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Plotting a function graph/ 4*sin(x+(pi/6))+2

Graphing y = 4*sin(x+(pi/6))+2

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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            /    pi\    
f(x) = 4*sin|x + --| + 2
            \    6 /
f(x)=4sin⁡(x+π6)+2f{\left(x \right)} = 4 \sin{\left(x + \frac{\pi}{6} \right)} + 2 
f = 4*sin(x + pi/6) + 2
The graph of the function
02468-8-6-4-2-1010-1010
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:
4sin⁡(x+π6)+2=04 \sin{\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=πx_{2} = \pi
Numerical solution
x1=−97.3893722612836x_{1} = -97.3893722612836
x2=−70.162235930172x_{2} = -70.162235930172
x3=−72.2566310325652x_{3} = -72.2566310325652
x4=−59.6902604182061x_{4} = -59.6902604182061
x5=−78.5398163397448x_{5} = -78.5398163397448
x6=97.3893722612836x_{6} = 97.3893722612836
x7=9.42477796076938x_{7} = 9.42477796076938
x8=99.4837673636768x_{8} = 99.4837673636768
x9=86.9173967493176x_{9} = 86.9173967493176
x10=−57.5958653158129x_{10} = -57.5958653158129
x11=−32.4631240870945x_{11} = -32.4631240870945
x12=84.8230016469244x_{12} = 84.8230016469244
x13=5.23598775598299x_{13} = 5.23598775598299
x14=−21.9911485751286x_{14} = -21.9911485751286
x15=3.14159265358979x_{15} = 3.14159265358979
x16=−95.2949771588904x_{16} = -95.2949771588904
x17=36.6519142918809x_{17} = 36.6519142918809
x18=−220.958683302482x_{18} = -220.958683302482
x19=28.2743338823081x_{19} = 28.2743338823081
x20=68.0678408277789x_{20} = 68.0678408277789
x21=−13.6135681655558x_{21} = -13.6135681655558
x22=−28.2743338823081x_{22} = -28.2743338823081
x23=−40.8407044966673x_{23} = -40.8407044966673
x24=−65.9734457253857x_{24} = -65.9734457253857
x25=−91.106186954104x_{25} = -91.106186954104
x26=42.9350995990605x_{26} = 42.9350995990605
x27=−76.4454212373516x_{27} = -76.4454212373516
x28=17.8023583703422x_{28} = 17.8023583703422
x29=80.634211442138x_{29} = 80.634211442138
x30=55.5014702134197x_{30} = 55.5014702134197
x31=30.3687289847013x_{31} = 30.3687289847013
x32=40.8407044966673x_{32} = 40.8407044966673
x33=−89.0117918517108x_{33} = -89.0117918517108
x34=49.2182849062401x_{34} = 49.2182849062401
x35=−53.4070751110265x_{35} = -53.4070751110265
x36=−3.14159265358979x_{36} = -3.14159265358979
x37=−7.33038285837618x_{37} = -7.33038285837618
x38=21.9911485751286x_{38} = 21.9911485751286
x39=−84.8230016469244x_{39} = -84.8230016469244
x40=34.5575191894877x_{40} = 34.5575191894877
x41=−45.0294947014537x_{41} = -45.0294947014537
x42=−9.42477796076938x_{42} = -9.42477796076938
x43=−15.707963267949x_{43} = -15.707963267949
x44=47.1238898038469x_{44} = 47.1238898038469
x45=93.2005820564972x_{45} = 93.2005820564972
x46=53.4070751110265x_{46} = 53.4070751110265
x47=74.3510261349584x_{47} = 74.3510261349584
x48=24.0855436775217x_{48} = 24.0855436775217
x49=61.7846555205993x_{49} = 61.7846555205993
x50=65.9734457253857x_{50} = 65.9734457253857
x51=91.106186954104x_{51} = 91.106186954104
x52=11.5191730631626x_{52} = 11.5191730631626
x53=59.6902604182061x_{53} = 59.6902604182061
x54=−82.7286065445312x_{54} = -82.7286065445312
x55=−26.1799387799149x_{55} = -26.1799387799149
x56=−63.8790506229925x_{56} = -63.8790506229925
x57=−1.0471975511966x_{57} = -1.0471975511966
x58=78.5398163397448x_{58} = 78.5398163397448
x59=15.707963267949x_{59} = 15.707963267949
x60=72.2566310325652x_{60} = 72.2566310325652
x61=−19.8967534727354x_{61} = -19.8967534727354
x62=−47.1238898038469x_{62} = -47.1238898038469
x63=−103.672557568463x_{63} = -103.672557568463
x64=−38.7463093942741x_{64} = -38.7463093942741
x65=−51.3126800086333x_{65} = -51.3126800086333
x66=−34.5575191894877x_{66} = -34.5575191894877
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 4*sin(x + pi/6) + 2.
4sin⁡(π6)+24 \sin{\left(\frac{\pi}{6} \right)} + 2
The result:
f(0)=4f{\left(0 \right)} = 4
The point:
(0, 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
4cos⁡(x+π6)=04 \cos{\left(x + \frac{\pi}{6} \right)} = 0
Solve this equation
The roots of this equation
x1=π3x_{1} = \frac{\pi}{3}
x2=4π3x_{2} = \frac{4 \pi}{3}
The values of the extrema at the points:
 pi           /pi   pi\ 
(--, 2 + 4*sin|-- + --|)
 3            \3    6 / 

