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Plotting a function graph/ sin(x-pi/3)

Graphing y = sin(x-pi/3)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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          /    pi\
f(x) = sin|x - --|
          \    3 /
f(x)=sin⁡(x−π3)f{\left(x \right)} = \sin{\left(x - \frac{\pi}{3} \right)} 
f = sin(x - pi/3)
The graph of the function
-3.0-2.5-2.0-1.5-1.0-0.50.00.51.01.52.02.53.02-2
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−π3)=0\sin{\left(x - \frac{\pi}{3} \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=π3x_{1} = \frac{\pi}{3}
x2=4π3x_{2} = \frac{4 \pi}{3}
Numerical solution
x1=−61.7846555205993x_{1} = -61.7846555205993
x2=−83.7758040957278x_{2} = -83.7758040957278
x3=−8.37758040957278x_{3} = -8.37758040957278
x4=−71.2094334813686x_{4} = -71.2094334813686
x5=−90.0589894029074x_{5} = -90.0589894029074
x6=63.8790506229925x_{6} = 63.8790506229925
x7=−86.9173967493176x_{7} = -86.9173967493176
x8=95.2949771588904x_{8} = 95.2949771588904
x9=−20.943951023932x_{9} = -20.943951023932
x10=73.3038285837618x_{10} = 73.3038285837618
x11=−5644.39480094966x_{11} = -5644.39480094966
x12=−39.7935069454707x_{12} = -39.7935069454707
x13=70.162235930172x_{13} = 70.162235930172
x14=98.4365698124802x_{14} = 98.4365698124802
x15=−99.4837673636768x_{15} = -99.4837673636768
x16=−17.8023583703422x_{16} = -17.8023583703422
x17=13.6135681655558x_{17} = 13.6135681655558
x18=82.7286065445312x_{18} = 82.7286065445312
x19=−77.4926187885482x_{19} = -77.4926187885482
x20=−80.634211442138x_{20} = -80.634211442138
x21=−42.9350995990605x_{21} = -42.9350995990605
x22=−49.2182849062401x_{22} = -49.2182849062401
x23=−52.3598775598299x_{23} = -52.3598775598299
x24=−14.6607657167524x_{24} = -14.6607657167524
x25=10.471975511966x_{25} = 10.471975511966
x26=32.4631240870945x_{26} = 32.4631240870945
x27=79.5870138909414x_{27} = 79.5870138909414
x28=−55.5014702134197x_{28} = -55.5014702134197
x29=−68.0678408277789x_{29} = -68.0678408277789
x30=51.3126800086333x_{30} = 51.3126800086333
x31=−46.0766922526503x_{31} = -46.0766922526503
x32=7.33038285837618x_{32} = 7.33038285837618
x33=104.71975511966x_{33} = 104.71975511966
x34=67.0206432765823x_{34} = 67.0206432765823
x35=41.8879020478639x_{35} = 41.8879020478639
x36=76.4454212373516x_{36} = 76.4454212373516
x37=−58.6430628670095x_{37} = -58.6430628670095
x38=54.4542726622231x_{38} = 54.4542726622231
x39=38.7463093942741x_{39} = 38.7463093942741
x40=−27.2271363311115x_{40} = -27.2271363311115
x41=−5.23598775598299x_{41} = -5.23598775598299
x42=57.5958653158129x_{42} = 57.5958653158129
x43=85.870199198121x_{43} = 85.870199198121
x44=23.0383461263252x_{44} = 23.0383461263252
x45=60.7374579694027x_{45} = 60.7374579694027
x46=29.3215314335047x_{46} = 29.3215314335047
x47=−11.5191730631626x_{47} = -11.5191730631626
x48=4.18879020478639x_{48} = 4.18879020478639
x49=48.1710873550435x_{49} = 48.1710873550435
x50=35.6047167406843x_{50} = 35.6047167406843
x51=−30.3687289847013x_{51} = -30.3687289847013
x52=−2.0943951023932x_{52} = -2.0943951023932
x53=−93.2005820564972x_{53} = -93.2005820564972
x54=92.1533845053006x_{54} = 92.1533845053006
x55=−24.0855436775217x_{55} = -24.0855436775217
x56=−96.342174710087x_{56} = -96.342174710087
x57=−64.9262481741891x_{57} = -64.9262481741891
x58=16.7551608191456x_{58} = 16.7551608191456
x59=45.0294947014537x_{59} = 45.0294947014537
x60=1.0471975511966x_{60} = 1.0471975511966
x61=26.1799387799149x_{61} = 26.1799387799149
x62=−33.5103216382911x_{62} = -33.5103216382911
x63=89.0117918517108x_{63} = 89.0117918517108
x64=−36.6519142918809x_{64} = -36.6519142918809
x65=19.8967534727354x_{65} = 19.8967534727354
x66=−74.3510261349584x_{66} = -74.3510261349584
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/3).
sin⁡(−π3)\sin{\left(- \frac{\pi}{3} \right)}
The result:
f(0)=−32f{\left(0 \right)} = - \frac{\sqrt{3}}{2}
The point:
(0, -sqrt(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−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=5π6x_{2} = \frac{5 \pi}{6}
The values of the extrema at the points:
 -pi       /pi   pi\ 
(----, -sin|-- + --|)
  6        \6    3 / 

 5*pi     /pi   pi\ 
(----, cos|-- - --|)
  6       \3    3 /


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=−π6x_{1} = - \frac{\pi}{6}
Maxima of the function at points:
x1=5π6x_{1} = \frac{5 \pi}{6}
Decreasing at intervals
[−π6,5π6]\left[- \frac{\pi}{6}, \frac{5 \pi}{6}\right]
Increasing at intervals
(−∞,−π6]∪[5π6,∞)\left(-\infty, - \frac{\pi}{6}\right] \cup \left[\frac{5 \pi}{6}, \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+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}

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
(−∞,π3]∪[4π3,∞)\left(-\infty, \frac{\pi}{3}\right] \cup \left[\frac{4 \pi}{3}, \infty\right)
Convex at the intervals
[π3,4π3]\left[\frac{\pi}{3}, \frac{4 \pi}{3}\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−π3)=⟨−1,1⟩\lim_{x \to -\infty} \sin{\left(x - \frac{\pi}{3} \right)} = \left\langle -1, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
lim⁡x→∞sin⁡(x−π3)=⟨−1,1⟩\lim_{x \to \infty} \sin{\left(x - \frac{\pi}{3} \right)} = \left\langle -1, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sin(x - pi/3), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡(x−π3)x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(x - \frac{\pi}{3} \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(sin⁡(x−π3)x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(x - \frac{\pi}{3} \right)}}{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−π3)=−sin⁡(x+π3)\sin{\left(x - \frac{\pi}{3} \right)} = - \sin{\left(x + \frac{\pi}{3} \right)}
- No
sin⁡(x−π3)=sin⁡(x+π3)\sin{\left(x - \frac{\pi}{3} \right)} = \sin{\left(x + \frac{\pi}{3} \right)}
- No
so, the function
not is
neither even, nor odd
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