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y=-tg(x/3+pi/6)
Plotting a function graph/ y=-tg(x/3+pi/6)

Graphing y = y=-tg(x/3+pi/6)

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The graph:

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Intersection points:

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Piecewise:

The solution

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           /x   pi\
f(x) = -tan|- + --|
           \3   6 /
f(x)=−tan⁡(x3+π6)f{\left(x \right)} = - \tan{\left(\frac{x}{3} + \frac{\pi}{6} \right)} 
f = -tan(x/3 + pi/6)
The graph of the function
010203040506070-10-20002000
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
−tan⁡(x3+π6)=0- \tan{\left(\frac{x}{3} + \frac{\pi}{6} \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−π2x_{1} = - \frac{\pi}{2}
Numerical solution
x1=64.4026493985908x_{1} = 64.4026493985908
x2=54.9778714378214x_{2} = 54.9778714378214
x3=73.8274273593601x_{3} = 73.8274273593601
x4=−20.4203522483337x_{4} = -20.4203522483337
x5=102.101761241668x_{5} = 102.101761241668
x6=−10.9955742875643x_{6} = -10.9955742875643
x7=7.85398163397448x_{7} = 7.85398163397448
x8=−67.5442420521806x_{8} = -67.5442420521806
x9=−48.6946861306418x_{9} = -48.6946861306418
x10=−76.9690200129499x_{10} = -76.9690200129499
x11=−29.845130209103x_{11} = -29.845130209103
x12=17.2787595947439x_{12} = 17.2787595947439
x13=45.553093477052x_{13} = 45.553093477052
x14=83.2522053201295x_{14} = 83.2522053201295
x15=−58.1194640914112x_{15} = -58.1194640914112
x16=−39.2699081698724x_{16} = -39.2699081698724
x17=−86.3937979737193x_{17} = -86.3937979737193
x18=92.6769832808989x_{18} = 92.6769832808989
x19=−95.8185759344887x_{19} = -95.8185759344887
x20=36.1283155162826x_{20} = 36.1283155162826
x21=26.7035375555132x_{21} = 26.7035375555132
x22=−1.5707963267949x_{22} = -1.5707963267949
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to -tan(x/3 + pi/6).
−tan⁡(03+π6)- \tan{\left(\frac{0}{3} + \frac{\pi}{6} \right)}
The result:
f(0)=−33f{\left(0 \right)} = - \frac{\sqrt{3}}{3}
The point:
(0, -sqrt(3)/3)
Extrema of the function
In order to find the extrema, we need to solve the equation
ddxf(x)=0\frac{d}{d x} f{\left(x \right)} = 0
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
the first derivative
−tan⁡2(x3+π6)3−13=0- \frac{\tan^{2}{\left(\frac{x}{3} + \frac{\pi}{6} \right)}}{3} - \frac{1}{3} = 0
Solve this equation
Solutions are not found,
function may have no extrema
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
−2(tan⁡2(2x+π6)+1)tan⁡(2x+π6)9=0- \frac{2 \left(\tan^{2}{\left(\frac{2 x + \pi}{6} \right)} + 1\right) \tan{\left(\frac{2 x + \pi}{6} \right)}}{9} = 0
Solve this equation
The roots of this equation
x1=−π2x_{1} = - \frac{\pi}{2}

Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
(−∞,−π2]\left(-\infty, - \frac{\pi}{2}\right]
Convex at the intervals
[−π2,∞)\left[- \frac{\pi}{2}, \infty\right)
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(−tan⁡(x3+π6))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(- \tan{\left(\frac{x}{3} + \frac{\pi}{6} \right)}\right) = \left\langle -\infty, \infty\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
lim⁡x→∞(−tan⁡(x3+π6))=⟨−∞,∞⟩\lim_{x \to \infty}\left(- \tan{\left(\frac{x}{3} + \frac{\pi}{6} \right)}\right) = \left\langle -\infty, \infty\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of -tan(x/3 + pi/6), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(−tan⁡(x3+π6)x)=lim⁡x→−∞(−tan⁡(x3+π6)x)\lim_{x \to -\infty}\left(- \frac{\tan{\left(\frac{x}{3} + \frac{\pi}{6} \right)}}{x}\right) = \lim_{x \to -\infty}\left(- \frac{\tan{\left(\frac{x}{3} + \frac{\pi}{6} \right)}}{x}\right)
Let's take the limit
so,
inclined asymptote equation on the left:
y=xlim⁡x→−∞(−tan⁡(x3+π6)x)y = x \lim_{x \to -\infty}\left(- \frac{\tan{\left(\frac{x}{3} + \frac{\pi}{6} \right)}}{x}\right)
lim⁡x→∞(−tan⁡(x3+π6)x)=lim⁡x→∞(−tan⁡(x3+π6)x)\lim_{x \to \infty}\left(- \frac{\tan{\left(\frac{x}{3} + \frac{\pi}{6} \right)}}{x}\right) = \lim_{x \to \infty}\left(- \frac{\tan{\left(\frac{x}{3} + \frac{\pi}{6} \right)}}{x}\right)
Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(−tan⁡(x3+π6)x)y = x \lim_{x \to \infty}\left(- \frac{\tan{\left(\frac{x}{3} + \frac{\pi}{6} \right)}}{x}\right)
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-x) и f = -f(-x).
So, check:
−tan⁡(x3+π6)=−cot⁡(x3+π3)- \tan{\left(\frac{x}{3} + \frac{\pi}{6} \right)} = - \cot{\left(\frac{x}{3} + \frac{\pi}{3} \right)}
- No
−tan⁡(x3+π6)=cot⁡(x3+π3)- \tan{\left(\frac{x}{3} + \frac{\pi}{6} \right)} = \cot{\left(\frac{x}{3} + \frac{\pi}{3} \right)}
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = y=-tg(x/3+pi/6)
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