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Plotting a function graph/ sin(x+pi/6)+1

Graphing y = sin(x+pi/6)+1

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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          /    pi\    
f(x) = sin|x + --| + 1
          \    6 /
f(x)=sin⁡(x+π6)+1f{\left(x \right)} = \sin{\left(x + \frac{\pi}{6} \right)} + 1 
f = sin(x + pi/6) + 1
The graph of the function
-3.0-2.5-2.0-1.5-1.0-0.53.00.00.51.01.52.02.504
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:
sin⁡(x+π6)+1=0\sin{\left(x + \frac{\pi}{6} \right)} + 1 = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−2π3x_{1} = - \frac{2 \pi}{3}
x2=4π3x_{2} = \frac{4 \pi}{3}
Numerical solution
x1=−39.7935064774491x_{1} = -39.7935064774491
x2=−77.4926189323297x_{2} = -77.4926189323297
x3=−96.3421752381095x_{3} = -96.3421752381095
x4=54.4542726557069x_{4} = 54.4542726557069
x5=54.4542731772212x_{5} = 54.4542731772212
x6=16.75516131708x_{6} = 16.75516131708
x7=98.4365697036444x_{7} = 98.4365697036444
x8=29.3215317794356x_{8} = 29.3215317794356
x9=85.8701992979325x_{9} = 85.8701992979325
x10=67.0206429182693x_{10} = 67.0206429182693
x11=−14.6607653773517x_{11} = -14.6607653773517
x12=10.4719753854359x_{12} = 10.4719753854359
x13=23.0383465634668x_{13} = 23.0383465634668
x14=23.0383457618207x_{14} = 23.0383457618207
x15=−90.0589898961183x_{15} = -90.0589898961183
x16=−77.4926186337816x_{16} = -77.4926186337816
x17=−33.510321121959x_{17} = -33.510321121959
x18=85.8701993668955x_{18} = 85.8701993668955
x19=−2.09439487552071x_{19} = -2.09439487552071
x20=73.3038281387027x_{20} = 73.3038281387027
x21=73.3038289361328x_{21} = 73.3038289361328
x22=−83.7758040750887x_{22} = -83.7758040750887
x23=35.604716971914x_{23} = 35.604716971914
x24=35.6047158906949x_{24} = 35.6047158906949
x25=−33.5103215424455x_{25} = -33.5103215424455
x26=−14.6607661705898x_{26} = -14.6607661705898
x27=98.4365697615429x_{27} = 98.4365697615429
x28=60.7374584698885x_{28} = 60.7374584698885
x29=79.5870133918867x_{29} = 79.5870133918867
x30=−46.0766920003146x_{30} = -46.0766920003146
x31=54.4542725444797x_{31} = 54.4542725444797
x32=41.8879021371507x_{32} = 41.8879021371507
x33=−71.2094329890238x_{33} = -71.2094329890238
x34=−52.3598781878037x_{34} = -52.3598781878037
x35=−2537730.37448219x_{35} = -2537730.37448219
x36=4.18878981363265x_{36} = 4.18878981363265
x37=−58.6430633250402x_{37} = -58.6430633250402
x38=−2.09439560156919x_{38} = -2.09439560156919
x39=79.5870129226152x_{39} = 79.5870129226152
x40=48.1710874369572x_{40} = 48.1710874369572
x41=10.4719760277638x_{41} = 10.4719760277638
x42=−46.0766921860726x_{42} = -46.0766921860726
x43=35.6047162396481x_{43} = 35.6047162396481
x44=−52.3598780642478x_{44} = -52.3598780642478
x45=79.5870141299736x_{45} = 79.5870141299736
x46=−20.9439513945345x_{46} = -20.9439513945345
x47=−20.9439505916939x_{47} = -20.9439505916939
x48=−83.7758044339049x_{48} = -83.7758044339049
x49=−8.37758018623408x_{49} = -8.37758018623408
x50=41.8879015462722x_{50} = 41.8879015462722
x51=−14.6607653512222x_{51} = -14.6607653512222
x52=85.870198694467x_{52} = 85.870198694467
x53=−96.3421745028061x_{53} = -96.3421745028061
x54=−90.0589893488743x_{54} = -90.0589893488743
x55=92.1533846224853x_{55} = 92.1533846224853
x56=41.8879022469632x_{56} = 41.8879022469632
x57=48.1710878089946x_{57} = 48.1710878089946
x58=60.7374577215862x_{58} = 60.7374577215862
x59=−33.5103217734977x_{59} = -33.5103217734977
x60=48.1710869517551x_{60} = 48.1710869517551
x61=92.1533849512619x_{61} = 92.1533849512619
x62=−46.0766927490392x_{62} = -46.0766927490392
x63=−83.775803621544x_{63} = -83.775803621544
x64=10.4719755542211x_{64} = 10.4719755542211
x65=−90.058989127203x_{65} = -90.058989127203
x66=−96.342175215893x_{66} = -96.342175215893
x67=−71.2094337514317x_{67} = -71.2094337514317
x68=−8.37758091239869x_{68} = -8.37758091239869
x69=29.3215315503859x_{69} = 29.3215315503859
x70=16.7551615795837x_{70} = 16.7551615795837
x71=16.7551602282542x_{71} = 16.7551602282542
x72=−27.227135835892x_{72} = -27.227135835892
x73=−64.9262477469699x_{73} = -64.9262477469699
x74=−58.6430625342279x_{74} = -58.6430625342279
x75=−52.359877344473x_{75} = -52.359877344473
x76=29.3215309839921x_{76} = 29.3215309839921
x77=−77.492618271937x_{77} = -77.492618271937
x78=−8.37758114523565x_{78} = -8.37758114523565
x79=67.0206437185101x_{79} = 67.0206437185101
x80=−83.7758044593608x_{80} = -83.7758044593608
x81=−2.09439502413559x_{81} = -2.09439502413559
x82=−39.7935069083105x_{82} = -39.7935069083105
x83=4.18879026163738x_{83} = 4.18879026163738
x84=−39.793507323436x_{84} = -39.793507323436
x85=−46.0766933186229x_{85} = -46.0766933186229
x86=98.4365703264027x_{86} = 98.4365703264027
x87=16.7551605638403x_{87} = 16.7551605638403
x88=4.18879066607671x_{88} = 4.18879066607671
x89=−108.908544294949x_{89} = -108.908544294949
x90=60.7374591685857x_{90} = 60.7374591685857
x91=−64.9262485507945x_{91} = -64.9262485507945
x92=−27.2271365938325x_{92} = -27.2271365938325
x93=92.1533840908503x_{93} = 92.1533840908503
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(x + pi/6) + 1.
sin⁡(π6)+1\sin{\left(\frac{\pi}{6} \right)} + 1
The result:
f(0)=32f{\left(0 \right)} = \frac{3}{2}
The point:
(0, 3/2)
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
cos⁡(x+π6)=0\cos{\left(x + \frac{\pi}{6} \right)} = 0
Solve this equation
The roots of this equation
x1=π3x_{1} = \frac{\pi}{3}
x2=4π3x_{2} = \frac{4 \pi}{3}
The values of the extrema at the points:
 pi         /pi   pi\ 
(--, 1 + sin|-- + --|)
 3          \3    6 / 

