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Plotting a function graph/ cos((x+pi)/3)-1

Graphing y = cos((x+pi)/3)-1

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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          /x + pi\    
f(x) = cos|------| - 1
          \  3   /
f(x)=cos⁡(x+π3)−1f{\left(x \right)} = \cos{\left(\frac{x + \pi}{3} \right)} - 1 
f = cos((x + pi)/3) - 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:
cos⁡(x+π3)−1=0\cos{\left(\frac{x + \pi}{3} \right)} - 1 = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−πx_{1} = - \pi
x2=5πx_{2} = 5 \pi
Numerical solution
x1=−474.380493870955x_{1} = -474.380493870955
x2=−59.6902589797336x_{2} = -59.6902589797336
x3=−3.14159276062015x_{3} = -3.14159276062015
x4=91.1061865286018x_{4} = 91.1061865286018
x5=53.4070735966335x_{5} = 53.4070735966335
x6=−59.690264170919x_{6} = -59.690264170919
x7=−3.14159134496669x_{7} = -3.14159134496669
x8=72.2566324813595x_{8} = 72.2566324813595
x9=−97.3893733028467x_{9} = -97.3893733028467
x10=−40.8407029507697x_{10} = -40.8407029507697
x11=53.4070756953521x_{11} = 53.4070756953521
x12=15.7079643832x_{12} = 15.7079643832
x13=34.5575205743468x_{13} = 34.5575205743468
x14=53.4070766577001x_{14} = 53.4070766577001
x15=−21.9911500132853x_{15} = -21.9911500132853
x16=53.4070738154238x_{16} = 53.4070738154238
x17=−59.6902596099374x_{17} = -59.6902596099374
x18=72.2566318621985x_{18} = 72.2566318621985
x19=15.7079643063945x_{19} = 15.7079643063945
x20=15.7079625798234x_{20} = 15.7079625798234
x21=53.4070745520074x_{21} = 53.4070745520074
x22=72.2566298163471x_{22} = 72.2566298163471
x23=34.5575206523395x_{23} = 34.5575206523395
x24=−78.5398163691121x_{24} = -78.5398163691121
x25=91.1061881917053x_{25} = 91.1061881917053
x26=91.1061869214634x_{26} = 91.1061869214634
x27=−3.14159419270109x_{27} = -3.14159419270109
x28=−59.6902613484779x_{28} = -59.6902613484779
x29=−40.8407032751831x_{29} = -40.8407032751831
x30=−3.14159113256562x_{30} = -3.14159113256562
x31=−78.5398160333126x_{31} = -78.5398160333126
x32=72.2566278965339x_{32} = 72.2566278965339
x33=−40.8407049460369x_{33} = -40.8407049460369
x34=−97.3893708890293x_{34} = -97.3893708890293
x35=−97.3893707853007x_{35} = -97.3893707853007
x36=34.5575181670294x_{36} = 34.5575181670294
x37=72.2566323921551x_{37} = 72.2566323921551
x38=−21.9911485864388x_{38} = -21.9911485864388
x39=−59.6902604578483x_{39} = -59.6902604578483
x40=−78.5398156708263x_{40} = -78.5398156708263
x41=53.4070751199802x_{41} = 53.4070751199802
x42=−78.5398150881882x_{42} = -78.5398150881882
x43=−40.8407044242062x_{43} = -40.8407044242062
x44=91.1061857467651x_{44} = 91.1061857467651
x45=91.1061854151691x_{45} = 91.1061854151691
x46=−21.9911459637955x_{46} = -21.9911459637955
x47=15.7079642483061x_{47} = 15.7079642483061
x48=72.2566301226729x_{48} = 72.2566301226729
x49=−78.539815215081x_{49} = -78.539815215081
x50=−40.8407057203169x_{50} = -40.8407057203169
x51=−97.3893715773867x_{51} = -97.3893715773867
x52=−3.14159207598184x_{52} = -3.14159207598184
x53=−78.5398147949047x_{53} = -78.5398147949047
x54=−78.5398169218883x_{54} = -78.5398169218883
x55=53.4070762534334x_{55} = 53.4070762534334
x56=15.7079647959253x_{56} = 15.7079647959253
x57=−59.6902615980313x_{57} = -59.6902615980313
x58=−21.9911499262929x_{58} = -21.9911499262929
x59=−21.9911470840848x_{59} = -21.9911470840848
x60=91.1061868853346x_{60} = 91.1061868853346
x61=−59.6902590363808x_{61} = -59.6902590363808
x62=34.5575165845019x_{62} = 34.5575165845019
x63=53.4070761151575x_{63} = 53.4070761151575
x64=−3.14159321181009x_{64} = -3.14159321181009
x65=−40.8407043841756x_{65} = -40.8407043841756
x66=15.7079634550617x_{66} = 15.7079634550617
x67=−59.6902619075171x_{67} = -59.6902619075171
x68=53.4070754421324x_{68} = 53.4070754421324
x69=−40.840704052337x_{69} = -40.840704052337
x70=−78.5398176498381x_{70} = -78.5398176498381
x71=−21.9911493872822x_{71} = -21.9911493872822
x72=−78.5398160333896x_{72} = -78.5398160333896
x73=34.5575182254197x_{73} = 34.5575182254197
x74=34.557519895729x_{74} = 34.557519895729
x75=34.5575190194442x_{75} = 34.5575190194442
x76=−97.3893724570633x_{76} = -97.3893724570633
x77=15.707961894931x_{77} = 15.707961894931
x78=−40.8407060333612x_{78} = -40.8407060333612
x79=−21.9911473480329x_{79} = -21.9911473480329
x80=91.1061884976653x_{80} = 91.1061884976653
x81=15.7079659774467x_{81} = 15.7079659774467
x82=−3.14159358974287x_{82} = -3.14159358974287
x83=−78.5398178465509x_{83} = -78.5398178465509
x84=72.2566295511795x_{84} = 72.2566295511795
x85=−21.9911476483818x_{85} = -21.9911476483818
x86=−3.14159296600646x_{86} = -3.14159296600646
x87=15.7079618120813x_{87} = 15.7079618120813
x88=72.2566310277172x_{88} = 72.2566310277172
x89=−3.14159378098184x_{89} = -3.14159378098184
x90=34.5575176700454x_{90} = 34.5575176700454
x91=91.1061874222829x_{91} = 91.1061874222829
x92=−97.3893745206956x_{92} = -97.3893745206956
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) - 1.
−1+cos⁡(π3)-1 + \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)3=0- \frac{\sin{\left(\frac{x + \pi}{3} \right)}}{3} = 0
Solve this equation
The roots of this equation
x1=−πx_{1} = - \pi
x2=2πx_{2} = 2 \pi
The values of the extrema at the points:
(-pi, 0)

