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absolute(cosx-1)
Plotting a function graph/ absolute(cosx-1)

Graphing y = absolute(cosx-1)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
f(x) = |cos(x) - 1|
f(x)=∣cos⁡(x)−1∣f{\left(x \right)} = \left|{\cos{\left(x \right)} - 1}\right| 
f = Abs(cos(x) - 1*1)
The graph of the function
0-60-50-40-30-20-10102004
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)−1∣=0\left|{\cos{\left(x \right)} - 1}\right| = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
x2=2πx_{2} = 2 \pi
Numerical solution
x1=12.5663704426592x_{1} = 12.5663704426592
x2=−62.8318534787248x_{2} = -62.8318534787248
x3=25.1327408328211x_{3} = 25.1327408328211
x4=100.530965157364x_{4} = 100.530965157364
x5=−43.9822976246252x_{5} = -43.9822976246252
x6=−18.8495563230046x_{6} = -18.8495563230046
x7=−94.2477797298079x_{7} = -94.2477797298079
x8=81.6814085860518x_{8} = 81.6814085860518
x9=−81.6814084945807x_{9} = -81.6814084945807
x10=56.5486682809363x_{10} = 56.5486682809363
x11=−25.1327415297174x_{11} = -25.1327415297174
x12=50.2654829439723x_{12} = 50.2654829439723
x13=−87.964593928489x_{13} = -87.964593928489
x14=−94.2477794452815x_{14} = -94.2477794452815
x15=−81.6814075578313x_{15} = -81.6814075578313
x16=−43.9822967932182x_{16} = -43.9822967932182
x17=−25.1327407505866x_{17} = -25.1327407505866
x18=6.28318626747926x_{18} = 6.28318626747926
x19=69.1150379887504x_{19} = 69.1150379887504
x20=−37.6991121287155x_{20} = -37.6991121287155
x21=−75.3982231045728x_{21} = -75.3982231045728
x22=94.2477800892631x_{22} = 94.2477800892631
x23=50.2654821322586x_{23} = 50.2654821322586
x24=−56.5486682426592x_{24} = -56.5486682426592
x25=−69.1150379045123x_{25} = -69.1150379045123
x26=18.8495564031971x_{26} = 18.8495564031971
x27=−18.8495552124105x_{27} = -18.8495552124105
x28=−31.4159260208155x_{28} = -31.4159260208155
x29=−31.4159267157965x_{29} = -31.4159267157965
x30=62.8318527849002x_{30} = 62.8318527849002
x31=6.28318528420851x_{31} = 6.28318528420851
x32=−6.2831851275477x_{32} = -6.2831851275477
x33=−6.28318555849548x_{33} = -6.28318555849548
x34=31.4159260648825x_{34} = 31.4159260648825
x35=−94.2477801171671x_{35} = -94.2477801171671
x36=−12.5663710889626x_{36} = -12.5663710889626
x37=−43.9822971745392x_{37} = -43.9822971745392
x38=−31.4159260507536x_{38} = -31.4159260507536
x39=56.5486676011951x_{39} = 56.5486676011951
x40=0x_{40} = 0
x41=−87.9645943586158x_{41} = -87.9645943586158
x42=−37.6991118772631x_{42} = -37.6991118772631
x43=−50.265482641087x_{43} = -50.265482641087
x44=75.3982232188727x_{44} = 75.3982232188727
x45=−50.2654822863493x_{45} = -50.2654822863493
x46=37.6991113348642x_{46} = 37.6991113348642
x47=−6.2831858160515x_{47} = -6.2831858160515
x48=−87.9645947692094x_{48} = -87.9645947692094
x49=50.2654824463392x_{49} = 50.2654824463392
