Graphing Tangent and Cotangent
One period of the graph of
is shown below. The vertical lines at
and
are vertical asymptotes for the graph. (These are lines that the graph cannot touch or cross.) These
asymptotes occur at the zeros of the cosine function, where the tangent function is undefined. The
tangent function is periodic with a period of  . (That is, tan(x   )  tan x .)
The interval
is considered to be a standard period for the tangent function. Besides the
asymptotes, important landmarks in this interval are the quarterpoints (
intercept at the midpoint (
) and (
) and the x-
. The tangent function is increasing between its asymptotes. The
domain of the tangent function is the set of all real numbers except for the odd multiples of , where it

has vertical asymptotes. (Formally, the domain is the set {x | x  (2k  1) } , where k is any integer.)
2
The range of the tangent function is all real numbers, the set { y |   y  } . The tangent graph has
no amplitude, since there is no maximum or minimum value.
The function
also has a period of π. A standard period, from x = 0 to x = π, is shown
below. The cotangent function has vertical asymptotes where the sine function equals 0, and it
decreases between the asymptotes. In the interval from 0 to π, the important landmarks besides the
asymptotes
and
are the quarterpoints (
) and (
) and the midpoint at (
).
The domain of the cotangent function is the set of all real numbers except the integer multiples of π
(the set {x | x  k } , where k is any integer.) The range is the set of all real numbers. Like the tangent
function, the cotangent function has no maximum or minimum value and therefore no amplitude.
The rules for transforming the graphs of sine and cosine:
Transforming the Graphs of Tangent and Cotangent
The graphs of
 
C 
y  k  A tan ( Bx  C )  k  A tan  B x   and
B 
 
 
C 
y  k  A cot(Bx  C )  k  A cot  B x   will have:
B 
 
1. Vertical stretch (or shrink) by |A|
If A< 0, the graph is reflected about the x-axis.
2. Period =

B
C
B
3. Phase Shift = 
4. Vertical Translation = k
EXAMPLE 1: Sketch the graph of y  2  4 tan x for 

2
x

2
.
and will be reflected about the x-axis if A < 0.
Solution: The graph will look exactly like the graph of the tangent function, except that it has been
stretched vertically by a factor of 4, reflected about the x-axis, and then translated 2 units up. The
transformations are shown in steps:
First, the stretch by a factor of 4 is shown by the values y  4 and y  4 at the quarterpoints.
8
4
2
4
4
2
-4
-8
Next, the graph is reflected about the x-axis and finally translated 2 units up. In the final graph, the
quarterpoints are at
and
changed. (They are still at
. The midpoint has moved to
and
8
4
2
4
4
-4
-8
2
. The asymptotes have not
EXAMPLE 2: Sketch one period of the graph of
Solution: This function has a period of
(
)
[ (
)].
and a phase shift of units to the right. Since the
standard period for cotangent is from 0 to π, one period for this function will go from
. There is no vertical stretch or shrink or vertical translation, so the midpoint is halfway
between the asymptotes at (
asymptote) are at (
) and (
), and the quarterpoints (halfway between the midpoint and each
).
or