determine amplitude, period, phase shift, vertical shift, asymptotes, domain & range

Question

for the function

g(x)=3cot(x+pi/6)+2 ; [-2pi/3, 2pi/3].

Answer 1

Continued --->

Between these two asymptotes x = - π/6 and x = 5π/6, plot a few points, including the x - intercept, as shown in the table.

x

First plotting the asymptotes.

The midpoint between two consecutive vertical asymptotes is an x - intercept of the graph. The period of the function y = A cot [ B (x - h) ]  is the distance between two consecutive vertical asymptotes. The amplitude of a cotangent function is not defined.

After plotting the asymptotes and the x - intercept, plot a few additional points between the two asymptotes and sketch one cycle. Finally, sketch one or two additional cycles to the left and right.

Plot these five points and fill in the graph of y = 3 cot [ 1 (x + π/6) ].

A vertical shift up 2 units gives the final graph of y = 2 + 3 cot [ 1 (x + π/6) ] ; [2π/3, - 2π/3] as shown in Figure.

The domain of cotangent function, y = cot (x) is - ∞ < x < ∞, where x not equal to integer multiplies of π or x ≠ kπ.

Observe the graph, the domain is and range is the set of all real numbers.

Answer 2

The finction is f(x) = 3 cot(x + π/6) + 2.

Compare the equation f(x) =3 cot(x + π/6) + 2 with y = a cot(bx - c) + d.

a = 3, b = 1, c = - π/6 and d = 2.

1). Amplitude = | a | = | 3 | = 3

2). Period = π/b = π/1 = π.

3). Phase shift = c/b = (- π/6)/1 = - π/6.

4). Vertical shift = d = 2.

5).To find the asymptotes of the function, graph the function over a period.

The solutions of the given functions are

x + π/6 = 0 and x + π/6 = π

⇒ x = - π/6 and x = π - π/6 = 5π/6.

Taking π as an interval difference plot the graph.

x	f(x) =3 cot(x + π/6) + 2
- 2π	f(x) =3 cot( - 2π + π/6) + 2 = 3 cot(- 11π/6) = 3√3
- π	f(x) =3 cot( - π + π/6) + 2 = 3 cot(- 5π/6) = 3√3
0	f(x) =3 cot( 0 + π/6) + 2 = 3 cot(π/6) = 3√3
π	f(x) =3 cot( π + π/6) + 2 = 3 cot(7π/6) = 3√3
2π	f(x) =3 cot(2π + π/6) + 2 = 3 cot(13π/6) = 3√3

Now plot these points

.6).

From the graph we can also say the domain and range of the function.

Domain is nπ , where n is an iinteger, and n is not equals to zero.

The range is .

Answer 3

Let the function is y = g(x) = 3 cot (x + π/6) + 2.

Compare the equation y = 2 + 3 cot [ 1 (x + π/6) ] with y = k + A cot [ B (x - h) ] where B > 0.

k = 2, A = 3, B = 1 and h = - π/6.

Period = π/B = π/1 = π.

Horizontal translation = Phase shift = h = - π/6.

Vertical translation = k = 2.

For cotangent functions, there is no concept of amplitude since the range of the cotangent function is (- ∞ , ∞) or the set of all real numbers. The value of | A | is the factor by which the basic graphs are expand or contracted vertically. If A < 0 the graph will be reflected about the x - axis.

The period of y = tan x is π, so the period of y = k + A cot [ B (x - h) ] is π/b = π/(1) = π, caused by the horizontal compression of the graph by a factor of 1/b = 1/1 = 1.

First draw the graph of y = 3 cot [ 1 (x + π/6) ].

Two consecutive vertical asymptotes can be found by solving the equations B (x - h) = 0 and B (x - h) = π.

x + π/6 = 0 ------> x = - π/6 and

x + π/6 = π -------> x = π - π/6 -----> x = (6π - π)/6 -----> x = 5π/6.

The two consecutive vertical asymptotes occur at x = - π/6 and x = 5π/6.

The interval [- π/6, 5π/6] corresponds to one cycle of the graph. The cycle begins with - π/6 and ends with 5π/6 and find the three middle values. Dividing this interval into four equal parts produces the key points.

one fourth of cycle is [5π/6 - (- π/6) ]/4 = (6π/6)(1/4) = π/4.

The x - coordinates of the five key points are

x = - π/6.

x = - π/6 + π/4 = (- 2π + 3π)/12 = π/12.

x = π/12 + π/4 = (π + 3π)/12 = 4π/12 = π/3.

x = π/3 + π/4 = (4π + 3π)/12 = 7π/12.

x = 7π/12 + π/4 = (7π + 3π)/12 = 10π/12 = 5π/6.