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

Graphing y = cos(x-pi/3)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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          /    pi\
f(x) = cos|x - --|
          \    3 /
f(x)=cos⁡(x−π3)f{\left(x \right)} = \cos{\left(x - \frac{\pi}{3} \right)} 
f = cos(x - pi/3)
The graph of the function
-3.0-2.5-2.0-1.5-1.0-0.53.00.00.51.01.52.02.52-2
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:
cos⁡(x−π3)=0\cos{\left(x - \frac{\pi}{3} \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−π6x_{1} = - \frac{\pi}{6}
x2=5π6x_{2} = \frac{5 \pi}{6}
Numerical solution
x1=68.5914396033772x_{1} = 68.5914396033772
x2=−63.3554518473942x_{2} = -63.3554518473942
x3=46.6002910282486x_{3} = 46.6002910282486
x4=43.4586983746588x_{4} = 43.4586983746588
x5=84.2994028713261x_{5} = 84.2994028713261
x6=−101.054563690472x_{6} = -101.054563690472
x7=−31.9395253114962x_{7} = -31.9395253114962
x8=71.733032256967x_{8} = 71.733032256967
x9=12.0427718387609x_{9} = 12.0427718387609
x10=87.4409955249159x_{10} = 87.4409955249159
x11=81.1578102177363x_{11} = 81.1578102177363
x12=−9.94837673636768x_{12} = -9.94837673636768
x13=−53.9306738866248x_{13} = -53.9306738866248
x14=−57.0722665402146x_{14} = -57.0722665402146
x15=−47.6474885794452x_{15} = -47.6474885794452
x16=−19.3731546971371x_{16} = -19.3731546971371
x17=37.1755130674792x_{17} = 37.1755130674792
x18=−69.6386371545737x_{18} = -69.6386371545737
x19=−60.2138591938044x_{19} = -60.2138591938044
x20=−75.9218224617533x_{20} = -75.9218224617533
x21=5.75958653158129x_{21} = 5.75958653158129
x22=131.423292675173x_{22} = 131.423292675173
x23=65.4498469497874x_{23} = 65.4498469497874
x24=40.317105721069x_{24} = 40.317105721069
x25=−22.5147473507269x_{25} = -22.5147473507269
x26=56.025068989018x_{26} = 56.025068989018
x27=−94.7713783832921x_{27} = -94.7713783832921
x28=8.90117918517108x_{28} = 8.90117918517108
x29=−88.4881930761125x_{29} = -88.4881930761125
x30=100.007366139275x_{30} = 100.007366139275
x31=15.1843644923507x_{31} = 15.1843644923507
x32=−13.0899693899575x_{32} = -13.0899693899575
x33=49.7418836818384x_{33} = 49.7418836818384
x34=96.8657734856853x_{34} = 96.8657734856853
x35=−41.3643032722656x_{35} = -41.3643032722656
x36=−38.2227106186758x_{36} = -38.2227106186758
x37=−82.2050077689329x_{37} = -82.2050077689329
x38=24.60914245312x_{38} = 24.60914245312
x39=90.5825881785057x_{39} = 90.5825881785057
x40=−91.6297857297023x_{40} = -91.6297857297023
x41=−97.9129710368819x_{41} = -97.9129710368819
x42=78.0162175641465x_{42} = 78.0162175641465
x43=−35.081117965086x_{43} = -35.081117965086
x44=93.7241808320955x_{44} = 93.7241808320955
x45=−3.66519142918809x_{45} = -3.66519142918809
x46=−66.497044500984x_{46} = -66.497044500984
x47=−44.5058959258554x_{47} = -44.5058959258554
x48=2.61799387799149x_{48} = 2.61799387799149
x49=−79.0634151153431x_{49} = -79.0634151153431
x50=−0.523598775598299x_{50} = -0.523598775598299
x51=34.0339204138894x_{51} = 34.0339204138894
x52=−16.2315620435473x_{52} = -16.2315620435473
x53=−820.479281362534x_{53} = -820.479281362534
x54=−72.7802298081635x_{54} = -72.7802298081635
x55=18.3259571459405x_{55} = 18.3259571459405
x56=−25.6563400043166x_{56} = -25.6563400043166
x57=−50.789081233035x_{57} = -50.789081233035
x58=−28.7979326579064x_{58} = -28.7979326579064
x59=30.8923277602996x_{59} = 30.8923277602996
x60=27.7507351067098x_{60} = 27.7507351067098
x61=−6.80678408277789x_{61} = -6.80678408277789
x62=52.8834763354282x_{62} = 52.8834763354282
x63=21.4675497995303x_{63} = 21.4675497995303
x64=59.1666616426078x_{64} = 59.1666616426078
x65=62.3082542961976x_{65} = 62.3082542961976
x66=74.8746249105567x_{66} = 74.8746249105567
x67=−85.3466004225227x_{67} = -85.3466004225227
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cos(x - pi/3).
cos⁡(−π3)\cos{\left(- \frac{\pi}{3} \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
−sin⁡(x−π3)=0- \sin{\left(x - \frac{\pi}{3} \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\ 
(--, cos|-- - --|)
 3      \3    3 / 

 4*pi      /pi   pi\ 
(----, -cos|-- - --|)
  3        \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=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→−∞cos⁡(x−π3)=⟨−1,1⟩\lim_{x \to -\infty} \cos{\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→∞cos⁡(x−π3)=⟨−1,1⟩\lim_{x \to \infty} \cos{\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 cos(x - pi/3), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(cos⁡(x−π3)x)=0\lim_{x \to -\infty}\left(\frac{\cos{\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→∞(cos⁡(x−π3)x)=0\lim_{x \to \infty}\left(\frac{\cos{\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:
cos⁡(x−π3)=cos⁡(x+π3)\cos{\left(x - \frac{\pi}{3} \right)} = \cos{\left(x + \frac{\pi}{3} \right)}
- No
cos⁡(x−π3)=−cos⁡(x+π3)\cos{\left(x - \frac{\pi}{3} \right)} = - \cos{\left(x + \frac{\pi}{3} \right)}
- No
so, the function
not is
neither even, nor odd
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