Tangent Function Graph
Explore the unique shape of the tangent graph, understand its asymptotes and period, and learn how transformations change the function.
The Basic Tangent Graph
The function y = tan x is fundamentally different from sine and cosine. Since tan x = sin xcos x, it is undefined wherever cos x = 0.
y = tan x
The dashed red lines show vertical asymptotes at x = π/2, 3π/2, etc.
Period = π
Unlike sin and cos (period 2π), the tangent function repeats every π units.
No Amplitude
The tangent function has no maximum or minimum value — its range is (−∞, +∞).
Asymptotes
Vertical asymptotes at x = π/2 + nπ (where n is any integer).
x-intercepts
The graph crosses the x-axis at x = nπ (where n is any integer).
Transformations of tan x
The general form of the tangent function is: y = a tan(bx − c) + d
| Parameter | Effect | Formula |
|---|---|---|
| a | Vertical stretch/compression. If negative, reflects in x-axis. | Dilation factor = |a| |
| b | Changes the period (how compressed horizontally). | Period = π/|b| |
| c | Horizontal shift (phase shift). | Shift = c/b |
| d | Vertical shift; moves the centre of the graph up or down. | Centre at y = d |
Important: The period of the tangent function is π/|b|, not 2π/|b| like sine and cosine. The asymptotes also shift accordingly.
Finding Asymptotes
For y = tan(bx − c), vertical asymptotes occur where the argument equals π/2 + nπ:
bx − c = π/2 + nπ
Solve for x to find each asymptote position.
For the basic y = tan x, this gives x = π/2 + nπ, i.e., x = ..., −3π/2, −π/2, π/2, 3π/2, ...
Key Vocabulary
Asymptote
A line that a curve approaches but never touches. For tan x, these are vertical lines.
Period
The horizontal distance for one full pattern. For tan x, the period is π (not 2π).
Inflection Point
The point where the tangent curve crosses the centre line, midway between asymptotes.
Undefined
A value that does not exist. tan x is undefined when cos x = 0 (division by zero).
Worked Examples
Find the period and asymptotes of y = tan(2x).
Step 1: Period = π/|b| = π/2.
Step 2: Asymptotes: 2x = π/2 + nπ, so x = π/4 + nπ/2.
Answer: Period = π/2. Asymptotes at x = π/4, 3π/4, 5π/4, ...
Describe the transformations of y = 3 tan(x − π/6).
Step 1: a = 3: Vertical stretch by factor 3 (steeper curve).
Step 2: b = 1: Period = π/1 = π (unchanged).
Step 3: Phase shift = π/6 to the right.
Answer: Stretched vertically by 3, shifted right by π/6, period π.
Find the first positive asymptote of y = tan(x − π/4).
Step 1: Set the argument equal to π/2: x − π/4 = π/2.
Step 2: Solve: x = π/2 + π/4 = 3π/4.
Answer: The first positive asymptote is at x = 3π/4.
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
What is the period of y = tan x?
Question 2
Where does the first positive vertical asymptote of y = tan x occur?
Question 3
What is the period of y = tan(3x)?
Question 4
What is the range of y = tan x?
Question 5
What is the value of tan(0)?
Key Concepts Summary
- ●tan x = sin x / cos x, and is undefined when cos x = 0.
- ●The period of y = tan(bx) is π/|b|.
- ●Vertical asymptotes occur at x = π/2 + nπ for the basic function.
- ●The tangent function has no amplitude — its range is all real numbers.
- ●The graph passes through the origin and has x-intercepts at x = nπ.