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Plotting a function graph/ |sinx|

Graphing y = |sinx|

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
f(x) = |sin(x)|
f(x)=∣sin⁡(x)∣f{\left(x \right)} = \left|{\sin{\left(x \right)}}\right| 
f = Abs(sin(x))
The graph of the function
02468-8-6-4-2-101002
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⁡(x)∣=0\left|{\sin{\left(x \right)}}\right| = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
x2=πx_{2} = \pi
Numerical solution
x1=40.8407044966673x_{1} = 40.8407044966673
x2=0x_{2} = 0
x3=−285.884931476671x_{3} = -285.884931476671
x4=97.3893722612836x_{4} = 97.3893722612836
x5=34.5575191894877x_{5} = 34.5575191894877
x6=−56.5486677646163x_{6} = -56.5486677646163
x7=53.4070751110265x_{7} = 53.4070751110265
x8=−97.3893722612836x_{8} = -97.3893722612836
x9=62.8318530717959x_{9} = 62.8318530717959
x10=87.9645943005142x_{10} = 87.9645943005142
x11=43.9822971502571x_{11} = 43.9822971502571
x12=37.6991118430775x_{12} = 37.6991118430775
x13=−21.9911485751286x_{13} = -21.9911485751286
x14=65.9734457253857x_{14} = 65.9734457253857
x15=−50.2654824574367x_{15} = -50.2654824574367
x16=−94.2477796076938x_{16} = -94.2477796076938
x17=−75.398223686155x_{17} = -75.398223686155
x18=−53.4070751110265x_{18} = -53.4070751110265
x19=12.5663706143592x_{19} = 12.5663706143592
x20=−9.42477796076938x_{20} = -9.42477796076938
x21=−34.5575191894877x_{21} = -34.5575191894877
x22=21.9911485751286x_{22} = 21.9911485751286
x23=−3760.48640634698x_{23} = -3760.48640634698
x24=−47.1238898038469x_{24} = -47.1238898038469
x25=−43.9822971502571x_{25} = -43.9822971502571
x26=28.2743338823081x_{26} = 28.2743338823081
x27=−31.4159265358979x_{27} = -31.4159265358979
x28=−3.14159265358979x_{28} = -3.14159265358979
x29=−6.28318530717959x_{29} = -6.28318530717959
x30=−25.1327412287183x_{30} = -25.1327412287183
x31=650.309679293087x_{31} = 650.309679293087
x32=31.4159265358979x_{32} = 31.4159265358979
x33=−65.9734457253857x_{33} = -65.9734457253857
x34=72.2566310325652x_{34} = 72.2566310325652
x35=−59.6902604182061x_{35} = -59.6902604182061
x36=−427.256600888212x_{36} = -427.256600888212
x37=94.2477796076938x_{37} = 94.2477796076938
x38=81.6814089933346x_{38} = 81.6814089933346
x39=−91.106186954104x_{39} = -91.106186954104
x40=−100.530964914873x_{40} = -100.530964914873
x41=59.6902604182061x_{41} = 59.6902604182061
x42=−12.5663706143592x_{42} = -12.5663706143592
x43=78.5398163397448x_{43} = 78.5398163397448
x44=56.5486677646163x_{44} = 56.5486677646163
x45=84.8230016469244x_{45} = 84.8230016469244
x46=100.530964914873x_{46} = 100.530964914873
x47=−69.1150383789755x_{47} = -69.1150383789755
x48=9.42477796076938x_{48} = 9.42477796076938
x49=−78.5398163397448x_{49} = -78.5398163397448
x50=−87.9645943005142x_{50} = -87.9645943005142
x51=−81.6814089933346x_{51} = -81.6814089933346
x52=15.707963267949x_{52} = 15.707963267949
x53=−28.2743338823081x_{53} = -28.2743338823081
x54=−15.707963267949x_{54} = -15.707963267949
x55=−37.6991118430775x_{55} = -37.6991118430775
x56=18.8495559215388x_{56} = 18.8495559215388
x57=50.2654824574367x_{57} = 50.2654824574367
x58=−72.2566310325652x_{58} = -72.2566310325652
x59=75.398223686155x_{59} = 75.398223686155
x60=6.28318530717959x_{60} = 6.28318530717959
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to Abs(sin(x)).
