check your work

• Substitute the central crossing and quarter-period inputs.
• Confirm undefined points solve $\cos(b(x-h))=0$.
• Verify branches repeat after one period.

Practice

1. Find the period of $y=\tan(4x)$.
2. Where are the nearest asymptotes of $y=\tan x$ around zero?
3. What is the range of $y=\tan x$?

Answers and brief solutions

Synthesis and transfer

A periodic slope signal can be modelled branch by branch; locating denominator zeros before plotting prevents the curve from being drawn through a forbidden input.

For $y=2\tan(3(x-\pi/6))+1$, each branch is centred at $(\pi/6,1)$ and repeats every $\pi/3$. The nearest vertical asymptotes lie half a period away, at $x=0$ and $x=\pi/3$. Quarter-period points have tangent arguments $\pm\pi/4$, so their outputs are $-1$ and $3$. No curve crosses an asymptote because those inputs make cosine zero. If the vertical factor were zero, the displayed formula would equal the constant midline wherever the tangent expression is defined, rather than having all real outputs. Identifying domain gaps before sketching prevents a constant-looking special case from being misdescribed.

Related topics

• Tangent
• Sinusoidal Functions
• Reciprocal Trigonometric Functions

Teaching and accessibility note