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Plotting a function graph/ 2sin(x/2+pi/6)+1

Graphing y = 2sin(x/2+pi/6)+1

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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            /x   pi\    
f(x) = 2*sin|- + --| + 1
            \2   6 /
f(x)=2sin⁡(x2+π6)+1f{\left(x \right)} = 2 \sin{\left(\frac{x}{2} + \frac{\pi}{6} \right)} + 1 
f = 2*sin(x/2 + pi/6) + 1
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:
2sin⁡(x2+π6)+1=02 \sin{\left(\frac{x}{2} + \frac{\pi}{6} \right)} + 1 = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−2π3x_{1} = - \frac{2 \pi}{3}
x2=2πx_{2} = 2 \pi
Numerical solution
x1=−1024.15920507027x_{1} = -1024.15920507027
x2=−18.8495559215388x_{2} = -18.8495559215388
x3=−56.5486677646163x_{3} = -56.5486677646163
x4=43.9822971502571x_{4} = 43.9822971502571
x5=69.1150383789755x_{5} = 69.1150383789755
x6=10.471975511966x_{6} = 10.471975511966
x7=−94.2477796076938x_{7} = -94.2477796076938
x8=182.212373908208x_{8} = 182.212373908208
x9=85.870199198121x_{9} = 85.870199198121
x10=−43.9822971502571x_{10} = -43.9822971502571
x11=60.7374579694027x_{11} = 60.7374579694027
x12=−90.0589894029074x_{12} = -90.0589894029074
x13=−31.4159265358979x_{13} = -31.4159265358979
x14=−14.6607657167524x_{14} = -14.6607657167524
x15=−6.28318530717959x_{15} = -6.28318530717959
x16=31.4159265358979x_{16} = 31.4159265358979
x17=45312.2380402768x_{17} = 45312.2380402768
x18=94.2477796076938x_{18} = 94.2477796076938
x19=81.6814089933346x_{19} = 81.6814089933346
x20=56.5486677646163x_{20} = 56.5486677646163
x21=−2.0943951023932x_{21} = -2.0943951023932
x22=35.6047167406843x_{22} = 35.6047167406843
x23=98.4365698124802x_{23} = 98.4365698124802
x24=23.0383461263252x_{24} = 23.0383461263252
x25=−69.1150383789755x_{25} = -69.1150383789755
x26=−52.3598775598299x_{26} = -52.3598775598299
x27=−102.625360017267x_{27} = -102.625360017267
x28=−27.2271363311115x_{28} = -27.2271363311115
x29=−81.6814089933346x_{29} = -81.6814089933346
x30=−64.9262481741891x_{30} = -64.9262481741891
x31=48.1710873550435x_{31} = 48.1710873550435
x32=−39.7935069454707x_{32} = -39.7935069454707
x33=18.8495559215388x_{33} = 18.8495559215388
x34=−77.4926187885482x_{34} = -77.4926187885482
x35=73.3038285837618x_{35} = 73.3038285837618
x36=6.28318530717959x_{36} = 6.28318530717959
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 2*sin(x/2 + pi/6) + 1.
2sin⁡(02+π6)+12 \sin{\left(\frac{0}{2} + \frac{\pi}{6} \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
cos⁡(x2+π6)=0\cos{\left(\frac{x}{2} + \frac{\pi}{6} \right)} = 0
Solve this equation
The roots of this equation
x1=2π3x_{1} = \frac{2 \pi}{3}
x2=8π3x_{2} = \frac{8 \pi}{3}
The values of the extrema at the points:
 2*pi           /pi   pi\ 
(----, 1 + 2*sin|-- + --|)
  3             \3    6 / 

 8*pi           /pi   pi\ 
(----, 1 - 2*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=8π3x_{1} = \frac{8 \pi}{3}
Maxima of the function at points:
x1=2π3x_{1} = \frac{2 \pi}{3}
Decreasing at intervals
(−∞,2π3]∪[8π3,∞)\left(-\infty, \frac{2 \pi}{3}\right] \cup \left[\frac{8 \pi}{3}, \infty\right)
Increasing at intervals
[2π3,8π3]\left[\frac{2 \pi}{3}, \frac{8 \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
−sin⁡(3x+π6)2=0- \frac{\sin{\left(\frac{3 x + \pi}{6} \right)}}{2} = 0
Solve this equation
The roots of this equation
x1=−π3x_{1} = - \frac{\pi}{3}
x2=5π3x_{2} = \frac{5 \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]∪[5π3,∞)\left(-\infty, - \frac{\pi}{3}\right] \cup \left[\frac{5 \pi}{3}, \infty\right)
Convex at the intervals
[−π3,5π3]\left[- \frac{\pi}{3}, \frac{5 \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→−∞(2sin⁡(x2+π6)+1)=⟨−1,3⟩\lim_{x \to -\infty}\left(2 \sin{\left(\frac{x}{2} + \frac{\pi}{6} \right)} + 1\right) = \left\langle -1, 3\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−1,3⟩y = \left\langle -1, 3\right\rangle
lim⁡x→∞(2sin⁡(x2+π6)+1)=⟨−1,3⟩\lim_{x \to \infty}\left(2 \sin{\left(\frac{x}{2} + \frac{\pi}{6} \right)} + 1\right) = \left\langle -1, 3\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−1,3⟩y = \left\langle -1, 3\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 2*sin(x/2 + pi/6) + 1, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(2sin⁡(x2+π6)+1x)=0\lim_{x \to -\infty}\left(\frac{2 \sin{\left(\frac{x}{2} + \frac{\pi}{6} \right)} + 1}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(2sin⁡(x2+π6)+1x)=0\lim_{x \to \infty}\left(\frac{2 \sin{\left(\frac{x}{2} + \frac{\pi}{6} \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:
2sin⁡(x2+π6)+1=1−2sin⁡(x2−π6)2 \sin{\left(\frac{x}{2} + \frac{\pi}{6} \right)} + 1 = 1 - 2 \sin{\left(\frac{x}{2} - \frac{\pi}{6} \right)}
- No
2sin⁡(x2+π6)+1=2sin⁡(x2−π6)−12 \sin{\left(\frac{x}{2} + \frac{\pi}{6} \right)} + 1 = 2 \sin{\left(\frac{x}{2} - \frac{\pi}{6} \right)} - 1
- No
so, the function
not is
neither even, nor odd
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