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tgx+2
Plotting a function graph/ tgx+2

Graphing y = tgx+2

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The graph:

from to

Intersection points:

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Piecewise:

The solution

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f(x) = tan(x) + 2
f(x)=tan⁡(x)+2f{\left(x \right)} = \tan{\left(x \right)} + 2 
f = tan(x) + 2
The graph of the function
0-20-101020304050607080-200200
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)+2=0\tan{\left(x \right)} + 2 = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−atan⁡(2)x_{1} = - \operatorname{atan}{\left(2 \right)}
Numerical solution
x1=14.6008145501549x_{1} = 14.6008145501549
x2=83.7158529291303x_{2} = 83.7158529291303
x3=−29.3814826001022x_{3} = -29.3814826001022
x4=30.3087778181038x_{4} = 30.3087778181038
x5=2.0344439357957x_{5} = 2.0344439357957
x6=99.4238161970793x_{6} = 99.4238161970793
x7=−19.9567046393328x_{7} = -19.9567046393328
x8=−101.638113632667x_{8} = -101.638113632667
x9=−13.6735193321533x_{9} = -13.6735193321533
x10=−4.24874137138388x_{10} = -4.24874137138388
x11=−57.6558164824104x_{11} = -57.6558164824104
x12=−117.346076900616x_{12} = -117.346076900616
x13=39.7335557788732x_{13} = 39.7335557788732
x14=−48.231038521641x_{14} = -48.231038521641
x15=−35.6646679072818x_{15} = -35.6646679072818
x16=74.2910749683609x_{16} = 74.2910749683609
x17=−16.8151119857431x_{17} = -16.8151119857431
x18=61.7247043540018x_{18} = 61.7247043540018
x19=−63.93900178959x_{19} = -63.93900178959
x20=−70.2221870967695x_{20} = -70.2221870967695
x21=−26.2398899465124x_{21} = -26.2398899465124
x22=−79.6469650575389x_{22} = -79.6469650575389
x23=68.0078896611814x_{23} = 68.0078896611814
x24=52.2999263932324x_{24} = 52.2999263932324
x25=−7.39033402497368x_{25} = -7.39033402497368
x26=−51.3726311752308x_{26} = -51.3726311752308
x27=−126.770854861386x_{27} = -126.770854861386
x28=36.5919631252834x_{28} = 36.5919631252834
x29=−76.5053724039491x_{29} = -76.5053724039491
x30=80.5742602755405x_{30} = 80.5742602755405
x31=17.7424072037447x_{31} = 17.7424072037447
x32=24.0255925109243x_{32} = 24.0255925109243
x33=58.583111700412x_{33} = 58.583111700412
x34=46.0167410860528x_{34} = 46.0167410860528
x35=96.2822235434895x_{35} = 96.2822235434895
x36=−89.0717430183083x_{36} = -89.0717430183083
x37=89.9990382363099x_{37} = 89.9990382363099
x38=−92.2133356718981x_{38} = -92.2133356718981
x39=8.31762924297529x_{39} = 8.31762924297529
x40=−41.9478532144614x_{40} = -41.9478532144614
x41=−85.9301503647185x_{41} = -85.9301503647185
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to tan(x) + 2.
tan⁡(0)+2\tan{\left(0 \right)} + 2
The result:
f(0)=2f{\left(0 \right)} = 2
The point:
(0, 2)
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)+2)=⟨−∞,∞⟩\lim_{x \to -\infty}\left(\tan{\left(x \right)} + 2\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)+2)=⟨−∞,∞⟩\lim_{x \to \infty}\left(\tan{\left(x \right)} + 2\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) + 2, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(tan⁡(x)+2x)=lim⁡x→−∞(tan⁡(x)+2x)\lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)} + 2}{x}\right) = \lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)} + 2}{x}\right)
Let's take the limit
so,
inclined asymptote equation on the left:
y=xlim⁡x→−∞(tan⁡(x)+2x)y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)} + 2}{x}\right)
lim⁡x→∞(tan⁡(x)+2x)=lim⁡x→∞(tan⁡(x)+2x)\lim_{x \to \infty}\left(\frac{\tan{\left(x \right)} + 2}{x}\right) = \lim_{x \to \infty}\left(\frac{\tan{\left(x \right)} + 2}{x}\right)
Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(tan⁡(x)+2x)y = x \lim_{x \to \infty}\left(\frac{\tan{\left(x \right)} + 2}{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)+2=−tan⁡(x)+2\tan{\left(x \right)} + 2 = - \tan{\left(x \right)} + 2
- No
tan⁡(x)+2=tan⁡(x)−2\tan{\left(x \right)} + 2 = \tan{\left(x \right)} - 2
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = tgx+2
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