x50=6.28318579821791x_{50} = 6.28318579821791
x51=100.530964759815x_{51} = 100.530964759815
x52=−62.831852673202x_{52} = -62.831852673202
x53=12.5663710110881x_{53} = 12.5663710110881
x54=43.9822971694647x_{54} = 43.9822971694647
x55=31.4159268459961x_{55} = 31.4159268459961
x56=87.9645938121814x_{56} = 87.9645938121814
x57=−37.6991113479743x_{57} = -37.6991113479743
x58=25.1327416384075x_{58} = 25.1327416384075
x59=−12.5663703112531x_{59} = -12.5663703112531
x60=81.6814091897036x_{60} = 81.6814091897036
x61=37.6991120311338x_{61} = 37.6991120311338
x62=69.115038794053x_{62} = 69.115038794053
x63=94.2477792651059x_{63} = 94.2477792651059
x64=43.9822966661001x_{64} = 43.9822966661001
x65=−81.6814090382277x_{65} = -81.6814090382277
x66=43.9822974733639x_{66} = 43.9822974733639
x67=62.8318535568358x_{67} = 62.8318535568358
x68=18.8495556275525x_{68} = 18.8495556275525
x69=−18.8495555173448x_{69} = -18.8495555173448
x70=−75.3982231720141x_{70} = -75.3982231720141
x71=6.28318500093652x_{71} = 6.28318500093652
x72=−56.5486674685864x_{72} = -56.5486674685864
x73=−75.3982238741744x_{73} = -75.3982238741744
x74=94.2477796093523x_{74} = 94.2477796093523
x75=−69.1150386869085x_{75} = -69.1150386869085
x76=56.5486668532011x_{76} = 56.5486668532011
x77=75.3982240031607x_{77} = 75.3982240031607
x78=87.964594335905x_{78} = 87.964594335905
x79=12.5663711301703x_{79} = 12.5663711301703
x80=87.9645946044253x_{80} = 87.9645946044253
x81=37.6991115173992x_{81} = 37.6991115173992
x82=81.6814084860076x_{82} = 81.6814084860076
x83=−100.530964626003x_{83} = -100.530964626003
x84=−50.2654829667315x_{84} = -50.2654829667315
x85=56.5486680806249x_{85} = 56.5486680806249
x86=−81.6814092565354x_{86} = -81.6814092565354
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to Abs(cos(x) - 1*1).
∣(−1)1+cos⁡(0)∣\left|{\left(-1\right) 1 + \cos{\left(0 \right)}}\right|
The result:
f(0)=0f{\left(0 \right)} = 0
The point:
(0, 0)
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)sign⁡(cos⁡(x)−1)=0- \sin{\left(x \right)} \operatorname{sign}{\left(\cos{\left(x \right)} - 1 \right)} = 0
Solve this equation
The roots of this equation
x1=37.6991118430775x_{1} = 37.6991118430775
x2=6.28318530717959x_{2} = 6.28318530717959
x3=87.9645943005142x_{3} = 87.9645943005142
x4=72.2566310325652x_{4} = 72.2566310325652
x5=9.42477796076938x_{5} = 9.42477796076938
x6=−87.9645943005142x_{6} = -87.9645943005142
x7=−59.6902604182061x_{7} = -59.6902604182061
x8=28.2743338823081x_{8} = 28.2743338823081
x9=43.9822971502571x_{9} = 43.9822971502571
x10=47.1238898038469x_{10} = 47.1238898038469
x11=84.8230016469244x_{11} = 84.8230016469244
x12=25.1327412287183x_{12} = 25.1327412287183
x13=91.106186954104x_{13} = 91.106186954104
x14=−18.8495559215388x_{14} = -18.8495559215388
x15=56.5486677646163x_{15} = 56.5486677646163
x16=−91.106186954104x_{16} = -91.106186954104
