Mister Exam

Other calculators

Plotting a function graph/ |3cosx-1|

Graphing y = |3cosx-1|

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
f(x) = |3*cos(x) - 1|
f(x)=∣3cos⁡(x)−1∣f{\left(x \right)} = \left|{3 \cos{\left(x \right)} - 1}\right| 
f = Abs(3*cos(x) - 1)
The graph of the function
02468-8-6-4-2-101005
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:
∣3cos⁡(x)−1∣=0\left|{3 \cos{\left(x \right)} - 1}\right| = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−acos⁡(13)+2πx_{1} = - \operatorname{acos}{\left(\frac{1}{3} \right)} + 2 \pi
x2=acos⁡(13)x_{2} = \operatorname{acos}{\left(\frac{1}{3} \right)}
Numerical solution
x1=45.2132565675979x_{1} = 45.2132565675979
x2=−49.0345230400959x_{2} = -49.0345230400959
x3=−64.0628124891366x_{3} = -64.0628124891366
x4=51.4964418747775x_{4} = 51.4964418747775
x5=−95.4787390250346x_{5} = -95.4787390250346
x6=−1.23095941734077x_{6} = -1.23095941734077
x7=−82.9123684106754x_{7} = -82.9123684106754
x8=5.05222588983881x_{8} = 5.05222588983881
x9=95.4787390250346x_{9} = 95.4787390250346
x10=340.522966005038x_{10} = 340.522966005038
x11=−70.3459977963162x_{11} = -70.3459977963162
x12=36.4681524257367x_{12} = 36.4681524257367
x13=23.9017818113776x_{13} = 23.9017818113776
x14=17.618596504198x_{14} = 17.618596504198
x15=−74.1672642688143x_{15} = -74.1672642688143
x16=−51.4964418747775x_{16} = -51.4964418747775
x17=−86.7336348831734x_{17} = -86.7336348831734
x18=−42.7513377329163x_{18} = -42.7513377329163
x19=−57.7796271819571x_{19} = -57.7796271819571
x20=74.1672642688143x_{20} = 74.1672642688143
x21=57.7796271819571x_{21} = 57.7796271819571
x22=61.6008936544551x_{22} = 61.6008936544551
x23=67.8840789616347x_{23} = 67.8840789616347
x24=−38.9300712604183x_{24} = -38.9300712604183
x25=89.195553717855x_{25} = 89.195553717855
x26=−89.195553717855x_{26} = -89.195553717855
x27=−17.618596504198x_{27} = -17.618596504198
x28=−30.1849671185572x_{28} = -30.1849671185572
x29=−26.3637006460591x_{29} = -26.3637006460591
x30=−11.3354111970184x_{30} = -11.3354111970184
x31=−55.3177083472755x_{31} = -55.3177083472755
x32=86.7336348831734x_{32} = 86.7336348831734
x33=−5.05222588983881x_{33} = -5.05222588983881
x34=93.016820190353x_{34} = 93.016820190353
x35=76.6291831034958x_{35} = 76.6291831034958
x36=−13.7973300316999x_{36} = -13.7973300316999
x37=49.0345230400959x_{37} = 49.0345230400959
x38=42.7513377329163x_{38} = 42.7513377329163
x39=1.23095941734077x_{39} = 1.23095941734077
x40=−32.6468859532387x_{40} = -32.6468859532387
x41=−7.51414472452036x_{41} = -7.51414472452036
x42=704.947713821454x_{42} = 704.947713821454
x43=80.4504495759938x_{43} = 80.4504495759938
x44=−36.4681524257367x_{44} = -36.4681524257367
x45=70.3459977963162x_{45} = 70.3459977963162
x46=30.1849671185572x_{46} = 30.1849671185572
x47=−93.016820190353x_{47} = -93.016820190353
x48=−99.3000054975326x_{48} = -99.3000054975326
x49=7.51414472452036x_{49} = 7.51414472452036
x50=13.7973300316999x_{50} = 13.7973300316999
x51=28438.9276597121x_{51} = 28438.9276597121
x52=26.3637006460591x_{52} = 26.3637006460591
x53=−20.0805153388795x_{53} = -20.0805153388795
x54=−45.2132565675979x_{54} = -45.2132565675979
x55=−61.6008936544551x_{55} = -61.6008936544551
x56=82.9123684106754x_{56} = 82.9123684106754
x57=−80.4504495759938x_{57} = -80.4504495759938
x58=32.6468859532387x_{58} = 32.6468859532387
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to Abs(3*cos(x) - 1).
∣−1+3cos⁡(0)∣\left|{-1 + 3 \cos{\left(0 \right)}}\right|
