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Plotting a function graph/ 2*sin(x/2)

Graphing y = 2*sin(x/2)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
            /x\
f(x) = 2*sin|-|
            \2/
f(x)=2sin⁡(x2)f{\left(x \right)} = 2 \sin{\left(\frac{x}{2} \right)} 
f = 2*sin(x/2)
The graph of the function
-10.0-7.5-5.0-2.50.02.55.07.510.05-5
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)=02 \sin{\left(\frac{x}{2} \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
x2=2πx_{2} = 2 \pi
Numerical solution
x1=12.5663706143592x_{1} = 12.5663706143592
x2=37.6991118430775x_{2} = 37.6991118430775
x3=−31.4159265358979x_{3} = -31.4159265358979
x4=0x_{4} = 0
x5=−50.2654824574367x_{5} = -50.2654824574367
x6=−94.2477796076938x_{6} = -94.2477796076938
x7=6.28318530717959x_{7} = 6.28318530717959
x8=69.1150383789755x_{8} = 69.1150383789755
x9=−69.1150383789755x_{9} = -69.1150383789755
x10=62.8318530717959x_{10} = 62.8318530717959
x11=50.2654824574367x_{11} = 50.2654824574367
x12=81.6814089933346x_{12} = 81.6814089933346
x13=100.530964914873x_{13} = 100.530964914873
x14=−87.9645943005142x_{14} = -87.9645943005142
x15=−62.8318530717959x_{15} = -62.8318530717959
x16=−18.8495559215388x_{16} = -18.8495559215388
x17=−56.5486677646163x_{17} = -56.5486677646163
x18=−37.6991118430775x_{18} = -37.6991118430775
x19=−25.1327412287183x_{19} = -25.1327412287183
x20=−100.530964914873x_{20} = -100.530964914873
x21=−75.398223686155x_{21} = -75.398223686155
x22=18.8495559215388x_{22} = 18.8495559215388
x23=−6.28318530717959x_{23} = -6.28318530717959
x24=25.1327412287183x_{24} = 25.1327412287183
x25=56.5486677646163x_{25} = 56.5486677646163
x26=−43.9822971502571x_{26} = -43.9822971502571
x27=−226.194671058465x_{27} = -226.194671058465
x28=31.4159265358979x_{28} = 31.4159265358979
x29=94.2477796076938x_{29} = 94.2477796076938
x30=−106.814150222053x_{30} = -106.814150222053
x31=−12.5663706143592x_{31} = -12.5663706143592
x32=75.398223686155x_{32} = 75.398223686155
x33=−81.6814089933346x_{33} = -81.6814089933346
x34=43.9822971502571x_{34} = 43.9822971502571
x35=87.9645943005142x_{35} = 87.9645943005142
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).
2sin⁡(02)2 \sin{\left(\frac{0}{2} \right)}
The result:
f(0)=0f{\left(0 \right)} = 0
The point:
(0, 0)
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)=0\cos{\left(\frac{x}{2} \right)} = 0
Solve this equation
The roots of this equation
x1=πx_{1} = \pi
x2=3πx_{2} = 3 \pi
The values of the extrema at the points:
(pi, 2)

(3*pi, -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=3πx_{1} = 3 \pi
Maxima of the function at points:
x1=πx_{1} = \pi
Decreasing at intervals
(−∞,π]∪[3π,∞)\left(-\infty, \pi\right] \cup \left[3 \pi, \infty\right)
Increasing at intervals
[π,3π]\left[\pi, 3 \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
−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

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