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Plotting a function graph/ sin(x/2)

Graphing y = sin(x/2)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          /x\
f(x) = sin|-|
          \2/
f(x)=sin⁡(x2)f{\left(x \right)} = \sin{\left(\frac{x}{2} \right)} 
f = sin(x/2)
The graph of the function
-1.0-0.8-0.6-0.4-0.21.00.00.20.40.60.81-1
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:
sin⁡(x2)=0\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=−18.8495559215388x_{1} = -18.8495559215388
x2=−37.6991118430775x_{2} = -37.6991118430775
x3=−226.194671058465x_{3} = -226.194671058465
x4=−56.5486677646163x_{4} = -56.5486677646163
x5=12.5663706143592x_{5} = 12.5663706143592
x6=−81.6814089933346x_{6} = -81.6814089933346
x7=−31.4159265358979x_{7} = -31.4159265358979
x8=94.2477796076938x_{8} = 94.2477796076938
x9=0x_{9} = 0
x10=−87.9645943005142x_{10} = -87.9645943005142
x11=81.6814089933346x_{11} = 81.6814089933346
x12=−75.398223686155x_{12} = -75.398223686155
x13=62.8318530717959x_{13} = 62.8318530717959
x14=100.530964914873x_{14} = 100.530964914873
x15=75.398223686155x_{15} = 75.398223686155
x16=−106.814150222053x_{16} = -106.814150222053
x17=6.28318530717959x_{17} = 6.28318530717959
x18=−6.28318530717959x_{18} = -6.28318530717959
x19=31.4159265358979x_{19} = 31.4159265358979
x20=25.1327412287183x_{20} = 25.1327412287183
x21=18.8495559215388x_{21} = 18.8495559215388
x22=−94.2477796076938x_{22} = -94.2477796076938
x23=56.5486677646163x_{23} = 56.5486677646163
x24=−25.1327412287183x_{24} = -25.1327412287183
x25=87.9645943005142x_{25} = 87.9645943005142
x26=−50.2654824574367x_{26} = -50.2654824574367
x27=−100.530964914873x_{27} = -100.530964914873
x28=−43.9822971502571x_{28} = -43.9822971502571
x29=50.2654824574367x_{29} = 50.2654824574367
x30=69.1150383789755x_{30} = 69.1150383789755
x31=−62.8318530717959x_{31} = -62.8318530717959
x32=−69.1150383789755x_{32} = -69.1150383789755
x33=−12.5663706143592x_{33} = -12.5663706143592
x34=37.6991118430775x_{34} = 37.6991118430775
x35=43.9822971502571x_{35} = 43.9822971502571
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(x/2).
sin⁡(02)\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)2=0\frac{\cos{\left(\frac{x}{2} \right)}}{2} = 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, 1)

(3*pi, -1)


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)4=0- \frac{\sin{\left(\frac{x}{2} \right)}}{4} = 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→−∞sin⁡(x2)=⟨−1,1⟩\lim_{x \to -\infty} \sin{\left(\frac{x}{2} \right)} = \left\langle -1, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
lim⁡x→∞sin⁡(x2)=⟨−1,1⟩\lim_{x \to \infty} \sin{\left(\frac{x}{2} \right)} = \left\langle -1, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sin(x/2), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡(x2)x)=0\lim_{x \to -\infty}\left(\frac{\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→∞(sin⁡(x2)x)=0\lim_{x \to \infty}\left(\frac{\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:
sin⁡(x2)=−sin⁡(x2)\sin{\left(\frac{x}{2} \right)} = - \sin{\left(\frac{x}{2} \right)}
- No
sin⁡(x2)=sin⁡(x2)\sin{\left(\frac{x}{2} \right)} = \sin{\left(\frac{x}{2} \right)}
- No
so, the function
not is
neither even, nor odd
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