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Plotting a function graph/ cos(x)-1

Graphing y = cos(x)-1

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = cos(x) - 1
f(x)=cos⁡(x)−1f{\left(x \right)} = \cos{\left(x \right)} - 1 
f = cos(x) - 1
The graph of the function
0.00.51.01.52.02.53.03.54.04.55.05.56.02-4
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)−1=0\cos{\left(x \right)} - 1 = 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=81.6814084860076x_{1} = 81.6814084860076
x2=18.8495564031971x_{2} = 18.8495564031971
x3=94.2477800892631x_{3} = 94.2477800892631
x4=6.28318579821791x_{4} = 6.28318579821791
x5=6.28318528420851x_{5} = 6.28318528420851
x6=−6.2831851275477x_{6} = -6.2831851275477
x7=75.3982240031607x_{7} = 75.3982240031607
x8=37.6991115173992x_{8} = 37.6991115173992
x9=−18.8495552124105x_{9} = -18.8495552124105
x10=−75.3982238741744x_{10} = -75.3982238741744
x11=−37.6991113479743x_{11} = -37.6991113479743
x12=−31.4159267157965x_{12} = -31.4159267157965
x13=−31.4159260208155x_{13} = -31.4159260208155
x14=75.3982232188727x_{14} = 75.3982232188727
x15=43.9822974733639x_{15} = 43.9822974733639
x16=−50.2654829667315x_{16} = -50.2654829667315
x17=−94.2477801171671x_{17} = -94.2477801171671
x18=37.6991120311338x_{18} = 37.6991120311338
x19=12.5663710110881x_{19} = 12.5663710110881
x20=−87.9645943586158x_{20} = -87.9645943586158
x21=62.8318535568358x_{21} = 62.8318535568358
x22=37.6991113348642x_{22} = 37.6991113348642
x23=−75.3982231720141x_{23} = -75.3982231720141
x24=100.530964759815x_{24} = 100.530964759815
x25=56.5486668532011x_{25} = 56.5486668532011
x26=−94.2477797298079x_{26} = -94.2477797298079
x27=100.530965157364x_{27} = 100.530965157364
x28=0x_{28} = 0
x29=50.2654821322586x_{29} = 50.2654821322586
x30=−81.6814090382277x_{30} = -81.6814090382277
x31=−56.5486674685864x_{31} = -56.5486674685864
x32=−6.2831858160515x_{32} = -6.2831858160515
x33=−25.1327415297174x_{33} = -25.1327415297174
x34=87.9645946044253x_{34} = 87.9645946044253
x35=6.28318626747926x_{35} = 6.28318626747926
x36=50.2654829439723x_{36} = 50.2654829439723
x37=56.5486676011951x_{37} = 56.5486676011951
x38=69.1150379887504x_{38} = 69.1150379887504
x39=−43.9822976246252x_{39} = -43.9822976246252
x40=12.5663711301703x_{40} = 12.5663711301703
x41=69.115038794053x_{41} = 69.115038794053
x42=−43.9822971745392x_{42} = -43.9822971745392
x43=−69.1150379045123x_{43} = -69.1150379045123
x44=43.9822971694647x_{44} = 43.9822971694647
x45=6.28318500093652x_{45} = 6.28318500093652
x46=31.4159268459961x_{46} = 31.4159268459961
x47=−87.964593928489x_{47} = -87.964593928489
x48=18.8495556275525x_{48} = 18.8495556275525
x49=−37.6991121287155x_{49} = -37.6991121287155
x50=25.1327416384075x_{50} = 25.1327416384075
x51=−62.8318534787248x_{51} = -62.8318534787248
x52=−56.5486682426592x_{52} = -56.5486682426592
x53=−81.6814075578313x_{53} = -81.6814075578313
x54=−12.5663710889626x_{54} = -12.5663710889626
x55=−12.5663703112531x_{55} = -12.5663703112531
x56=87.964594335905x_{56} = 87.964594335905
x57=−62.831852673202x_{57} = -62.831852673202
x58=50.2654824463392x_{58} = 50.2654824463392
x59=−100.530964626003x_{59} = -100.530964626003
x60=−81.6814084945807x_{60} = -81.6814084945807
x61=81.6814085860518x_{61} = 81.6814085860518
x62=43.9822966661001x_{62} = 43.9822966661001
x63=−43.9822967932182x_{63} = -43.9822967932182
x64=−37.6991118772631x_{64} = -37.6991118772631
x65=−18.8495555173448x_{65} = -18.8495555173448
x66=81.6814091897036x_{66} = 81.6814091897036
x67=94.2477792651059x_{67} = 94.2477792651059
x68=−87.9645947692094x_{68} = -87.9645947692094
x69=−50.2654822863493x_{69} = -50.2654822863493
x70=56.5486682809363x_{70} = 56.5486682809363
x71=−81.6814092565354x_{71} = -81.6814092565354
x72=−50.265482641087x_{72} = -50.265482641087
x73=−94.2477794452815x_{73} = -94.2477794452815
x74=31.4159260648825x_{74} = 31.4159260648825
x75=−6.28318555849548x_{75} = -6.28318555849548
x76=87.9645938121814x_{76} = 87.9645938121814
x77=12.5663704426592x_{77} = 12.5663704426592
x78=56.5486680806249x_{78} = 56.5486680806249
x79=−18.8495563230046x_{79} = -18.8495563230046
x80=62.8318527849002x_{80} = 62.8318527849002
x81=−69.1150386869085x_{81} = -69.1150386869085
x82=25.1327408328211x_{82} = 25.1327408328211
x83=94.2477796093523x_{83} = 94.2477796093523
x84=−75.3982231045728x_{84} = -75.3982231045728
x85=−25.1327407505866x_{85} = -25.1327407505866
x86=−31.4159260507536x_{86} = -31.4159260507536
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cos(x) - 1.
−1+cos⁡(0)-1 + \cos{\left(0 \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)=0- \sin{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=πx_{2} = \pi
The values of the extrema at the points:
(0, 0)

(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=πx_{1} = \pi
Maxima of the function at points:
x1=0x_{1} = 0
Decreasing at intervals
(−∞,0]∪[π,∞)\left(-\infty, 0\right] \cup \left[\pi, \infty\right)
Increasing at intervals
[0,π]\left[0, \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)=0- \cos{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=π2x_{1} = \frac{\pi}{2}
x2=3π2x_{2} = \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
[π2,3π2]\left[\frac{\pi}{2}, \frac{3 \pi}{2}\right]
Convex at the intervals
(−∞,π2]∪[3π2,∞)\left(-\infty, \frac{\pi}{2}\right] \cup \left[\frac{3 \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)−1)=⟨−2,0⟩\lim_{x \to -\infty}\left(\cos{\left(x \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)−1)=⟨−2,0⟩\lim_{x \to \infty}\left(\cos{\left(x \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) - 1, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(cos⁡(x)−1x)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)} - 1}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(cos⁡(x)−1x)=0\lim_{x \to \infty}\left(\frac{\cos{\left(x \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)−1=cos⁡(x)−1\cos{\left(x \right)} - 1 = \cos{\left(x \right)} - 1
- Yes
cos⁡(x)−1=1−cos⁡(x)\cos{\left(x \right)} - 1 = 1 - \cos{\left(x \right)}
- No
so, the function
is
even
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