∣sin⁡(0)∣\left|{\sin{\left(0 \right)}}\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⁡(x)sign⁡(sin⁡(x))=0\cos{\left(x \right)} \operatorname{sign}{\left(\sin{\left(x \right)} \right)} = 0
Solve this equation
The roots of this equation
x1=7.85398163397448x_{1} = 7.85398163397448
x2=−73.8274273593601x_{2} = -73.8274273593601
x3=−54.9778714378214x_{3} = -54.9778714378214
x4=73.8274273593601x_{4} = 73.8274273593601
x5=0x_{5} = 0
x6=−26.7035375555132x_{6} = -26.7035375555132
x7=−1.5707963267949x_{7} = -1.5707963267949
x8=−306.305283725005x_{8} = -306.305283725005
x9=−95.8185759344887x_{9} = -95.8185759344887
x10=−39.2699081698724x_{10} = -39.2699081698724
x11=−4.71238898038469x_{11} = -4.71238898038469
x12=14.1371669411541x_{12} = 14.1371669411541
x13=10.9955742875643x_{13} = 10.9955742875643
x14=58.1194640914112x_{14} = 58.1194640914112
x15=70.6858347057703x_{15} = 70.6858347057703
x16=−36.1283155162826x_{16} = -36.1283155162826
x17=54.9778714378214x_{17} = 54.9778714378214
x18=23.5619449019235x_{18} = 23.5619449019235
x19=237.190245346029x_{19} = 237.190245346029
x20=−92.6769832808989x_{20} = -92.6769832808989
x21=−86.3937979737193x_{21} = -86.3937979737193
x22=−10.9955742875643x_{22} = -10.9955742875643
x23=92.6769832808989x_{23} = 92.6769832808989
x24=39.2699081698724x_{24} = 39.2699081698724
x25=−32.9867228626928x_{25} = -32.9867228626928
x26=98.9601685880785x_{26} = 98.9601685880785
x27=36.1283155162826x_{27} = 36.1283155162826
x28=−7.85398163397448x_{28} = -7.85398163397448
x29=−58.1194640914112x_{29} = -58.1194640914112
x30=−67.5442420521806x_{30} = -67.5442420521806
x31=−61.261056745001x_{31} = -61.261056745001
x32=26.7035375555132x_{32} = 26.7035375555132
x33=86.3937979737193x_{33} = 86.3937979737193
x34=−48.6946861306418x_{34} = -48.6946861306418
x35=51.8362787842316x_{35} = 51.8362787842316
x36=−42.4115008234622x_{36} = -42.4115008234622
x37=−89.5353906273091x_{37} = -89.5353906273091
x38=−98.9601685880785x_{38} = -98.9601685880785
x39=−14.1371669411541x_{39} = -14.1371669411541
x40=80.1106126665397x_{40} = 80.1106126665397
x41=−64.4026493985908x_{41} = -64.4026493985908
x42=95.8185759344887x_{42} = 95.8185759344887
x43=1.5707963267949x_{43} = 1.5707963267949
x44=45.553093477052x_{44} = 45.553093477052
x45=−17.2787595947439x_{45} = -17.2787595947439
x46=4.71238898038469x_{46} = 4.71238898038469
x47=48.6946861306418x_{47} = 48.6946861306418
x48=76.9690200129499x_{48} = 76.9690200129499
x49=−45.553093477052x_{49} = -45.553093477052
x50=−183.783170235003x_{50} = -183.783170235003
x51=20.4203522483337x_{51} = 20.4203522483337
x52=17.2787595947439x_{52} = 17.2787595947439
x53=−83.2522053201295x_{53} = -83.2522053201295
x54=−20.4203522483337x_{54} = -20.4203522483337
x55=−80.1106126665397x_{55} = -80.1106126665397
x56=61.261056745001x_{56} = 61.261056745001
x57=32.9867228626928x_{57} = 32.9867228626928
x58=64.4026493985908x_{58} = 64.4026493985908
x59=−23.5619449019235x_{59} = -23.5619449019235
x60=29.845130209103x_{60} = 29.845130209103
x61=42.4115008234622x_{61} = 42.4115008234622
x62=89.5353906273091x_{62} = 89.5353906273091
x63=−51.8362787842316x_{63} = -51.8362787842316
x64=−70.6858347057703x_{64} = -70.6858347057703
x65=83.2522053201295x_{65} = 83.2522053201295
x66=67.5442420521806x_{66} = 67.5442420521806
x67=−29.845130209103x_{67} = -29.845130209103
x68=−76.9690200129499x_{68} = -76.9690200129499
x69=−2279.22547017939x_{69} = -2279.22547017939
The values of the extrema at the points:
(7.853981633974483, 1)