The result:
f(0)=2f{\left(0 \right)} = 2
The point:
(0, 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
−3sin⁡(x)sign⁡(3cos⁡(x)−1)=0- 3 \sin{\left(x \right)} \operatorname{sign}{\left(3 \cos{\left(x \right)} - 1 \right)} = 0
Solve this equation
The roots of this equation
x1=−75.398223686155x_{1} = -75.398223686155
x2=47.1238898038469x_{2} = 47.1238898038469
x3=−31.4159265358979x_{3} = -31.4159265358979
x4=9.42477796076938x_{4} = 9.42477796076938
x5=−34.5575191894877x_{5} = -34.5575191894877
x6=−97.3893722612836x_{6} = -97.3893722612836
x7=−62.8318530717959x_{7} = -62.8318530717959
x8=87.9645943005142x_{8} = 87.9645943005142
x9=−358.141562509236x_{9} = -358.141562509236
x10=−87.9645943005142x_{10} = -87.9645943005142
x11=−3.14159265358979x_{11} = -3.14159265358979
x12=6.28318530717959x_{12} = 6.28318530717959
x13=59.6902604182061x_{13} = 59.6902604182061
x14=−47.1238898038469x_{14} = -47.1238898038469
x15=−40.8407044966673x_{15} = -40.8407044966673
x16=100.530964914873x_{16} = 100.530964914873
x17=62.8318530717959x_{17} = 62.8318530717959
x18=3.14159265358979x_{18} = 3.14159265358979
x19=28.2743338823081x_{19} = 28.2743338823081
x20=−69.1150383789755x_{20} = -69.1150383789755
x21=97.3893722612836x_{21} = 97.3893722612836
x22=12.5663706143592x_{22} = 12.5663706143592
x23=94.2477796076938x_{23} = 94.2477796076938
x24=31.4159265358979x_{24} = 31.4159265358979
x25=25.1327412287183x_{25} = 25.1327412287183
x26=−37.6991118430775x_{26} = -37.6991118430775
x27=−94.2477796076938x_{27} = -94.2477796076938
x28=−59.6902604182061x_{28} = -59.6902604182061
x29=−56.5486677646163x_{29} = -56.5486677646163
x30=81.6814089933346x_{30} = 81.6814089933346
x31=43.9822971502571x_{31} = 43.9822971502571
x32=−91.106186954104x_{32} = -91.106186954104
x33=15.707963267949x_{33} = 15.707963267949
x34=34.5575191894877x_{34} = 34.5575191894877
x35=21.9911485751286x_{35} = 21.9911485751286
x36=40.8407044966673x_{36} = 40.8407044966673
x37=69.1150383789755x_{37} = 69.1150383789755
x38=37.6991118430775x_{38} = 37.6991118430775
x39=65.9734457253857x_{39} = 65.9734457253857
x40=−72.2566310325652x_{40} = -72.2566310325652
x41=−131.946891450771x_{41} = -131.946891450771
x42=−21.9911485751286x_{42} = -21.9911485751286
x43=91.106186954104x_{43} = 91.106186954104
x44=53.4070751110265x_{44} = 53.4070751110265
x45=−28.2743338823081x_{45} = -28.2743338823081
x46=56.5486677646163x_{46} = 56.5486677646163
x47=−65.9734457253857x_{47} = -65.9734457253857
x48=−18.8495559215388x_{48} = -18.8495559215388
x49=−100.530964914873x_{49} = -100.530964914873
x50=−53.4070751110265x_{50} = -53.4070751110265
x51=−15.707963267949x_{51} = -15.707963267949
x52=84.8230016469244x_{52} = 84.8230016469244
x53=72.2566310325652x_{53} = 72.2566310325652
x54=18.8495559215388x_{54} = 18.8495559215388
x55=0x_{55} = 0
x56=−43.9822971502571x_{56} = -43.9822971502571
x57=−84.8230016469244x_{57} = -84.8230016469244
x58=−78.5398163397448x_{58} = -78.5398163397448
x59=−12.5663706143592x_{59} = -12.5663706143592
x60=75.398223686155x_{60} = 75.398223686155
x61=−6.28318530717959x_{61} = -6.28318530717959
x62=78.5398163397448x_{62} = 78.5398163397448
x63=−50.2654824574367x_{63} = -50.2654824574367
x64=−81.6814089933346x_{64} = -81.6814089933346
x65=50.2654824574367x_{65} = 50.2654824574367
x66=−9.42477796076938x_{66} = -9.42477796076938
x67=−292.168116783851x_{67} = -292.168116783851
x68=−25.1327412287183x_{68} = -25.1327412287183
The values of the extrema at the points:
(-75.39822368615503, 2)

