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Plotting a function graph/ -2cos(x-1)+2

Graphing y = -2cos(x-1)+2

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = -2*cos(x - 1) + 2
f(x)=2−2cos⁡(x−1)f{\left(x \right)} = 2 - 2 \cos{\left(x - 1 \right)} 
f = 2 - 2*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:
2−2cos⁡(x−1)=02 - 2 \cos{\left(x - 1 \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=1x_{1} = 1
x2=1+2πx_{2} = 1 + 2 \pi
Numerical solution
x1=−36.6991114896456x_{1} = -36.6991114896456
x2=38.6991123485095x_{2} = 38.6991123485095
x3=19.8495560331216x_{3} = 19.8495560331216
x4=−80.6814094421164x_{4} = -80.6814094421164
x5=−5.28318554617199x_{5} = -5.28318554617199
x6=−17.8495557968376x_{6} = -17.8495557968376
x7=101.530964423296x_{7} = 101.530964423296
x8=88.964593962401x_{8} = 88.964593962401
x9=95.2477791775533x_{9} = 95.2477791775533
x10=−30.4159270346054x_{10} = -30.4159270346054
x11=13.5663695105623x_{11} = 13.5663695105623
x12=13.5663701178499x_{12} = 13.5663701178499
x13=19.8495554159042x_{13} = 19.8495554159042
x14=−42.9822975075327x_{14} = -42.9822975075327
x15=63.8318531931265x_{15} = 63.8318531931265
x16=7.28318566694506x_{16} = 7.28318566694506
x17=82.6814095007727x_{17} = 82.6814095007727
x18=32.4159264380897x_{18} = 32.4159264380897
x19=82.6814100849983x_{19} = 82.6814100849983
x20=57.5486680202546x_{20} = 57.5486680202546
x21=13.5663708623807x_{21} = 13.5663708623807
x22=−55.5486678808888x_{22} = -55.5486678808888
x23=−17.8495554769708x_{23} = -17.8495554769708
x24=70.115038805167x_{24} = 70.115038805167
x25=82.6814087697046x_{25} = 82.6814087697046
x26=7.28318486747256x_{26} = 7.28318486747256
x27=−24.1327417326819x_{27} = -24.1327417326819
x28=−93.2477791093241x_{28} = -93.2477791093241
x29=51.2654820224528x_{29} = 51.2654820224528
x30=1.00000044735768x_{30} = 1.00000044735768
x31=38.6991116117304x_{31} = 38.6991116117304
x32=−24.1327411271999x_{32} = -24.1327411271999
x33=−86.9645946639967x_{33} = -86.9645946639967
x34=−74.3982234536233x_{34} = -74.3982234536233
x35=−74.3982241868753x_{35} = -74.3982241868753
x36=−86.9645938625694x_{36} = -86.9645938625694
x37=57.5486672704854x_{37} = 57.5486672704854
x38=−68.1150382879088x_{38} = -68.1150382879088
x39=−24.1327410652463x_{39} = -24.1327410652463
x40=26.1327414173968x_{40} = 26.1327414173968
x41=−55.5486672497774x_{41} = -55.5486672497774
x42=−49.2654827040023x_{42} = -49.2654827040023
x43=32.4159270482023x_{43} = 32.4159270482023
x44=76.3982235519432x_{44} = 76.3982235519432
x45=70.1150379433459x_{45} = 70.1150379433459
x46=−93.2477787150689x_{46} = -93.2477787150689
x47=−17.849556337754x_{47} = -17.849556337754
x48=−93.2477798617602x_{48} = -93.2477798617602
x49=−80.681408426653x_{49} = -80.681408426653
x50=−11.5663706722839x_{50} = -11.5663706722839
x51=70.1150387480058x_{51} = 70.1150387480058
x52=−61.8318534770036x_{52} = -61.8318534770036
x53=−5.28318480393057x_{53} = -5.28318480393057
x54=95.2477799798464x_{54} = 95.2477799798464
x55=−11.5663707217028x_{55} = -11.5663707217028
x56=−36.6991122873847x_{56} = -36.6991122873847
x57=26.1327416644778x_{57} = 26.1327416644778
x58=26.132740802384x_{58} = 26.132740802384
x59=−30.4159262955782x_{59} = -30.4159262955782
x60=51.2654828234429x_{60} = 51.2654828234429
x61=−80.6814086463278x_{61} = -80.6814086463278
x62=−61.8318529849279x_{62} = -61.8318529849279
x63=76.3982241967455x_{63} = 76.3982241967455
x64=76.3982235970974x_{64} = 76.3982235970974
x65=−30.4159275239453x_{65} = -30.4159275239453
x66=19.8495560513666x_{66} = 19.8495560513666
x67=63.8318525648093x_{67} = 63.8318525648093
x68=−68.1150388809337x_{68} = -68.1150388809337
x69=−99.5309643991898x_{69} = -99.5309643991898
x70=44.9822976021389x_{70} = 44.9822976021389
x71=44.9822968056677x_{71} = 44.9822968056677
x72=−991.74327913164x_{72} = -991.74327913164
x73=0.999999649022634x_{73} = 0.999999649022634
x74=88.9645947567955x_{74} = 88.9645947567955
x75=−11.5663701006429x_{75} = -11.5663701006429
x76=−42.9822967075459x_{76} = -42.9822967075459
x77=−55.5486677788x_{77} = -55.5486677788
x78=−99.5309650399519x_{78} = -99.5309650399519
x79=−61.8318526191255x_{79} = -61.8318526191255
x80=32.4159264376201x_{80} = 32.4159264376201
x81=−74.3982245556249x_{81} = -74.3982245556249
x82=−49.2654819565432x_{82} = -49.2654819565432
x83=63.8318531648007x_{83} = 63.8318531648007
x84=−68.1150381847295x_{84} = -68.1150381847295
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to -2*cos(x - 1) + 2.
2−2cos⁡(−1)2 - 2 \cos{\left(-1 \right)}
The result:
f(0)=2−2cos⁡(1)f{\left(0 \right)} = 2 - 2 \cos{\left(1 \right)}
The point:
(0, 2 - 2*cos(1))
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
2sin⁡(x−1)=02 \sin{\left(x - 1 \right)} = 0
Solve this equation
The roots of this equation
x1=1x_{1} = 1
x2=1+πx_{2} = 1 + \pi
The values of the extrema at the points:
(1, 0)