 4*pi           /pi   pi\ 
(----, 2 - 4*sin|-- + --|)
  3             \3    6 /


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=4π3x_{1} = \frac{4 \pi}{3}
Maxima of the function at points:
x1=π3x_{1} = \frac{\pi}{3}
Decreasing at intervals
(−∞,π3]∪[4π3,∞)\left(-\infty, \frac{\pi}{3}\right] \cup \left[\frac{4 \pi}{3}, \infty\right)
Increasing at intervals
[π3,4π3]\left[\frac{\pi}{3}, \frac{4 \pi}{3}\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
−4sin⁡(x+π6)=0- 4 \sin{\left(x + \frac{\pi}{6} \right)} = 0
Solve this equation
The roots of this equation
x1=−π6x_{1} = - \frac{\pi}{6}
x2=5π6x_{2} = \frac{5 \pi}{6}

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
(−∞,−π6]∪[5π6,∞)\left(-\infty, - \frac{\pi}{6}\right] \cup \left[\frac{5 \pi}{6}, \infty\right)
Convex at the intervals
[−π6,5π6]\left[- \frac{\pi}{6}, \frac{5 \pi}{6}\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(4sin⁡(x+π6)+2)=⟨−2,6⟩\lim_{x \to -\infty}\left(4 \sin{\left(x + \frac{\pi}{6} \right)} + 2\right) = \left\langle -2, 6\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−2,6⟩y = \left\langle -2, 6\right\rangle
lim⁡x→∞(4sin⁡(x+π6)+2)=⟨−2,6⟩\lim_{x \to \infty}\left(4 \sin{\left(x + \frac{\pi}{6} \right)} + 2\right) = \left\langle -2, 6\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−2,6⟩y = \left\langle -2, 6\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 4*sin(x + pi/6) + 2, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(4sin⁡(x+π6)+2x)=0\lim_{x \to -\infty}\left(\frac{4 \sin{\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→∞(4sin⁡(x+π6)+2x)=0\lim_{x \to \infty}\left(\frac{4 \sin{\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:
4sin⁡(x+π6)+2=2−4sin⁡(x−π6)4 \sin{\left(x + \frac{\pi}{6} \right)} + 2 = 2 - 4 \sin{\left(x - \frac{\pi}{6} \right)}
- No
4sin⁡(x+π6)+2=4sin⁡(x−π6)−24 \sin{\left(x + \frac{\pi}{6} \right)} + 2 = 4 \sin{\left(x - \frac{\pi}{6} \right)} - 2
- No
so, the function
not is
neither even, nor odd
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