Section 4.5 Graphing Tangent and Cotangent (part 2)
Discovering the graphs of y = tanx and y = cotx: Answer the
following questions.
Fist, be sure your calculator is in radians and your xoom is in
ZTrig.
1. Graph y = tanx. Describe what the graph of y = tanx looks
like. How long is one period of the graph of y = tanx?
2. y = tyanx is an asymptotic function. Where are the
asymptotes located? Describe the domain of y = tanx.
3. Where will the graph of y = tanx have zeros?
4. What is the range of y = tanx?
5. Graph y = cotx. Describe what the graph of y = tanx looks
like. How long is one period of the graph of y = cotx?
6. y = cotx is also asymptotic. Where are its asymptotes
located? Describe the domain of y = cotx.
7. Where will the graph of y = cotx have zeros?
8. What is the range of y = cotx?
As you have seen….. Both the Tangent and Cotangent functions
are asymptotic.

Tangent is undefined for all x = 2 +nπ and Cotangent is
undefined for all x = 0 + nπ, where n is an integer.
Within each period, tangent goes down on the left (-∞) and up
on the right (∞). Cotangent does the opposite. The periods of
each are broken up into four parts, defined by 5 key points, like
those of sine and cosine. For tangent and cotangent, the 5 key
points are: asymptote, inflection point, intercept, inflection
point, asymptote.
Tangent: y = atan(bx ± c) ± d
 │a│ determines the distance between the points of
inflection and the x-axis.
 π/b is the period (horizontal stretch or shrink)
 c determines phase shift (left or right shift)
 to find the new starting point for a period, use the first

asymptote of y = tanx, which is x  2 .
 d determines vertical shift.
Cotangent: y = acot(bx ± c) ± d
 to find the new starting point for a period, use the first
asymptote of y = cotx, which is x = 0
Examples: Graph two full periods
1. y = (1/3)tanx
2. y = (1/4)tan3x
3. y = -2tanπx

4. y = 2cot(x - 2 )
Assignment: text p.401
#2,4,5,6,9,10,13,14,21,22,24,25,27,28,31,34,35,37,40,46,51
..and sketch the graphs of:
 y = 2tan(0.5x),


y


cot
x




2

 Y = 3tan2x + 1
Warm-up 04/25/2012
1. Evaluate cos(15π/2).
2. For what value of x, on [0,2π) would the graph of
y = csc(x – π/4) have a local maximum? A local minimum?
3. For the function given in #2, describe where its vertical
asymptotes would be located.
4. Use your graphing calculator. Graph y = secx and y = cscx
in the same viewing screen. For what values of x will the
graphs intersect?
5. Using the same graph from #4, describe how you could
obtain the graph of y = secx from the graph of y = cscx and
vice versa.