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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
02468-8-6-4-2-10102-4
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}
Numerical solution
x1=−5.23598801500526x_{1} = -5.23598801500526
x2=13.6135673734423x_{2} = 13.6135673734423
x3=8181.75446884145x_{3} = 8181.75446884145
x4=−24.0855441753518x_{4} = -24.0855441753518
x5=70.1622363802125x_{5} = 70.1622363802125
x6=19.8967535565785x_{6} = 19.8967535565785
x7=32.4631246025263x_{7} = 32.4631246025263
x8=−11.5191729947456x_{8} = -11.5191729947456
x9=−55.5014696969167x_{9} = -55.5014696969167
x10=38.746309893505x_{10} = 38.746309893505
x11=−93.2005815656485x_{11} = -93.2005815656485
x12=95.2949767161054x_{12} = 95.2949767161054
x13=32.4631241043919x_{13} = 32.4631241043919
x14=26.1799383825674x_{14} = 26.1799383825674
x15=70.1622355211858x_{15} = 70.1622355211858
x16=63.8790507175974x_{16} = 63.8790507175974
x17=57.5958655509547x_{17} = 57.5958655509547
x18=−36.6519132139351x_{18} = -36.6519132139351
x19=−24.0855436050114x_{19} = -24.0855436050114
x20=−17.8023583238349x_{20} = -17.8023583238349
x21=101.578161968076x_{21} = 101.578161968076
x22=−30.3687294883483x_{22} = -30.3687294883483
x23=51.3126795613314x_{23} = 51.3126795613314
x24=82.7286070462301x_{24} = 82.7286070462301
x25=51.3126803577955x_{25} = 51.3126803577955
x26=89.0117914965281x_{26} = 89.0117914965281
x27=−17.8023587550388x_{27} = -17.8023587550388
x28=7.33038273028594x_{28} = 7.33038273028594
x29=−17.8023579056127x_{29} = -17.8023579056127
x30=−86.9173963246505x_{30} = -86.9173963246505
x31=−74.3510259236276x_{31} = -74.3510259236276
x32=63.8790501203265x_{32} = 63.8790501203265
x33=13.6135683928511x_{33} = 13.6135683928511
x34=−74.3510267119586x_{34} = -74.3510267119586
x35=32.4631239649434x_{35} = 32.4631239649434
x36=63.8790508072645x_{36} = 63.8790508072645
x37=−36.6519147478316x_{37} = -36.6519147478316
x38=−11.5191731940441x_{38} = -11.5191731940441
x39=−68.0678407673484x_{39} = -68.0678407673484
x40=19.8967529723062x_{40} = 19.8967529723062
x41=7.33038240668485x_{41} = 7.33038240668485
x42=−74.3510266400963x_{42} = -74.3510266400963
x43=−30.3687296655956x_{43} = -30.3687296655956
x44=82.7286063004863x_{44} = 82.7286063004863
x45=−80.6342119022151x_{45} = -80.6342119022151
x46=38.7463104599544x_{46} = 38.7463104599544
x47=82.7286077977356x_{47} = 82.7286077977356
x48=−99.4837675117098x_{48} = -99.4837675117098
x49=57.5958644069028x_{49} = 57.5958644069028
x50=−55.5014703529259x_{50} = -55.5014703529259
x51=−68.0678413226289x_{51} = -68.0678413226289
x52=−61.7846550494277x_{52} = -61.7846550494277
x53=13.6135676636004x_{53} = 13.6135676636004
x54=−24.085543437643x_{54} = -24.085543437643
x55=26.1799388484168x_{55} = 26.1799388484168
x56=−86.9173969295634x_{56} = -86.9173969295634
x57=38.7463091427042x_{57} = 38.7463091427042
x58=−86.9173971288882x_{58} = -86.9173971288882
x59=−42.9350999726765x_{59} = -42.9350999726765
x60=89.011792295987x_{60} = 89.011792295987
x61=19.8967536860244x_{61} = 19.8967536860244
x62=−93.2005823302035x_{62} = -93.2005823302035
x63=26.1799392376142x_{63} = 26.1799392376142
x64=−99.483766847019x_{64} = -99.483766847019
x65=−5.23598787206093x_{65} = -5.23598787206093
x66=−36.6519139557788x_{66} = -36.6519139557788
x67=45.0294943400334x_{67} = 45.0294943400334
x68=57.5958648157438x_{68} = 57.5958648157438
x69=−11.5191725470651x_{69} = -11.5191725470651
x70=76.4454211240462x_{70} = 76.4454211240462
x71=−61.7846558915474x_{71} = -61.7846558915474
x72=45.0294951410032x_{72} = 45.0294951410032
x73=1.0471971836313x_{73} = 1.0471971836313
x74=−68.0678405635095x_{74} = -68.0678405635095
x75=−5.23598725938577x_{75} = -5.23598725938577
x76=−49.2182851726414x_{76} = -49.2182851726414
x77=1.04719798590098x_{77} = 1.04719798590098
x78=−80.6342111126989x_{78} = -80.6342111126989
x79=−68.0678392493177x_{79} = -68.0678392493177
x80=−55.5014700888133x_{80} = -55.5014700888133
x81=−30.368728765342x_{81} = -30.368728765342
x82=−61.7846554920087x_{82} = -61.7846554920087
x83=95.2949775144473x_{83} = 95.2949775144473
x84=76.4454212081088x_{84} = 76.4454212081088
x85=−49.2182844124382x_{85} = -49.2182844124382
x86=70.162236027948x_{86} = 70.162236027948
x87=−42.9350991693176x_{87} = -42.9350991693176
x88=76.4454217518471x_{88} = 76.4454217518471
x89=7.33038320105311x_{89} = 7.33038320105311
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.
−1+sin⁡(π6)-1 + \sin{\left(\frac{\pi}{6} \right)}
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=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)=⟨−2,0⟩\lim_{x \to -\infty}\left(\sin{\left(x + \frac{\pi}{6} \right)} - 1\right) = \left\langle -2, 0\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−2,0⟩y = \left\langle -2, 0\right\rangle
lim⁡x→∞(sin⁡(x+π6)−1)=⟨−2,0⟩\lim_{x \to \infty}\left(\sin{\left(x + \frac{\pi}{6} \right)} - 1\right) = \left\langle -2, 0\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−2,0⟩y = \left\langle -2, 0\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=−sin⁡(x−π6)−1\sin{\left(x + \frac{\pi}{6} \right)} - 1 = - \sin{\left(x - \frac{\pi}{6} \right)} - 1
- 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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