x17=18.8495559215388x_{17} = 18.8495559215388
x18=−75.398223686155x_{18} = -75.398223686155
x19=−75.398223686155x_{19} = -75.398223686155
x20=−94.2477796076938x_{20} = -94.2477796076938
x21=−40.8407044966673x_{21} = -40.8407044966673
x22=97.3893722612836x_{22} = 97.3893722612836
x23=3.14159265358979x_{23} = 3.14159265358979
x24=−43.9822971502571x_{24} = -43.9822971502571
x25=−84.8230016469244x_{25} = -84.8230016469244
x26=−3.14159265358979x_{26} = -3.14159265358979
x27=0x_{27} = 0
x28=−6.28318530717957x_{28} = -6.28318530717957
x29=−50.2654824574367x_{29} = -50.2654824574367
x30=−97.3893722612836x_{30} = -97.3893722612836
x31=−25.1327412287183x_{31} = -25.1327412287183
x32=56.5486677646163x_{32} = 56.5486677646163
x33=−62.8318530717959x_{33} = -62.8318530717959
x34=−2642.07942166902x_{34} = -2642.07942166902
x35=−78.5398163397448x_{35} = -78.5398163397448
x36=21.9911485751286x_{36} = 21.9911485751286
x37=−72.2566310325652x_{37} = -72.2566310325652
x38=69.1150383789755x_{38} = 69.1150383789755
x39=50.2654824574367x_{39} = 50.2654824574367
x40=100.530964914873x_{40} = 100.530964914873
x41=53.4070751110265x_{41} = 53.4070751110265
x42=12.5663706143592x_{42} = 12.5663706143592
x43=−113.097335529233x_{43} = -113.097335529233
x44=75.398223686155x_{44} = 75.398223686155
x45=−31.4159265358979x_{45} = -31.4159265358979
x46=−37.6991118430775x_{46} = -37.6991118430775
x47=−56.5486677646163x_{47} = -56.5486677646163
x48=−21.9911485751286x_{48} = -21.9911485751286
x49=−15.707963267949x_{49} = -15.707963267949
x50=81.6814089933346x_{50} = 81.6814089933346
x51=−232.477856365645x_{51} = -232.477856365645
x52=−69.1150383789755x_{52} = -69.1150383789755
x53=34.5575191894877x_{53} = 34.5575191894877
x54=−87.9645943005142x_{54} = -87.9645943005142
x55=6.28318530717959x_{55} = 6.28318530717959
x56=18.8495559215388x_{56} = 18.8495559215388
x57=−9.42477796076938x_{57} = -9.42477796076938
x58=78.5398163397448x_{58} = 78.5398163397448
x59=−53.4070751110265x_{59} = -53.4070751110265
x60=−34.5575191894877x_{60} = -34.5575191894877
x61=−100.530964914873x_{61} = -100.530964914873
x62=31.4159265358979x_{62} = 31.4159265358979
x63=−267.035375555132x_{63} = -267.035375555132
x64=−25.1327412287183x_{64} = -25.1327412287183
x65=−12.5663706143592x_{65} = -12.5663706143592
x66=40.8407044966673x_{66} = 40.8407044966673
x67=−47.1238898038469x_{67} = -47.1238898038469
x68=−81.6814089933346x_{68} = -81.6814089933346
x69=−28.2743338823081x_{69} = -28.2743338823081
x70=−65.9734457253857x_{70} = -65.9734457253857
x71=65.9734457253857x_{71} = 65.9734457253857
x72=15.707963267949x_{72} = 15.707963267949
x73=94.2477796076938x_{73} = 94.2477796076938
x74=62.8318530717959x_{74} = 62.8318530717959
x75=−31.4159265358979x_{75} = -31.4159265358979
x76=59.6902604182061x_{76} = 59.6902604182061
The values of the extrema at the points:
(37.6991118430775, 0)

