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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
-5.0-4.0-3.0-2.0-1.05.00.01.02.03.04.004
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=−π3x_{1} = - \frac{\pi}{3}
x2=5π3x_{2} = \frac{5 \pi}{3}
Numerical solution
x1=−38.7463098260451x_{1} = -38.7463098260451
x2=−13.6135682483753x_{2} = -13.6135682483753
x3=86.9173974067754x_{3} = 86.9173974067754
x4=24.0855440859231x_{4} = 24.0855440859231
x5=36.6519148022553x_{5} = 36.6519148022553
x6=30.3687294917281x_{6} = 30.3687294917281
x7=−26.1799387133702x_{7} = -26.1799387133702
x8=93.2005824445644x_{8} = 93.2005824445644
x9=55.5014704889569x_{9} = 55.5014704889569
x10=86.9173964288891x_{10} = 86.9173964288891
x11=36.6519151871741x_{11} = 36.6519151871741
x12=30.368728796664x_{12} = 30.368728796664
x13=99.4837668797295x_{13} = 99.4837668797295
x14=80.6342122207782x_{14} = 80.6342122207782
x15=99.4837676465256x_{15} = 99.4837676465256
x16=−63.8790501935183x_{16} = -63.8790501935183
x17=30.3687289114122x_{17} = 30.3687289114122
x18=24.0855429847921x_{18} = 24.0855429847921
x19=−32.4631245795436x_{19} = -32.4631245795436
x20=−51.3126802348481x_{20} = -51.3126802348481
x21=−63.8790510556062x_{21} = -63.8790510556062
x22=17.8023583939831x_{22} = 17.8023583939831
x23=−76.4454224109362x_{23} = -76.4454224109362
x24=−95.2949766541678x_{24} = -95.2949766541678
x25=55.501469168008x_{25} = 55.501469168008
x26=68.0678403709668x_{26} = 68.0678403709668
x27=17.8023578617743x_{27} = 17.8023578617743
x28=−89.0117914013259x_{28} = -89.0117914013259
x29=24.0855432280792x_{29} = 24.0855432280792
x30=49.2182852884197x_{30} = 49.2182852884197
x31=−95.2949759363908x_{31} = -95.2949759363908
x32=−38.7463090241633x_{32} = -38.7463090241633
x33=−13.6135676562812x_{33} = -13.6135676562812
x34=42.9351000623883x_{34} = 42.9351000623883
x35=99.483766182202x_{35} = 99.483766182202
x36=−51.3126788801149x_{36} = -51.3126788801149
x37=−7.33038234970242x_{37} = -7.33038234970242
x38=−19.8967530533671x_{38} = -19.8967530533671
x39=−19.8967525986102x_{39} = -19.8967525986102
x40=−101.578161953261x_{40} = -101.578161953261
x41=−1.047197884948x_{41} = -1.047197884948
x42=−70.162235824498x_{42} = -70.162235824498
x43=68.0678412242824x_{43} = 68.0678412242824
x44=−32.4631238264292x_{44} = -32.4631238264292
x45=−82.7286069811057x_{45} = -82.7286069811057
x46=5.23598732913431x_{46} = 5.23598732913431
x47=86.9173968009572x_{47} = 86.9173968009572
x48=−76.4454209842472x_{48} = -76.4454209842472
x49=5.23598813217661x_{49} = 5.23598813217661
x50=61.7846550114646x_{50} = 61.7846550114646
x51=74.3510259182365x_{51} = 74.3510259182365
x52=42.9350992719286x_{52} = 42.9350992719286
x53=11.5191733313054x_{53} = 11.5191733313054
x54=−57.5958658923729x_{54} = -57.5958658923729
x55=11.5191725733813x_{55} = 11.5191725733813
x56=−57.5958654382348x_{56} = -57.5958654382348
x57=42.935098044129x_{57} = 42.935098044129
x58=86.9173965904982x_{58} = 86.9173965904982
x59=−7.33038184400076x_{59} = -7.33038184400076
x60=−45.0294942466274x_{60} = -45.0294942466274
x61=−19.8967539141283x_{61} = -19.8967539141283
x62=−70.1622358120646x_{62} = -70.1622358120646
x63=61.784655497151x_{63} = 61.784655497151
x64=−13.6135683222534x_{64} = -13.6135683222534
x65=74.3510260714291x_{65} = 74.3510260714291
x66=−26.1799386522404x_{66} = -26.1799386522404
x67=17.8023585073476x_{67} = 17.8023585073476
x68=−51.3126795018451x_{68} = -51.3126795018451
x69=−76.4454217322964x_{69} = -76.4454217322964
x70=−57.5958654079676x_{70} = -57.5958654079676
x71=80.6342119540229x_{71} = 80.6342119540229
x72=74.3510266393966x_{72} = 74.3510266393966
x73=11.5191718122226x_{73} = 11.5191718122226
x74=80.6342112394353x_{74} = 80.6342112394353
x75=−95.2949773927983x_{75} = -95.2949773927983
x76=55.5014697264737x_{76} = 55.5014697264737
x77=−26.1799392876319x_{77} = -26.1799392876319
x78=−57.5958648046158x_{78} = -57.5958648046158
x79=−70.1622364367418x_{79} = -70.1622364367418
x80=93.2005816398396x_{80} = 93.2005816398396
x81=−45.029495041739x_{81} = -45.029495041739
x82=68.067840622896x_{82} = 68.067840622896
x83=−82.7286061805984x_{83} = -82.7286061805984
x84=−7.33038307682589x_{84} = -7.33038307682589
x85=−89.011792198443x_{85} = -89.011792198443
x86=36.6519140812684x_{86} = 36.6519140812684
x87=49.21828448443x_{87} = 49.21828448443
x88=−1.04719709205516x_{88} = -1.04719709205516
x89=86.9173972167005x_{89} = 86.9173972167005
x90=−63.8790503300893x_{90} = -63.8790503300893
x91=−32.4631253376294x_{91} = -32.4631253376294
x92=61.7846556667327x_{92} = 61.7846556667327
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)=12f{\left(0 \right)} = \frac{1}{2}
The point:
(0, 1/2)
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=2π3x_{2} = \frac{2 \pi}{3}
The values of the extrema at the points:
 -pi          /pi   pi\ 
(----, 1 - sin|-- + --|)
  3           \3    6 / 

 2*pi         /pi   pi\ 
(----, 1 + cos|-- - --|)
  3           \6    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=−π3x_{1} = - \frac{\pi}{3}
Maxima of the function at points:
x1=2π3x_{1} = \frac{2 \pi}{3}
Decreasing at intervals
[−π3,2π3]\left[- \frac{\pi}{3}, \frac{2 \pi}{3}\right]
Increasing at intervals
(−∞,−π3]∪[2π3,∞)\left(-\infty, - \frac{\pi}{3}\right] \cup \left[\frac{2 \pi}{3}, \infty\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
cos⁡(x+π3)=0\cos{\left(x + \frac{\pi}{3} \right)} = 0
Solve this equation
The roots of this equation
x1=π6x_{1} = \frac{\pi}{6}
x2=7π6x_{2} = \frac{7 \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]∪[7π6,∞)\left(-\infty, \frac{\pi}{6}\right] \cup \left[\frac{7 \pi}{6}, \infty\right)
Convex at the intervals
[π6,7π6]\left[\frac{\pi}{6}, \frac{7 \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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