(1 + pi, 4)


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=1x_{1} = 1
Maxima of the function at points:
x1=1+πx_{1} = 1 + \pi
Decreasing at intervals
[1,1+π]\left[1, 1 + \pi\right]
Increasing at intervals
(−∞,1]∪[1+π,∞)\left(-\infty, 1\right] \cup \left[1 + \pi, \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
2cos⁡(x−1)=02 \cos{\left(x - 1 \right)} = 0
Solve this equation
The roots of this equation
x1=1+π2x_{1} = 1 + \frac{\pi}{2}
x2=1+3π2x_{2} = 1 + \frac{3 \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
(−∞,1+π2]∪[1+3π2,∞)\left(-\infty, 1 + \frac{\pi}{2}\right] \cup \left[1 + \frac{3 \pi}{2}, \infty\right)
Convex at the intervals
[1+π2,1+3π2]\left[1 + \frac{\pi}{2}, 1 + \frac{3 \pi}{2}\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(2−2cos⁡(x−1))=⟨0,4⟩\lim_{x \to -\infty}\left(2 - 2 \cos{\left(x - 1 \right)}\right) = \left\langle 0, 4\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨0,4⟩y = \left\langle 0, 4\right\rangle
lim⁡x→∞(2−2cos⁡(x−1))=⟨0,4⟩\lim_{x \to \infty}\left(2 - 2 \cos{\left(x - 1 \right)}\right) = \left\langle 0, 4\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨0,4⟩y = \left\langle 0, 4\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of -2*cos(x - 1) + 2, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(2−2cos⁡(x−1)x)=0\lim_{x \to -\infty}\left(\frac{2 - 2 \cos{\left(x - 1 \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(2−2cos⁡(x−1)x)=0\lim_{x \to \infty}\left(\frac{2 - 2 \cos{\left(x - 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:
2−2cos⁡(x−1)=2−2cos⁡(x+1)2 - 2 \cos{\left(x - 1 \right)} = 2 - 2 \cos{\left(x + 1 \right)}
- No
2−2cos⁡(x−1)=2cos⁡(x+1)−22 - 2 \cos{\left(x - 1 \right)} = 2 \cos{\left(x + 1 \right)} - 2
- No
so, the function
not is
neither even, nor odd
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