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

Graphing y = cos(x/2)+1

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          /x\    
f(x) = cos|-| + 1
          \2/
f(x)=cos⁡(x2)+1f{\left(x \right)} = \cos{\left(\frac{x}{2} \right)} + 1 
f = cos(x/2) + 1
The graph of the function
02468-8-6-4-2-101004
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⁡(x2)+1=0\cos{\left(\frac{x}{2} \right)} + 1 = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=2πx_{1} = 2 \pi
Numerical solution
x1=31.4159255107699x_{1} = 31.4159255107699
x2=−18.8495543136467x_{2} = -18.8495543136467
x3=−81.6814098544517x_{3} = -81.6814098544517
x4=−18.8495554800568x_{4} = -18.8495554800568
x5=56.548668769019x_{5} = 56.548668769019
x6=973.893720162307x_{6} = 973.893720162307
x7=−420.973417050673x_{7} = -420.973417050673
x8=−6.28318627375063x_{8} = -6.28318627375063
x9=69.1150394039813x_{9} = 69.1150394039813
x10=−6.28318500560576x_{10} = -6.28318500560576
x11=−31.4159274639505x_{11} = -31.4159274639505
x12=−94.2477790629797x_{12} = -94.2477790629797
x13=43.9822963919738x_{13} = 43.9822963919738
x14=18.8495549415754x_{14} = 18.8495549415754
x15=69.115037955532x_{15} = 69.115037955532
x16=6.28318445599649x_{16} = 6.28318445599649
x17=56.5486675992574x_{17} = 56.5486675992574
x18=−69.1150393645777x_{18} = -69.1150393645777
x19=56.5486681672153x_{19} = 56.5486681672153
x20=−56.5486677946696x_{20} = -56.5486677946696
x21=31.4159275256971x_{21} = 31.4159275256971
x22=43.9822980146158x_{22} = 43.9822980146158
x23=−31.4159267183799x_{23} = -31.4159267183799
x24=−69.1150405916551x_{24} = -69.1150405916551
x25=−31.4159258676764x_{25} = -31.4159258676764
x26=18.8495569537229x_{26} = 18.8495569537229
x27=−18.8495563601837x_{27} = -18.8495563601837
x28=−31.4159269573538x_{28} = -31.4159269573538
x29=−69.115038701061x_{29} = -69.115038701061
x30=−69.1150378269834x_{30} = -69.1150378269834
x31=81.6814079898679x_{31} = 81.6814079898679
x32=56.5486684458067x_{32} = 56.5486684458067
x33=−18.8495548993085x_{33} = -18.8495548993085
x34=69.1150368714384x_{34} = 69.1150368714384
x35=−6.28318597231491x_{35} = -6.28318597231491
x36=6.28318624191326x_{36} = 6.28318624191326
x37=56.5486668446775x_{37} = 56.5486668446775
x38=−6.28318512498487x_{38} = -6.28318512498487
x39=81.681409932803x_{39} = 81.681409932803
x40=56.5486672894687x_{40} = 56.5486672894687
x41=94.2477803922665x_{41} = 94.2477803922665
x42=94.2477805288104x_{42} = 94.2477805288104
x43=6.28318528419795x_{43} = 6.28318528419795
x44=−56.5486687887842x_{44} = -56.5486687887842
x45=81.6814094388795x_{45} = 81.6814094388795
x46=−81.68140804614x_{46} = -81.68140804614
x47=−81.6814090382823x_{47} = -81.6814090382823
x48=43.9822979229286x_{48} = 43.9822979229286
x49=−69.115037346238x_{49} = -69.115037346238
x50=31.415926404003x_{50} = 31.415926404003
x51=43.9822962232996x_{51} = 43.9822962232996
x52=−6.28318630492991x_{52} = -6.28318630492991
x53=−43.9822963521153x_{53} = -43.9822963521153
x54=−56.5486667805662x_{54} = -56.5486667805662
x55=−6.28318437561383x_{55} = -6.28318437561383
x56=31.4159259869291x_{56} = 31.4159259869291
x57=18.8495561010751x_{57} = 18.8495561010751
x58=6.28318608026041x_{58} = 6.28318608026041
x59=81.6814083413212x_{59} = 81.6814083413212
x60=−94.2477794433785x_{60} = -94.2477794433785
x61=−94.2477802857943x_{61} = -94.2477802857943
x62=43.982297169474x_{62} = 43.982297169474
x63=−31.4159267991785x_{63} = -31.4159267991785
x64=94.2477796093523x_{64} = 94.2477796093523
x65=69.1150397628024x_{65} = 69.1150397628024
x66=−56.5486683277755x_{66} = -56.5486683277755
x67=−56.548667456373x_{67} = -56.548667456373
x68=81.6814091930517x_{68} = 81.6814091930517
x69=−18.8495572113627x_{69} = -18.8495572113627
x70=−6.28318480992034x_{70} = -6.28318480992034
x71=94.2477787666659x_{71} = 94.2477787666659
x72=−18.8495569383691x_{72} = -18.8495569383691
x73=−81.6814097251289x_{73} = -81.6814097251289
x74=−31.4159255258029x_{74} = -31.4159255258029
x75=69.1150373651011x_{75} = 69.1150373651011
x76=18.8495554920234x_{76} = 18.8495554920234
x77=−94.2477786842917x_{77} = -94.2477786842917
x78=−43.9822979202134x_{78} = -43.9822979202134
x79=6.28318454684724x_{79} = 6.28318454684724
x80=18.8495564877336x_{80} = 18.8495564877336
x81=18.8495556154405x_{81} = 18.8495556154405
x82=−43.9822971745293x_{82} = -43.9822971745293
x83=31.4159268602638x_{83} = 31.4159268602638
x84=−43.9822980632848x_{84} = -43.9822980632848
x85=−81.6814082319334x_{85} = -81.6814082319334
x86=69.115038835587x_{86} = 69.115038835587
x87=−43.982296295854x_{87} = -43.982296295854
x88=−94.2477805989545x_{88} = -94.2477805989545
x89=81.6814069775521x_{89} = 81.6814069775521
x90=−69.1150380827831x_{90} = -69.1150380827831
x91=94.2477788210844x_{91} = 94.2477788210844
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cos(x/2) + 1.
cos⁡(02)+1\cos{\left(\frac{0}{2} \right)} + 1
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
−sin⁡(x2)2=0- \frac{\sin{\left(\frac{x}{2} \right)}}{2} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=2πx_{2} = 2 \pi
The values of the extrema at the points:
(0, 2)