(6.28318530717959, 0)

(87.9645943005142, 0)

(72.2566310325652, 2)

(9.42477796076938, 2)

(-87.9645943005142, 0)

(-59.6902604182061, 2)

(28.2743338823081, 2)

(43.9822971502571, 0)

(47.1238898038469, 2)

(84.8230016469244, 2)

(25.1327412287183, 0)

(91.106186954104, 2)

(-18.8495559215388, 0)

(56.5486677646163, 0)

(-91.106186954104, 2)

(18.8495559215388, 0)

(-75.398223686155, 0)

(-75.398223686155, 0)

(-94.2477796076938, 0)

(-40.8407044966673, 2)

(97.3893722612836, 2)

(3.14159265358979, 2)

(-43.9822971502571, 0)

(-84.8230016469244, 2)

(-3.14159265358979, 2)

(0, 0)

(-6.28318530717957, 0)

(-50.2654824574367, 0)

(-97.3893722612836, 2)

(-25.1327412287183, 0)

(56.5486677646163, 0)

(-62.8318530717959, 0)

(-2642.07942166902, 2)

(-78.5398163397448, 2)

(21.9911485751286, 2)

(-72.2566310325652, 2)

(69.1150383789755, 0)

(50.2654824574367, 0)

(100.530964914873, 0)

(53.4070751110265, 2)

(12.5663706143592, 0)

(-113.097335529233, 0)

(75.398223686155, 0)

(-31.4159265358979, 0)

(-37.6991118430775, 0)

(-56.5486677646163, 0)

(-21.9911485751286, 2)

(-15.707963267949, 2)

(81.6814089933346, 0)

(-232.477856365645, 0)

(-69.1150383789755, 0)

(34.5575191894877, 2)

(-87.9645943005142, 0)

(6.28318530717959, 0)

(18.8495559215388, 0)

(-9.42477796076938, 2)

(78.5398163397448, 2)

(-53.4070751110265, 2)

(-34.5575191894877, 2)

(-100.530964914873, 0)

(31.4159265358979, 0)

(-267.035375555132, 2)

(-25.1327412287183, 0)

(-12.5663706143592, 0)

(40.8407044966673, 2)

(-47.1238898038469, 2)

(-81.6814089933346, 0)

(-28.2743338823081, 2)

(-65.9734457253857, 2)

(65.9734457253857, 2)

(15.707963267949, 2)

(94.2477796076938, 0)

(62.8318530717959, 0)

(-31.4159265358979, 0)

