The cotangent graph has vertical asymptotes at each value of
where
we show these in the graph below with dashed lines. Since the cotangent is the reciprocal of the tangent,
has vertical asymptotes at all values of
where
and
at all values of
where
has its vertical asymptotes.
The cotangent function
Features of the graph of
y =
A Cot(
Bx )
The stretching factor is
The period is
The domain is
where
is an integer.
The range is
The asymptotes occur at
where
is an integer.
is an odd function.
Graphing variations of
y = cot
x
We can transform the graph of the cotangent in much the same way as we did for the tangent. The equation becomes the following.
Properties of the graph of
y =
A Cot(
Bx −c)+
D
The stretching factor is
The period is
The domain is
where
is an integer.
The range is
The vertical asymptotes occur at
where
is an integer.
There is no amplitude.
is an odd function because it is the quotient of even and odd functions (cosine and sine, respectively)
Given a modified cotangent function of the form
graph one period.
Express the function in the form
Identify the stretching factor,
Identify the period,
Draw the graph of
Plot any two reference points.
Use the reciprocal relationship between tangent and cotangent to draw the graph of
Sketch the asymptotes.
Graphing variations of the cotangent function
Determine the stretching factor, period, and phase shift of
and then sketch a graph.
Step 1. Expressing the function in the form
gives
Step 2. The stretching factor is
Step 3. The period is
Step 4. Sketch the graph of
Step 5. Plot two reference points. Two such points are
and
Step 6. Use the reciprocal relationship to draw
Step 7. Sketch the asymptotes,
The orange graph in
[link] shows
and the blue graph shows
Given a modified cotangent function of the form
graph one period.
Express the function in the form
Identify the stretching factor,
Identify the period,
Identify the phase shift,
Draw the graph of
shifted to the right by
and up by
Sketch the asymptotes
where
is an integer.
Plot any three reference points and draw the graph through these points.
Graphing a modified cotangent
Sketch a graph of one period of the function
Step 1. The function is already written in the general form
Step 2.
so the stretching factor is 4.
Step 3.
so the period is
Step 4.
so the phase shift is
Step 5. We draw
Step 6-7. Three points we can use to guide the graph are
and
We use the reciprocal relationship of tangent and cotangent to draw
