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Plotting a function graph/ -0.5*cos(x+pi/6)

Graphing y = -0.5*cos(x+pi/6)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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           /    pi\ 
       -cos|x + --| 
           \    6 / 
f(x) = -------------
             2
f(x)=−cos⁡(x+π6)2f{\left(x \right)} = - \frac{\cos{\left(x + \frac{\pi}{6} \right)}}{2} 
f = -cos(x + pi/6)/2
The graph of the function
02468-8-6-4-2-10101-1
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+π6)2=0- \frac{\cos{\left(x + \frac{\pi}{6} \right)}}{2} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=π3x_{1} = \frac{\pi}{3}
x2=4π3x_{2} = \frac{4 \pi}{3}
Numerical solution
x1=−96.342174710087x_{1} = -96.342174710087
x2=45.0294947014537x_{2} = 45.0294947014537
x3=98.4365698124802x_{3} = 98.4365698124802
x4=−46.0766922526503x_{4} = -46.0766922526503
x5=−17.8023583703422x_{5} = -17.8023583703422
x6=85.870199198121x_{6} = 85.870199198121
x7=−71.2094334813686x_{7} = -71.2094334813686
x8=35.6047167406843x_{8} = 35.6047167406843
x9=−80.634211442138x_{9} = -80.634211442138
x10=79.5870138909414x_{10} = 79.5870138909414
x11=−99.4837673636768x_{11} = -99.4837673636768
x12=−83.7758040957278x_{12} = -83.7758040957278
x13=−93.2005820564972x_{13} = -93.2005820564972
x14=−33.5103216382911x_{14} = -33.5103216382911
x15=−5644.39480094966x_{15} = -5644.39480094966
x16=26.1799387799149x_{16} = 26.1799387799149
x17=4.18879020478639x_{17} = 4.18879020478639
x18=89.0117918517108x_{18} = 89.0117918517108
x19=92.1533845053006x_{19} = 92.1533845053006
x20=82.7286065445312x_{20} = 82.7286065445312
x21=−14.6607657167524x_{21} = -14.6607657167524
x22=67.0206432765823x_{22} = 67.0206432765823
x23=−55.5014702134197x_{23} = -55.5014702134197
x24=−20.943951023932x_{24} = -20.943951023932
x25=−36.6519142918809x_{25} = -36.6519142918809
x26=60.7374579694027x_{26} = 60.7374579694027
x27=48.1710873550435x_{27} = 48.1710873550435
x28=104.71975511966x_{28} = 104.71975511966
x29=76.4454212373516x_{29} = 76.4454212373516
x30=16.7551608191456x_{30} = 16.7551608191456
x31=−5.23598775598299x_{31} = -5.23598775598299
x32=−77.4926187885482x_{32} = -77.4926187885482
x33=7.33038285837618x_{33} = 7.33038285837618
x34=−11.5191730631626x_{34} = -11.5191730631626
x35=57.5958653158129x_{35} = 57.5958653158129
x36=73.3038285837618x_{36} = 73.3038285837618
x37=32.4631240870945x_{37} = 32.4631240870945
x38=13.6135681655558x_{38} = 13.6135681655558
x39=−24.0855436775217x_{39} = -24.0855436775217
x40=63.8790506229925x_{40} = 63.8790506229925
x41=10.471975511966x_{41} = 10.471975511966
x42=−2.0943951023932x_{42} = -2.0943951023932
x43=−52.3598775598299x_{43} = -52.3598775598299
x44=−30.3687289847013x_{44} = -30.3687289847013
x45=19.8967534727354x_{45} = 19.8967534727354
x46=−68.0678408277789x_{46} = -68.0678408277789
x47=95.2949771588904x_{47} = 95.2949771588904
x48=−27.2271363311115x_{48} = -27.2271363311115
x49=−64.9262481741891x_{49} = -64.9262481741891
x50=−90.0589894029074x_{50} = -90.0589894029074
x51=−49.2182849062401x_{51} = -49.2182849062401
x52=41.8879020478639x_{52} = 41.8879020478639
x53=54.4542726622231x_{53} = 54.4542726622231
x54=51.3126800086333x_{54} = 51.3126800086333
x55=23.0383461263252x_{55} = 23.0383461263252
x56=−61.7846555205993x_{56} = -61.7846555205993
x57=29.3215314335047x_{57} = 29.3215314335047
x58=−58.6430628670095x_{58} = -58.6430628670095
x59=70.162235930172x_{59} = 70.162235930172
x60=1.0471975511966x_{60} = 1.0471975511966
x61=−8.37758040957278x_{61} = -8.37758040957278
x62=−39.7935069454707x_{62} = -39.7935069454707
x63=−74.3510261349584x_{63} = -74.3510261349584
x64=38.7463093942741x_{64} = 38.7463093942741
x65=−86.9173967493176x_{65} = -86.9173967493176
x66=−42.9350995990605x_{66} = -42.9350995990605
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to -cos(x + pi/6)/2.
−cos⁡(π6)2- \frac{\cos{\left(\frac{\pi}{6} \right)}}{2}
The result:
f(0)=−34f{\left(0 \right)} = - \frac{\sqrt{3}}{4}
The point:
(0, -sqrt(3)/4)
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+π6)2=0\frac{\sin{\left(x + \frac{\pi}{6} \right)}}{2} = 0
Solve this equation
The roots of this equation
x1=−π6x_{1} = - \frac{\pi}{6}
x2=5π6x_{2} = \frac{5 \pi}{6}
The values of the extrema at the points:
           /pi   pi\  
       -cos|-- - --|  
 -pi       \6    6 /  
(----, --------------)
  6          2        