(47.1238898038469, 4)

(-31.41592653589793, 2)

(9.42477796076938, 4)

(-34.55751918948773, 4)

(-97.3893722612836, 4)

(-62.83185307179586, 2)

(87.96459430051421, 2)

(-358.14156250923645, 2)

(-87.96459430051421, 2)

(-3.141592653589793, 4)

(6.283185307179586, 2)

(59.69026041820607, 4)

(-47.1238898038469, 4)

(-40.840704496667314, 4)

(100.53096491487338, 2)

(62.83185307179586, 2)

(3.141592653589793, 4)

(28.274333882308138, 4)

(-69.11503837897546, 2)

(97.3893722612836, 4)

(12.566370614359172, 2)

(94.2477796076938, 2)

(31.41592653589793, 2)

(25.132741228718345, 2)

(-37.69911184307752, 2)

(-94.2477796076938, 2)

(-59.69026041820607, 4)

(-56.548667764616276, 2)

(81.68140899333463, 2)

(43.982297150257104, 2)

(-91.106186954104, 4)

(15.707963267948966, 4)

(34.55751918948773, 4)

(21.991148575128552, 4)

(40.840704496667314, 4)

(69.11503837897546, 2)

(37.69911184307752, 2)

(65.97344572538566, 4)

(-72.25663103256524, 4)

(-131.94689145077132, 2)

(-21.991148575128552, 4)

(91.106186954104, 4)

(53.40707511102649, 4)

(-28.274333882308138, 4)

(56.548667764616276, 2)

(-65.97344572538566, 4)

(-18.84955592153876, 2)

(-100.53096491487338, 2)

(-53.40707511102649, 4)

(-15.707963267948966, 4)

(84.82300164692441, 4)

(72.25663103256524, 4)

(18.84955592153876, 2)

(0, 2)

(-43.982297150257104, 2)

(-84.82300164692441, 4)

(-78.53981633974483, 4)

(-12.566370614359172, 2)

(75.39822368615503, 2)

(-6.283185307179586, 2)

(78.53981633974483, 4)

(-50.26548245743669, 2)

(-81.68140899333463, 2)

(50.26548245743669, 2)

(-9.42477796076938, 4)

(-292.1681167838508, 4)