 4*pi         /pi   pi\ 
(----, 1 - sin|-- + --|)
  3           \3    6 /


Intervals of increase and decrease of the function:
Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
Minima of the function at points:
x1=4π3x_{1} = \frac{4 \pi}{3}
Maxima of the function at points:
x1=π3x_{1} = \frac{\pi}{3}
Decreasing at intervals
(−∞,π3]∪[4π3,∞)\left(-\infty, \frac{\pi}{3}\right] \cup \left[\frac{4 \pi}{3}, \infty\right)
Increasing at intervals
[π3,4π3]\left[\frac{\pi}{3}, \frac{4 \pi}{3}\right]
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
−sin⁡(x+π6)=0- \sin{\left(x + \frac{\pi}{6} \right)} = 0
Solve this equation
The roots of this equation
x1=−π6x_{1} = - \frac{\pi}{6}
x2=5π6x_{2} = \frac{5 \pi}{6}

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
(−∞,−π6]∪[5π6,∞)\left(-\infty, - \frac{\pi}{6}\right] \cup \left[\frac{5 \pi}{6}, \infty\right)
Convex at the intervals
[−π6,5π6]\left[- \frac{\pi}{6}, \frac{5 \pi}{6}\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(sin⁡(x+π6)+1)=⟨0,2⟩\lim_{x \to -\infty}\left(\sin{\left(x + \frac{\pi}{6} \right)} + 1\right) = \left\langle 0, 2\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨0,2⟩y = \left\langle 0, 2\right\rangle
lim⁡x→∞(sin⁡(x+π6)+1)=⟨0,2⟩\lim_{x \to \infty}\left(\sin{\left(x + \frac{\pi}{6} \right)} + 1\right) = \left\langle 0, 2\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨0,2⟩y = \left\langle 0, 2\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sin(x + pi/6) + 1, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡(x+π6)+1x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(x + \frac{\pi}{6} \right)} + 1}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(sin⁡(x+π6)+1x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(x + \frac{\pi}{6} \right)} + 1}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
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:
sin⁡(x+π6)+1=1−sin⁡(x−π6)\sin{\left(x + \frac{\pi}{6} \right)} + 1 = 1 - \sin{\left(x - \frac{\pi}{6} \right)}
- No
sin⁡(x+π6)+1=sin⁡(x−π6)−1\sin{\left(x + \frac{\pi}{6} \right)} + 1 = \sin{\left(x - \frac{\pi}{6} \right)} - 1
- No
so, the function
not is
neither even, nor odd
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