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y=tgx+1
Plotting a function graph/ y=tgx+1

Graphing y = y=tgx+1

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The graph:

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Intersection points:

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Piecewise:

The solution

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f(x) = tan(x) + 1
f(x)=tan⁡(x)+1f{\left(x \right)} = \tan{\left(x \right)} + 1 
f = tan(x) + 1
The graph of the function
05-20-15-10-5101520253035404550-200200
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⁡(x)+1=0\tan{\left(x \right)} + 1 = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−π4x_{1} = - \frac{\pi}{4}
Numerical solution
x1=−22.776546738526x_{1} = -22.776546738526
x2=40.0553063332699x_{2} = 40.0553063332699
x3=52.621676947629x_{3} = 52.621676947629
x4=−85.6083998103219x_{4} = -85.6083998103219
x5=58.9048622548086x_{5} = 58.9048622548086
x6=62.0464549083984x_{6} = 62.0464549083984
x7=−73.0420291959627x_{7} = -73.0420291959627
x8=−76.1836218495525x_{8} = -76.1836218495525
x9=−25.9181393921158x_{9} = -25.9181393921158
x10=−44.7676953136546x_{10} = -44.7676953136546
x11=30.6305283725005x_{11} = 30.6305283725005
x12=−98.174770424681x_{12} = -98.174770424681
x13=5.49778714378214x_{13} = 5.49778714378214
x14=−38.484510006475x_{14} = -38.484510006475
x15=−16.4933614313464x_{15} = -16.4933614313464
x16=−88.7499924639117x_{16} = -88.7499924639117
x17=14.9225651045515x_{17} = 14.9225651045515
x18=−101.316363078271x_{18} = -101.316363078271
x19=71.4712328691678x_{19} = 71.4712328691678
x20=−13.3517687777566x_{20} = -13.3517687777566
x21=−57.3340659280137x_{21} = -57.3340659280137
x22=−95.0331777710912x_{22} = -95.0331777710912
x23=−54.1924732744239x_{23} = -54.1924732744239
x24=−3.92699081698724x_{24} = -3.92699081698724
x25=90.3207887907066x_{25} = 90.3207887907066
x26=27.4889357189107x_{26} = 27.4889357189107
x27=46.3384916404494x_{27} = 46.3384916404494
x28=18.0641577581413x_{28} = 18.0641577581413
x29=−29.0597320457056x_{29} = -29.0597320457056
x30=2.35619449019234x_{30} = 2.35619449019234
x31=−82.4668071567321x_{31} = -82.4668071567321
x32=24.3473430653209x_{32} = 24.3473430653209
x33=−19.6349540849362x_{33} = -19.6349540849362
x34=−0.785398163397448x_{34} = -0.785398163397448
x35=96.6039740978861x_{35} = 96.6039740978861
x36=55.7632696012188x_{36} = 55.7632696012188
x37=11.7809724509617x_{37} = 11.7809724509617
x38=80.8960108299372x_{38} = 80.8960108299372
x39=84.037603483527x_{39} = 84.037603483527
x40=74.6128255227576x_{40} = 74.6128255227576
x41=−35.3429173528852x_{41} = -35.3429173528852
x42=99.7455667514759x_{42} = 99.7455667514759
x43=87.1791961371168x_{43} = 87.1791961371168
x44=49.4800842940392x_{44} = 49.4800842940392
x45=21.2057504117311x_{45} = 21.2057504117311
x46=−47.9092879672443x_{46} = -47.9092879672443
x47=33.7721210260903x_{47} = 33.7721210260903
x48=−60.4756585816035x_{48} = -60.4756585816035
x49=43.1968989868597x_{49} = 43.1968989868597
x50=93.4623814442964x_{50} = 93.4623814442964
x51=36.9137136796801x_{51} = 36.9137136796801
x52=−63.6172512351933x_{52} = -63.6172512351933
x53=−32.2013246992954x_{53} = -32.2013246992954
x54=−91.8915851175014x_{54} = -91.8915851175014
x55=−7.06858347057703x_{55} = -7.06858347057703
x56=−69.9004365423729x_{56} = -69.9004365423729
x57=65.1880475619882x_{57} = 65.1880475619882
x58=−41.6261026600648x_{58} = -41.6261026600648
x59=−10.2101761241668x_{59} = -10.2101761241668
x60=−51.0508806208341x_{60} = -51.0508806208341
x61=−79.3252145031423x_{61} = -79.3252145031423
x62=−66.7588438887831x_{62} = -66.7588438887831
x63=77.7544181763474x_{63} = 77.7544181763474
x64=68.329640215578x_{64} = 68.329640215578
x65=8.63937979737193x_{65} = 8.63937979737193
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to tan(x) + 1.
tan⁡(0)+1\tan{\left(0 \right)} + 1
The result:
f(0)=1f{\left(0 \right)} = 1
The point:
(0, 1)
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(x)+1=0\tan^{2}{\left(x \right)} + 1 = 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(x)+1)tan⁡(x)=02 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0

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
[0,∞)\left[0, \infty\right)
Convex at the intervals
(−∞,0]\left(-\infty, 0\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⁡(x)+1)=⟨−∞,∞⟩\lim_{x \to -\infty}\left(\tan{\left(x \right)} + 1\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⁡(x)+1)=⟨−∞,∞⟩\lim_{x \to \infty}\left(\tan{\left(x \right)} + 1\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) + 1, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(tan⁡(x)+1x)=lim⁡x→−∞(tan⁡(x)+1x)\lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)} + 1}{x}\right) = \lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)} + 1}{x}\right)
Let's take the limit
so,
inclined asymptote equation on the left:
y=xlim⁡x→−∞(tan⁡(x)+1x)y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)} + 1}{x}\right)
lim⁡x→∞(tan⁡(x)+1x)=lim⁡x→∞(tan⁡(x)+1x)\lim_{x \to \infty}\left(\frac{\tan{\left(x \right)} + 1}{x}\right) = \lim_{x \to \infty}\left(\frac{\tan{\left(x \right)} + 1}{x}\right)
Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(tan⁡(x)+1x)y = x \lim_{x \to \infty}\left(\frac{\tan{\left(x \right)} + 1}{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⁡(x)+1=−tan⁡(x)+1\tan{\left(x \right)} + 1 = - \tan{\left(x \right)} + 1
- No
tan⁡(x)+1=tan⁡(x)−1\tan{\left(x \right)} + 1 = \tan{\left(x \right)} - 1
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = y=tgx+1
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