(-25.132741228718345, 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:
The function has no minima
Maxima of the function at points:
x68=−75.398223686155x_{68} = -75.398223686155
x68=47.1238898038469x_{68} = 47.1238898038469
x68=−31.4159265358979x_{68} = -31.4159265358979
x68=9.42477796076938x_{68} = 9.42477796076938
x68=−34.5575191894877x_{68} = -34.5575191894877
x68=−97.3893722612836x_{68} = -97.3893722612836
x68=−62.8318530717959x_{68} = -62.8318530717959
x68=87.9645943005142x_{68} = 87.9645943005142
x68=−358.141562509236x_{68} = -358.141562509236
x68=−87.9645943005142x_{68} = -87.9645943005142
x68=−3.14159265358979x_{68} = -3.14159265358979
x68=6.28318530717959x_{68} = 6.28318530717959
x68=59.6902604182061x_{68} = 59.6902604182061
x68=−47.1238898038469x_{68} = -47.1238898038469
x68=−40.8407044966673x_{68} = -40.8407044966673
x68=100.530964914873x_{68} = 100.530964914873
x68=62.8318530717959x_{68} = 62.8318530717959
x68=3.14159265358979x_{68} = 3.14159265358979
x68=28.2743338823081x_{68} = 28.2743338823081
x68=−69.1150383789755x_{68} = -69.1150383789755
x68=97.3893722612836x_{68} = 97.3893722612836
x68=12.5663706143592x_{68} = 12.5663706143592
x68=94.2477796076938x_{68} = 94.2477796076938
x68=31.4159265358979x_{68} = 31.4159265358979
x68=25.1327412287183x_{68} = 25.1327412287183
x68=−37.6991118430775x_{68} = -37.6991118430775
x68=−94.2477796076938x_{68} = -94.2477796076938
x68=−59.6902604182061x_{68} = -59.6902604182061
x68=−56.5486677646163x_{68} = -56.5486677646163
x68=81.6814089933346x_{68} = 81.6814089933346
x68=43.9822971502571x_{68} = 43.9822971502571
x68=−91.106186954104x_{68} = -91.106186954104
x68=15.707963267949x_{68} = 15.707963267949
x68=34.5575191894877x_{68} = 34.5575191894877
x68=21.9911485751286x_{68} = 21.9911485751286
x68=40.8407044966673x_{68} = 40.8407044966673
x68=69.1150383789755x_{68} = 69.1150383789755
x68=37.6991118430775x_{68} = 37.6991118430775
x68=65.9734457253857x_{68} = 65.9734457253857
x68=−72.2566310325652x_{68} = -72.2566310325652
x68=−131.946891450771x_{68} = -131.946891450771
x68=−21.9911485751286x_{68} = -21.9911485751286
x68=91.106186954104x_{68} = 91.106186954104
x68=53.4070751110265x_{68} = 53.4070751110265
x68=−28.2743338823081x_{68} = -28.2743338823081
x68=56.5486677646163x_{68} = 56.5486677646163
x68=−65.9734457253857x_{68} = -65.9734457253857
x68=−18.8495559215388x_{68} = -18.8495559215388
x68=−100.530964914873x_{68} = -100.530964914873
x68=−53.4070751110265x_{68} = -53.4070751110265
x68=−15.707963267949x_{68} = -15.707963267949
x68=84.8230016469244x_{68} = 84.8230016469244
x68=72.2566310325652x_{68} = 72.2566310325652
x68=18.8495559215388x_{68} = 18.8495559215388
x68=0x_{68} = 0
x68=−43.9822971502571x_{68} = -43.9822971502571
x68=−84.8230016469244x_{68} = -84.8230016469244
x68=−78.5398163397448x_{68} = -78.5398163397448
x68=−12.5663706143592x_{68} = -12.5663706143592
x68=75.398223686155x_{68} = 75.398223686155
x68=−6.28318530717959x_{68} = -6.28318530717959
x68=78.5398163397448x_{68} = 78.5398163397448
x68=−50.2654824574367x_{68} = -50.2654824574367
x68=−81.6814089933346x_{68} = -81.6814089933346
x68=50.2654824574367x_{68} = 50.2654824574367
x68=−9.42477796076938x_{68} = -9.42477796076938
x68=−292.168116783851x_{68} = -292.168116783851
x68=−25.1327412287183x_{68} = -25.1327412287183
Decreasing at intervals
(−∞,−358.141562509236]\left(-\infty, -358.141562509236\right]
Increasing at intervals
[100.530964914873,∞)\left[100.530964914873, \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
3(6sin⁡2(x)δ(3cos⁡(x)−1)−cos⁡(x)sign⁡(3cos⁡(x)−1))=03 \left(6 \sin^{2}{\left(x \right)} \delta\left(3 \cos{\left(x \right)} - 1\right) - \cos{\left(x \right)} \operatorname{sign}{\left(3 \cos{\left(x \right)} - 1 \right)}\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→−∞∣3cos⁡(x)−1∣=∣⟨−4,2⟩∣\lim_{x \to -\infty} \left|{3 \cos{\left(x \right)} - 1}\right| = \left|{\left\langle -4, 2\right\rangle}\right|
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=∣⟨−4,2⟩∣y = \left|{\left\langle -4, 2\right\rangle}\right|
lim⁡x→∞∣3cos⁡(x)−1∣=∣⟨−4,2⟩∣\lim_{x \to \infty} \left|{3 \cos{\left(x \right)} - 1}\right| = \left|{\left\langle -4, 2\right\rangle}\right|
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=∣⟨−4,2⟩∣y = \left|{\left\langle -4, 2\right\rangle}\right|
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of Abs(3*cos(x) - 1), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(∣3cos⁡(x)−1∣x)=0\lim_{x \to -\infty}\left(\frac{\left|{3 \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→∞(∣3cos⁡(x)−1∣x)=0\lim_{x \to \infty}\left(\frac{\left|{3 \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:
∣3cos⁡(x)−1∣=∣3cos⁡(x)−1∣\left|{3 \cos{\left(x \right)} - 1}\right| = \left|{3 \cos{\left(x \right)} - 1}\right|
- Yes
∣3cos⁡(x)−1∣=−∣3cos⁡(x)−1∣\left|{3 \cos{\left(x \right)} - 1}\right| = - \left|{3 \cos{\left(x \right)} - 1}\right|
- No
so, the function
is
even
    © Mister Exam – Calculator