(2*pi, 0)


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=0x_{1} = 0
Decreasing at intervals
(−∞,0]∪[2π,∞)\left(-\infty, 0\right] \cup \left[2 \pi, \infty\right)
Increasing at intervals
[0,2π]\left[0, 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⁡(x2)4=0- \frac{\cos{\left(\frac{x}{2} \right)}}{4} = 0
Solve this equation
The roots of this equation
x1=πx_{1} = \pi
x2=3πx_{2} = 3 \pi

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
[π,3π]\left[\pi, 3 \pi\right]
Convex at the intervals
(−∞,π]∪[3π,∞)\left(-\infty, \pi\right] \cup \left[3 \pi, \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⁡(x2)+1)=⟨0,2⟩\lim_{x \to -\infty}\left(\cos{\left(\frac{x}{2} \right)} + 1\right) = \left\langle 0, 2\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨0,2⟩y = \left\langle 0, 2\right\rangle
lim⁡x→∞(cos⁡(x2)+1)=⟨0,2⟩\lim_{x \to \infty}\left(\cos{\left(\frac{x}{2} \right)} + 1\right) = \left\langle 0, 2\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨0,2⟩y = \left\langle 0, 2\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cos(x/2) + 1, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(cos⁡(x2)+1x)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(\frac{x}{2} \right)} + 1}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(cos⁡(x2)+1x)=0\lim_{x \to \infty}\left(\frac{\cos{\left(\frac{x}{2} \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⁡(x2)+1=cos⁡(x2)+1\cos{\left(\frac{x}{2} \right)} + 1 = \cos{\left(\frac{x}{2} \right)} + 1
- No
cos⁡(x2)+1=−cos⁡(x2)−1\cos{\left(\frac{x}{2} \right)} + 1 = - \cos{\left(\frac{x}{2} \right)} - 1
- No
so, the function
not is
neither even, nor odd
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