(-73.82742735936014, 1)

(-54.977871437821385, 1)

(73.82742735936014, 1)

(0, 0)

(-26.703537555513243, 1)

(-1.5707963267948966, 1)

(-306.3052837250048, 1)

(-95.81857593448869, 1)

(-39.269908169872416, 1)

(-4.71238898038469, 1)

(14.137166941154069, 1)

(10.995574287564276, 1)

(58.119464091411174, 1)

(70.68583470577035, 1)

(-36.12831551628262, 1)

(54.977871437821385, 1)

(23.56194490192345, 1)

(237.1902453460294, 1)

(-92.6769832808989, 1)

(-86.39379797371932, 1)

(-10.995574287564276, 1)

(92.6769832808989, 1)

(39.269908169872416, 1)

(-32.98672286269283, 1)

(98.96016858807849, 1)

(36.12831551628262, 1)

(-7.853981633974483, 1)

(-58.119464091411174, 1)

(-67.54424205218055, 1)

(-61.26105674500097, 1)

(26.703537555513243, 1)

(86.39379797371932, 1)

(-48.6946861306418, 1)

(51.83627878423159, 1)

(-42.411500823462205, 1)

(-89.53539062730911, 1)

(-98.96016858807849, 1)

(-14.137166941154069, 1)

(80.11061266653972, 1)

(-64.40264939859077, 1)

(95.81857593448869, 1)

(1.5707963267948966, 1)

(45.553093477052, 1)

(-17.278759594743864, 1)

(4.71238898038469, 1)

(48.6946861306418, 1)

(76.96902001294994, 1)

(-45.553093477052, 1)

(-183.7831702350029, 1)

(20.420352248333657, 1)

(17.278759594743864, 1)

(-83.25220532012952, 1)

(-20.420352248333657, 1)

(-80.11061266653972, 1)

(61.26105674500097, 1)

(32.98672286269283, 1)

(64.40264939859077, 1)

(-23.56194490192345, 1)

(29.845130209103036, 1)

(42.411500823462205, 1)

(89.53539062730911, 1)

(-51.83627878423159, 1)

(-70.68583470577035, 1)

(83.25220532012952, 1)

(67.54424205218055, 1)

(-29.845130209103036, 1)

(-76.96902001294994, 1)