(59.6902604182061, 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=37.6991118430775x_{1} = 37.6991118430775
x2=6.28318530717959x_{2} = 6.28318530717959
x3=87.9645943005142x_{3} = 87.9645943005142
x4=−87.9645943005142x_{4} = -87.9645943005142
x5=43.9822971502571x_{5} = 43.9822971502571
x6=25.1327412287183x_{6} = 25.1327412287183
x7=−18.8495559215388x_{7} = -18.8495559215388
x8=56.5486677646163x_{8} = 56.5486677646163
x9=18.8495559215388x_{9} = 18.8495559215388
x10=−75.398223686155x_{10} = -75.398223686155
x11=−75.398223686155x_{11} = -75.398223686155
x12=−94.2477796076938x_{12} = -94.2477796076938
x13=−43.9822971502571x_{13} = -43.9822971502571
x14=0x_{14} = 0
x15=−6.28318530717957x_{15} = -6.28318530717957
x16=−50.2654824574367x_{16} = -50.2654824574367
x17=−25.1327412287183x_{17} = -25.1327412287183
x18=56.5486677646163x_{18} = 56.5486677646163
x19=−62.8318530717959x_{19} = -62.8318530717959
x20=69.1150383789755x_{20} = 69.1150383789755
x21=50.2654824574367x_{21} = 50.2654824574367
x22=100.530964914873x_{22} = 100.530964914873
x23=12.5663706143592x_{23} = 12.5663706143592
x24=−113.097335529233x_{24} = -113.097335529233
x25=75.398223686155x_{25} = 75.398223686155
x26=−31.4159265358979x_{26} = -31.4159265358979
x27=−37.6991118430775x_{27} = -37.6991118430775
x28=−56.5486677646163x_{28} = -56.5486677646163
x29=81.6814089933346x_{29} = 81.6814089933346
x30=−232.477856365645x_{30} = -232.477856365645
x31=−69.1150383789755x_{31} = -69.1150383789755
x32=−87.9645943005142x_{32} = -87.9645943005142
x33=6.28318530717959x_{33} = 6.28318530717959
x34=18.8495559215388x_{34} = 18.8495559215388
x35=−100.530964914873x_{35} = -100.530964914873
x36=31.4159265358979x_{36} = 31.4159265358979
x37=−25.1327412287183x_{37} = -25.1327412287183
x38=−12.5663706143592x_{38} = -12.5663706143592
x39=−81.6814089933346x_{39} = -81.6814089933346
x40=94.2477796076938x_{40} = 94.2477796076938
x41=62.8318530717959x_{41} = 62.8318530717959
x42=−31.4159265358979x_{42} = -31.4159265358979
Maxima of the function at points:
x42=72.2566310325652x_{42} = 72.2566310325652
x42=9.42477796076938x_{42} = 9.42477796076938
x42=−59.6902604182061x_{42} = -59.6902604182061
x42=28.2743338823081x_{42} = 28.2743338823081
x42=47.1238898038469x_{42} = 47.1238898038469
x42=84.8230016469244x_{42} = 84.8230016469244
x42=91.106186954104x_{42} = 91.106186954104
x42=−91.106186954104x_{42} = -91.106186954104
x42=−40.8407044966673x_{42} = -40.8407044966673
x42=97.3893722612836x_{42} = 97.3893722612836
x42=3.14159265358979x_{42} = 3.14159265358979
x42=−84.8230016469244x_{42} = -84.8230016469244
x42=−3.14159265358979x_{42} = -3.14159265358979
x42=−97.3893722612836x_{42} = -97.3893722612836
x42=−2642.07942166902x_{42} = -2642.07942166902
x42=−78.5398163397448x_{42} = -78.5398163397448
x42=21.9911485751286x_{42} = 21.9911485751286
x42=−72.2566310325652x_{42} = -72.2566310325652
x42=53.4070751110265x_{42} = 53.4070751110265
x42=−21.9911485751286x_{42} = -21.9911485751286
x42=−15.707963267949x_{42} = -15.707963267949
x42=34.5575191894877x_{42} = 34.5575191894877
x42=−9.42477796076938x_{42} = -9.42477796076938
x42=78.5398163397448x_{42} = 78.5398163397448
x42=−53.4070751110265x_{42} = -53.4070751110265
x42=−34.5575191894877x_{42} = -34.5575191894877
x42=−267.035375555132x_{42} = -267.035375555132
x42=40.8407044966673x_{42} = 40.8407044966673
x42=−47.1238898038469x_{42} = -47.1238898038469
x42=−28.2743338823081x_{42} = -28.2743338823081
x42=−65.9734457253857x_{42} = -65.9734457253857
x42=65.9734457253857x_{42} = 65.9734457253857
x42=15.707963267949x_{42} = 15.707963267949
x42=59.6902604182061x_{42} = 59.6902604182061
Decreasing at intervals
[100.530964914873,∞)\left[100.530964914873, \infty\right)
Increasing at intervals
(−∞,−232.477856365645]\left(-\infty, -232.477856365645\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
2sin⁡2(x)δ(cos⁡(x)−1)−cos⁡(x)sign⁡(cos⁡(x)−1)=02 \sin^{2}{\left(x \right)} \delta\left(\cos{\left(x \right)} - 1\right) - \cos{\left(x \right)} \operatorname{sign}{\left(\cos{\left(x \right)} - 1 \right)} = 0
Solve this equation
Solutions are not found,
maybe, the function has no inflections
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)−1∣=∣⟨−2,0⟩∣\lim_{x \to -\infty} \left|{\cos{\left(x \right)} - 1}\right| = \left|{\left\langle -2, 0\right\rangle}\right|
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=∣⟨−2,0⟩∣y = \left|{\left\langle -2, 0\right\rangle}\right|
lim⁡x→∞∣cos⁡(x)−1∣=∣⟨−2,0⟩∣\lim_{x \to \infty} \left|{\cos{\left(x \right)} - 1}\right| = \left|{\left\langle -2, 0\right\rangle}\right|
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=∣⟨−2,0⟩∣y = \left|{\left\langle -2, 0\right\rangle}\right|
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of Abs(cos(x) - 1*1), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(∣cos⁡(x)−1∣x)=0\lim_{x \to -\infty}\left(\frac{\left|{\cos{\left(x \right)} - 1}\right|}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(∣cos⁡(x)−1∣x)=0\lim_{x \to \infty}\left(\frac{\left|{\cos{\left(x \right)} - 1}\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)−1∣=∣cos⁡(x)−1∣\left|{\cos{\left(x \right)} - 1}\right| = \left|{\cos{\left(x \right)} - 1}\right|
- Yes
∣cos⁡(x)−1∣=−∣cos⁡(x)−1∣\left|{\cos{\left(x \right)} - 1}\right| = - \left|{\cos{\left(x \right)} - 1}\right|
- No
so, the function
is
even
The graph
Graphing y = absolute(cosx-1)
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