(2*pi, -2)


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=2πx_{1} = 2 \pi
Maxima of the function at points:
x1=−πx_{1} = - \pi
Decreasing at intervals
(−∞,−π]∪[2π,∞)\left(-\infty, - \pi\right] \cup \left[2 \pi, \infty\right)
Increasing at intervals
[−π,2π]\left[- \pi, 2 \pi\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)9=0- \frac{\cos{\left(\frac{x + \pi}{3} \right)}}{9} = 0
Solve this equation
The roots of this equation
x1=π2x_{1} = \frac{\pi}{2}
x2=7π2x_{2} = \frac{7 \pi}{2}

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
[π2,7π2]\left[\frac{\pi}{2}, \frac{7 \pi}{2}\right]
Convex at the intervals
(−∞,π2]∪[7π2,∞)\left(-\infty, \frac{\pi}{2}\right] \cup \left[\frac{7 \pi}{2}, \infty\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)=⟨−2,0⟩\lim_{x \to -\infty}\left(\cos{\left(\frac{x + \pi}{3} \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→∞(cos⁡(x+π3)−1)=⟨−2,0⟩\lim_{x \to \infty}\left(\cos{\left(\frac{x + \pi}{3} \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 cos((x + pi)/3) - 1, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(cos⁡(x+π3)−1x)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(\frac{x + \pi}{3} \right)} - 1}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(cos⁡(x+π3)−1x)=0\lim_{x \to \infty}\left(\frac{\cos{\left(\frac{x + \pi}{3} \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:
cos⁡(x+π3)−1=sin⁡(x3+π6)−1\cos{\left(\frac{x + \pi}{3} \right)} - 1 = \sin{\left(\frac{x}{3} + \frac{\pi}{6} \right)} - 1
- No
cos⁡(x+π3)−1=1−sin⁡(x3+π6)\cos{\left(\frac{x + \pi}{3} \right)} - 1 = 1 - \sin{\left(\frac{x}{3} + \frac{\pi}{6} \right)}
- No
so, the function
not is
neither even, nor odd
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