(-2279.225470179395, 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=0x_{1} = 0
Maxima of the function at points:
x1=7.85398163397448x_{1} = 7.85398163397448
x1=−73.8274273593601x_{1} = -73.8274273593601
x1=−54.9778714378214x_{1} = -54.9778714378214
x1=73.8274273593601x_{1} = 73.8274273593601
x1=−26.7035375555132x_{1} = -26.7035375555132
x1=−1.5707963267949x_{1} = -1.5707963267949
x1=−306.305283725005x_{1} = -306.305283725005
x1=−95.8185759344887x_{1} = -95.8185759344887
x1=−39.2699081698724x_{1} = -39.2699081698724
x1=−4.71238898038469x_{1} = -4.71238898038469
x1=14.1371669411541x_{1} = 14.1371669411541
x1=10.9955742875643x_{1} = 10.9955742875643
x1=58.1194640914112x_{1} = 58.1194640914112
x1=70.6858347057703x_{1} = 70.6858347057703
x1=−36.1283155162826x_{1} = -36.1283155162826
x1=54.9778714378214x_{1} = 54.9778714378214
x1=23.5619449019235x_{1} = 23.5619449019235
x1=237.190245346029x_{1} = 237.190245346029
x1=−92.6769832808989x_{1} = -92.6769832808989
x1=−86.3937979737193x_{1} = -86.3937979737193
x1=−10.9955742875643x_{1} = -10.9955742875643
x1=92.6769832808989x_{1} = 92.6769832808989
x1=39.2699081698724x_{1} = 39.2699081698724
x1=−32.9867228626928x_{1} = -32.9867228626928
x1=98.9601685880785x_{1} = 98.9601685880785
x1=36.1283155162826x_{1} = 36.1283155162826
x1=−7.85398163397448x_{1} = -7.85398163397448
x1=−58.1194640914112x_{1} = -58.1194640914112
x1=−67.5442420521806x_{1} = -67.5442420521806
x1=−61.261056745001x_{1} = -61.261056745001
x1=26.7035375555132x_{1} = 26.7035375555132
x1=86.3937979737193x_{1} = 86.3937979737193
x1=−48.6946861306418x_{1} = -48.6946861306418
x1=51.8362787842316x_{1} = 51.8362787842316
x1=−42.4115008234622x_{1} = -42.4115008234622
x1=−89.5353906273091x_{1} = -89.5353906273091
x1=−98.9601685880785x_{1} = -98.9601685880785
x1=−14.1371669411541x_{1} = -14.1371669411541
x1=80.1106126665397x_{1} = 80.1106126665397
x1=−64.4026493985908x_{1} = -64.4026493985908
x1=95.8185759344887x_{1} = 95.8185759344887
x1=1.5707963267949x_{1} = 1.5707963267949
x1=45.553093477052x_{1} = 45.553093477052
x1=−17.2787595947439x_{1} = -17.2787595947439
x1=4.71238898038469x_{1} = 4.71238898038469
x1=48.6946861306418x_{1} = 48.6946861306418
x1=76.9690200129499x_{1} = 76.9690200129499
x1=−45.553093477052x_{1} = -45.553093477052
x1=−183.783170235003x_{1} = -183.783170235003
x1=20.4203522483337x_{1} = 20.4203522483337
x1=17.2787595947439x_{1} = 17.2787595947439
x1=−83.2522053201295x_{1} = -83.2522053201295
x1=−20.4203522483337x_{1} = -20.4203522483337
x1=−80.1106126665397x_{1} = -80.1106126665397
x1=61.261056745001x_{1} = 61.261056745001
x1=32.9867228626928x_{1} = 32.9867228626928
x1=64.4026493985908x_{1} = 64.4026493985908
x1=−23.5619449019235x_{1} = -23.5619449019235
x1=29.845130209103x_{1} = 29.845130209103
x1=42.4115008234622x_{1} = 42.4115008234622
x1=89.5353906273091x_{1} = 89.5353906273091
x1=−51.8362787842316x_{1} = -51.8362787842316
x1=−70.6858347057703x_{1} = -70.6858347057703
x1=83.2522053201295x_{1} = 83.2522053201295
x1=67.5442420521806x_{1} = 67.5442420521806
x1=−29.845130209103x_{1} = -29.845130209103
x1=−76.9690200129499x_{1} = -76.9690200129499
x1=−2279.22547017939x_{1} = -2279.22547017939
Decreasing at intervals
(−∞,−2279.22547017939]∪[0,∞)\left(-\infty, -2279.22547017939\right] \cup \left[0, \infty\right)
Increasing at intervals
(−∞,0]∪[237.190245346029,∞)\left(-\infty, 0\right] \cup \left[237.190245346029, \infty\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⁡(x)sign⁡(sin⁡(x))+2cos⁡2(x)δ(sin⁡(x))=0- \sin{\left(x \right)} \operatorname{sign}{\left(\sin{\left(x \right)} \right)} + 2 \cos^{2}{\left(x \right)} \delta\left(\sin{\left(x \right)}\right) = 0
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞∣sin⁡(x)∣=∣⟨−1,1⟩∣\lim_{x \to -\infty} \left|{\sin{\left(x \right)}}\right| = \left|{\left\langle -1, 1\right\rangle}\right|
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=∣⟨−1,1⟩∣y = \left|{\left\langle -1, 1\right\rangle}\right|
lim⁡x→∞∣sin⁡(x)∣=∣⟨−1,1⟩∣\lim_{x \to \infty} \left|{\sin{\left(x \right)}}\right| = \left|{\left\langle -1, 1\right\rangle}\right|
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=∣⟨−1,1⟩∣y = \left|{\left\langle -1, 1\right\rangle}\right|
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of Abs(sin(x)), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(∣sin⁡(x)∣x)=0\lim_{x \to -\infty}\left(\frac{\left|{\sin{\left(x \right)}}\right|}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(∣sin⁡(x)∣x)=0\lim_{x \to \infty}\left(\frac{\left|{\sin{\left(x \right)}}\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⁡(x)∣=∣sin⁡(x)∣\left|{\sin{\left(x \right)}}\right| = \left|{\sin{\left(x \right)}}\right|
- Yes
∣sin⁡(x)∣=−∣sin⁡(x)∣\left|{\sin{\left(x \right)}}\right| = - \left|{\sin{\left(x \right)}}\right|
- No
so, the function
is
even
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