          /pi   pi\ 
       sin|-- + --| 
 5*pi     \3    6 / 
(----, ------------)
  6         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=−π6x_{1} = - \frac{\pi}{6}
Maxima of the function at points:
x1=5π6x_{1} = \frac{5 \pi}{6}
Decreasing at intervals
[−π6,5π6]\left[- \frac{\pi}{6}, \frac{5 \pi}{6}\right]
Increasing at intervals
(−∞,−π6]∪[5π6,∞)\left(-\infty, - \frac{\pi}{6}\right] \cup \left[\frac{5 \pi}{6}, \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
cos⁡(x+π6)2=0\frac{\cos{\left(x + \frac{\pi}{6} \right)}}{2} = 0
Solve this equation
The roots of this equation
x1=π3x_{1} = \frac{\pi}{3}
x2=4π3x_{2} = \frac{4 \pi}{3}

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
(−∞,π3]∪[4π3,∞)\left(-\infty, \frac{\pi}{3}\right] \cup \left[\frac{4 \pi}{3}, \infty\right)
Convex at the intervals
[π3,4π3]\left[\frac{\pi}{3}, \frac{4 \pi}{3}\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+π6)2)=⟨−12,12⟩\lim_{x \to -\infty}\left(- \frac{\cos{\left(x + \frac{\pi}{6} \right)}}{2}\right) = \left\langle - \frac{1}{2}, \frac{1}{2}\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−12,12⟩y = \left\langle - \frac{1}{2}, \frac{1}{2}\right\rangle
lim⁡x→∞(−cos⁡(x+π6)2)=⟨−12,12⟩\lim_{x \to \infty}\left(- \frac{\cos{\left(x + \frac{\pi}{6} \right)}}{2}\right) = \left\langle - \frac{1}{2}, \frac{1}{2}\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−12,12⟩y = \left\langle - \frac{1}{2}, \frac{1}{2}\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of -cos(x + pi/6)/2, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(−cos⁡(x+π6)2x)=0\lim_{x \to -\infty}\left(- \frac{\cos{\left(x + \frac{\pi}{6} \right)}}{2 x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(−cos⁡(x+π6)2x)=0\lim_{x \to \infty}\left(- \frac{\cos{\left(x + \frac{\pi}{6} \right)}}{2 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+π6)2=−cos⁡(x−π6)2- \frac{\cos{\left(x + \frac{\pi}{6} \right)}}{2} = - \frac{\cos{\left(x - \frac{\pi}{6} \right)}}{2}
- No
−cos⁡(x+π6)2=cos⁡(x−π6)2- \frac{\cos{\left(x + \frac{\pi}{6} \right)}}{2} = \frac{\cos{\left(x - \frac{\pi}{6} \right)}}{2}
- No
so, the function
not is
neither even, nor odd
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