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Plotting a function graph/ 2*sin(x-1)

Graphing y = 2*sin(x-1)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = 2*sin(x - 1)
f(x)=2sin⁡(x−1)f{\left(x \right)} = 2 \sin{\left(x - 1 \right)} 
f = 2*sin(x - 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⁡(x−1)=02 \sin{\left(x - 1 \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=1x_{1} = 1
x2=1+πx_{2} = 1 + \pi
Numerical solution
x1=22.9911485751286x_{1} = 22.9911485751286
x2=−90.106186954104x_{2} = -90.106186954104
x3=−49.2654824574367x_{3} = -49.2654824574367
x4=76.398223686155x_{4} = 76.398223686155
x5=−52.4070751110265x_{5} = -52.4070751110265
x6=38.6991118430775x_{6} = 38.6991118430775
x7=32.4159265358979x_{7} = 32.4159265358979
x8=88.9645943005142x_{8} = 88.9645943005142
x9=−24.1327412287183x_{9} = -24.1327412287183
x10=4.14159265358979x_{10} = 4.14159265358979
x11=−55.5486677646163x_{11} = -55.5486677646163
x12=−64.9734457253857x_{12} = -64.9734457253857
x13=−115.238928182822x_{13} = -115.238928182822
x14=95.2477796076938x_{14} = 95.2477796076938
x15=−68.1150383789755x_{15} = -68.1150383789755
x16=−58.6902604182061x_{16} = -58.6902604182061
x17=51.2654824574367x_{17} = 51.2654824574367
x18=−61.8318530717959x_{18} = -61.8318530717959
x19=−39.8407044966673x_{19} = -39.8407044966673
x20=−33.5575191894877x_{20} = -33.5575191894877
x21=73.2566310325652x_{21} = 73.2566310325652
x22=16.707963267949x_{22} = 16.707963267949
x23=−74.398223686155x_{23} = -74.398223686155
x24=−46.1238898038469x_{24} = -46.1238898038469
x25=60.6902604182061x_{25} = 60.6902604182061
x26=−551.920307031804x_{26} = -551.920307031804
x27=−124.663706143592x_{27} = -124.663706143592
x28=−80.6814089933346x_{28} = -80.6814089933346
x29=−27.2743338823081x_{29} = -27.2743338823081
x30=19.8495559215388x_{30} = 19.8495559215388
x31=1x_{31} = 1
x32=−30.4159265358979x_{32} = -30.4159265358979
x33=−71.2566310325652x_{33} = -71.2566310325652
x34=63.8318530717959x_{34} = 63.8318530717959
x35=−77.5398163397448x_{35} = -77.5398163397448
x36=92.106186954104x_{36} = 92.106186954104
x37=−93.2477796076938x_{37} = -93.2477796076938
x38=70.1150383789755x_{38} = 70.1150383789755
x39=48.1238898038469x_{39} = 48.1238898038469
x40=−8.42477796076938x_{40} = -8.42477796076938
x41=82.6814089933346x_{41} = 82.6814089933346
x42=−86.9645943005142x_{42} = -86.9645943005142
x43=41.8407044966673x_{43} = 41.8407044966673
x44=29.2743338823081x_{44} = 29.2743338823081
x45=−2.14159265358979x_{45} = -2.14159265358979
x46=−20.9911485751286x_{46} = -20.9911485751286
x47=44.9822971502571x_{47} = 44.9822971502571
x48=66.9734457253857x_{48} = 66.9734457253857
x49=54.4070751110265x_{49} = 54.4070751110265
x50=85.8230016469244x_{50} = 85.8230016469244
x51=98.3893722612836x_{51} = 98.3893722612836
x52=−14.707963267949x_{52} = -14.707963267949
x53=−96.3893722612836x_{53} = -96.3893722612836
x54=−83.8230016469244x_{54} = -83.8230016469244
x55=−36.6991118430775x_{55} = -36.6991118430775
x56=−5.28318530717959x_{56} = -5.28318530717959
x57=1229.36272755361x_{57} = 1229.36272755361
x58=13.5663706143592x_{58} = 13.5663706143592
x59=−102.672557568463x_{59} = -102.672557568463
x60=10.4247779607694x_{60} = 10.4247779607694
x61=35.5575191894877x_{61} = 35.5575191894877
x62=−11.5663706143592x_{62} = -11.5663706143592
x63=−17.8495559215388x_{63} = -17.8495559215388
x64=7.28318530717959x_{64} = 7.28318530717959
x65=−99.5309649148734x_{65} = -99.5309649148734
x66=57.5486677646163x_{66} = 57.5486677646163
x67=79.5398163397448x_{67} = 79.5398163397448
x68=−42.9822971502571x_{68} = -42.9822971502571
x69=26.1327412287183x_{69} = 26.1327412287183
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 - 1).
2sin⁡(−1)2 \sin{\left(-1 \right)}
The result:
f(0)=−2sin⁡(1)f{\left(0 \right)} = - 2 \sin{\left(1 \right)}
The point:
(0, -2*sin(1))
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
2cos⁡(x−1)=02 \cos{\left(x - 1 \right)} = 0
Solve this equation
The roots of this equation
x1=1+π2x_{1} = 1 + \frac{\pi}{2}
x2=1+3π2x_{2} = 1 + \frac{3 \pi}{2}
The values of the extrema at the points:
     pi    
(1 + --, 2)
     2     

     3*pi     
(1 + ----, -2)
      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=1+3π2x_{1} = 1 + \frac{3 \pi}{2}
Maxima of the function at points:
x1=1+π2x_{1} = 1 + \frac{\pi}{2}
Decreasing at intervals
(−∞,1+π2]∪[1+3π2,∞)\left(-\infty, 1 + \frac{\pi}{2}\right] \cup \left[1 + \frac{3 \pi}{2}, \infty\right)
Increasing at intervals
[1+π2,1+3π2]\left[1 + \frac{\pi}{2}, 1 + \frac{3 \pi}{2}\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
−2sin⁡(x−1)=0- 2 \sin{\left(x - 1 \right)} = 0
Solve this equation
The roots of this equation
x1=1x_{1} = 1
x2=1+πx_{2} = 1 + \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
(−∞,1]∪[1+π,∞)\left(-\infty, 1\right] \cup \left[1 + \pi, \infty\right)
Convex at the intervals
[1,1+π]\left[1, 1 + \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⁡(x−1))=⟨−2,2⟩\lim_{x \to -\infty}\left(2 \sin{\left(x - 1 \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⁡(x−1))=⟨−2,2⟩\lim_{x \to \infty}\left(2 \sin{\left(x - 1 \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 - 1), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(2sin⁡(x−1)x)=0\lim_{x \to -\infty}\left(\frac{2 \sin{\left(x - 1 \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(2sin⁡(x−1)x)=0\lim_{x \to \infty}\left(\frac{2 \sin{\left(x - 1 \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⁡(x−1)=−2sin⁡(x+1)2 \sin{\left(x - 1 \right)} = - 2 \sin{\left(x + 1 \right)}
- No
2sin⁡(x−1)=2sin⁡(x+1)2 \sin{\left(x - 1 \right)} = 2 \sin{\left(x + 1 \right)}
- No
so, the function
not is
neither even, nor odd
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