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  • Graphing y =:
  • -x^3-27x
  • x^3-3x^2+2x x^3-3x^2+2x
  • x^4+8x^2-9 x^4+8x^2-9
  • e^x/x

  • Identical expressions

  • one .5cos(x-п/ six)
  • 1.5 co sinus of e of (x minus п divide by 6)
  • one .5 co sinus of e of (x minus п divide by six)
  • 1.5cosx-п/6
  • 1.5cos(x-п divide by 6)

  • Similar expressions

  • 1.5cos(x+п/6)
Plotting a function graph/ 1.5cos(x-п/6)

Graphing y = 1.5cos(x-п/6)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
            /    pi\
       3*cos|x - --|
            \    6 /
f(x) = -------------
             2
f(x)=3cos⁡(x−π6)2f{\left(x \right)} = \frac{3 \cos{\left(x - \frac{\pi}{6} \right)}}{2} 
f = 3*cos(x - pi/6)/2
The graph of the function
02468-8-6-4-2-10105-5
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−π6)2=0\frac{3 \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=2π3x_{2} = \frac{2 \pi}{3}
Numerical solution
x1=−29.3215314335047x_{1} = -29.3215314335047
x2=−70.162235930172x_{2} = -70.162235930172
x3=36.6519142918809x_{3} = 36.6519142918809
x4=27.2271363311115x_{4} = 27.2271363311115
x5=−4.18879020478639x_{5} = -4.18879020478639
x6=−85.870199198121x_{6} = -85.870199198121
x7=11.5191730631626x_{7} = 11.5191730631626
x8=−32.4631240870945x_{8} = -32.4631240870945
x9=99.4837673636768x_{9} = 99.4837673636768
x10=−67.0206432765823x_{10} = -67.0206432765823
x11=49.2182849062401x_{11} = 49.2182849062401
x12=−7.33038285837618x_{12} = -7.33038285837618
x13=71.2094334813686x_{13} = 71.2094334813686
x14=52.3598775598299x_{14} = 52.3598775598299
x15=−95.2949771588904x_{15} = -95.2949771588904
x16=17.8023583703422x_{16} = 17.8023583703422
x17=−60.7374579694027x_{17} = -60.7374579694027
x18=83.7758040957278x_{18} = 83.7758040957278
x19=58.6430628670095x_{19} = 58.6430628670095
x20=−73.3038285837618x_{20} = -73.3038285837618
x21=−26.1799387799149x_{21} = -26.1799387799149
x22=77.4926187885482x_{22} = 77.4926187885482
x23=−79.5870138909414x_{23} = -79.5870138909414
x24=−48.1710873550435x_{24} = -48.1710873550435
x25=−10.471975511966x_{25} = -10.471975511966
x26=80.634211442138x_{26} = 80.634211442138
x27=20.943951023932x_{27} = 20.943951023932
x28=2.0943951023932x_{28} = 2.0943951023932
x29=24.0855436775217x_{29} = 24.0855436775217
x30=96.342174710087x_{30} = 96.342174710087
x31=−19.8967534727354x_{31} = -19.8967534727354
x32=33.5103216382911x_{32} = 33.5103216382911
x33=−13.6135681655558x_{33} = -13.6135681655558
x34=5.23598775598299x_{34} = 5.23598775598299
x35=39.7935069454707x_{35} = 39.7935069454707
x36=−45.0294947014537x_{36} = -45.0294947014537
x37=−23.0383461263252x_{37} = -23.0383461263252
x38=4579.39489138272x_{38} = 4579.39489138272
x39=−1.0471975511966x_{39} = -1.0471975511966
x40=64.9262481741891x_{40} = 64.9262481741891
x41=90.0589894029074x_{41} = 90.0589894029074
x42=−57.5958653158129x_{42} = -57.5958653158129
x43=8.37758040957278x_{43} = 8.37758040957278
x44=42.9350995990605x_{44} = 42.9350995990605
x45=86.9173967493176x_{45} = 86.9173967493176
x46=−92.1533845053006x_{46} = -92.1533845053006
x47=−89.0117918517108x_{47} = -89.0117918517108
x48=55.5014702134197x_{48} = 55.5014702134197
x49=−38.7463093942741x_{49} = -38.7463093942741
x50=−51.3126800086333x_{50} = -51.3126800086333
x51=68.0678408277789x_{51} = 68.0678408277789
x52=−154.985237577096x_{52} = -154.985237577096
x53=−76.4454212373516x_{53} = -76.4454212373516
x54=−63.8790506229925x_{54} = -63.8790506229925
x55=−54.4542726622231x_{55} = -54.4542726622231
x56=−16.7551608191456x_{56} = -16.7551608191456
x57=61.7846555205993x_{57} = 61.7846555205993
x58=−35.6047167406843x_{58} = -35.6047167406843
x59=14.6607657167524x_{59} = 14.6607657167524
x60=−41.8879020478639x_{60} = -41.8879020478639
x61=74.3510261349584x_{61} = 74.3510261349584
x62=102.625360017267x_{62} = 102.625360017267
x63=93.2005820564972x_{63} = 93.2005820564972
x64=−98.4365698124802x_{64} = -98.4365698124802
x65=−82.7286065445312x_{65} = -82.7286065445312
x66=46.0766922526503x_{66} = 46.0766922526503
x67=30.3687289847013x_{67} = 30.3687289847013
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 3*cos(x - pi/6)/2.
3cos⁡(−π6)2\frac{3 \cos{\left(- \frac{\pi}{6} \right)}}{2}
The result:
f(0)=334f{\left(0 \right)} = \frac{3 \sqrt{3}}{4}
The point:
(0, 3*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
−3sin⁡(x−π6)2=0- \frac{3 \sin{\left(x - \frac{\pi}{6} \right)}}{2} = 0
Solve this equation
The roots of this equation
x1=π6x_{1} = \frac{\pi}{6}
x2=7π6x_{2} = \frac{7 \pi}{6}
The values of the extrema at the points:
          /pi   pi\ 
     3*cos|-- - --| 
 pi       \6    6 / 
(--, --------------)
 6         2        

             /pi   pi\ 
       -3*cos|-- - --| 
 7*pi        \6    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=7π6x_{1} = \frac{7 \pi}{6}
Maxima of the function at points:
x1=π6x_{1} = \frac{\pi}{6}
Decreasing at intervals
(−∞,π6]∪[7π6,∞)\left(-\infty, \frac{\pi}{6}\right] \cup \left[\frac{7 \pi}{6}, \infty\right)
Increasing at intervals
[π6,7π6]\left[\frac{\pi}{6}, \frac{7 \pi}{6}\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
−3sin⁡(x+π3)2=0- \frac{3 \sin{\left(x + \frac{\pi}{3} \right)}}{2} = 0
Solve this equation
The roots of this equation
x1=−π3x_{1} = - \frac{\pi}{3}
x2=2π3x_{2} = \frac{2 \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]∪[2π3,∞)\left(-\infty, - \frac{\pi}{3}\right] \cup \left[\frac{2 \pi}{3}, \infty\right)
Convex at the intervals
[−π3,2π3]\left[- \frac{\pi}{3}, \frac{2 \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→−∞(3cos⁡(x−π6)2)=⟨−32,32⟩\lim_{x \to -\infty}\left(\frac{3 \cos{\left(x - \frac{\pi}{6} \right)}}{2}\right) = \left\langle - \frac{3}{2}, \frac{3}{2}\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−32,32⟩y = \left\langle - \frac{3}{2}, \frac{3}{2}\right\rangle
lim⁡x→∞(3cos⁡(x−π6)2)=⟨−32,32⟩\lim_{x \to \infty}\left(\frac{3 \cos{\left(x - \frac{\pi}{6} \right)}}{2}\right) = \left\langle - \frac{3}{2}, \frac{3}{2}\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−32,32⟩y = \left\langle - \frac{3}{2}, \frac{3}{2}\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 3*cos(x - pi/6)/2, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(3cos⁡(x−π6)2x)=0\lim_{x \to -\infty}\left(\frac{3 \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→∞(3cos⁡(x−π6)2x)=0\lim_{x \to \infty}\left(\frac{3 \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:
3cos⁡(x−π6)2=3cos⁡(x+π6)2\frac{3 \cos{\left(x - \frac{\pi}{6} \right)}}{2} = \frac{3 \cos{\left(x + \frac{\pi}{6} \right)}}{2}
- No
3cos⁡(x−π6)2=−3cos⁡(x+π6)2\frac{3 \cos{\left(x - \frac{\pi}{6} \right)}}{2} = - \frac{3 \cos{\left(x + \frac{\pi}{6} \right)}}{2}
- No
so, the function
not is
neither even, nor odd
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