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-sin(x)
Plotting a function graph/ -sin(x)

Graphing y = -sin(x)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = -sin(x)
f(x)=−sin⁡(x)f{\left(x \right)} = - \sin{\left(x \right)} 
f = -sin(x)
The graph of the function
0102030405060708090-102-2
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- \sin{\left(x \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=72.2566310325652x_{1} = 72.2566310325652
x2=94.2477796076938x_{2} = 94.2477796076938
x3=−3.14159265358979x_{3} = -3.14159265358979
x4=40.8407044966673x_{4} = 40.8407044966673
x5=−91.106186954104x_{5} = -91.106186954104
x6=−15.707963267949x_{6} = -15.707963267949
x7=−37.6991118430775x_{7} = -37.6991118430775
x8=−31.4159265358979x_{8} = -31.4159265358979
x9=−81.6814089933346x_{9} = -81.6814089933346
x10=6.28318530717959x_{10} = 6.28318530717959
x11=62.8318530717959x_{11} = 62.8318530717959
x12=53.4070751110265x_{12} = 53.4070751110265
x13=81.6814089933346x_{13} = 81.6814089933346
x14=100.530964914873x_{14} = 100.530964914873
x15=50.2654824574367x_{15} = 50.2654824574367
x16=84.8230016469244x_{16} = 84.8230016469244
x17=−47.1238898038469x_{17} = -47.1238898038469
x18=−267.035375555132x_{18} = -267.035375555132
x19=−56.5486677646163x_{19} = -56.5486677646163
x20=−53.4070751110265x_{20} = -53.4070751110265
x21=−113.097335529233x_{21} = -113.097335529233
x22=3.14159265358979x_{22} = 3.14159265358979
x23=25.1327412287183x_{23} = 25.1327412287183
x24=−9.42477796076938x_{24} = -9.42477796076938
x25=−232.477856365645x_{25} = -232.477856365645
x26=−87.9645943005142x_{26} = -87.9645943005142
x27=15.707963267949x_{27} = 15.707963267949
x28=91.106186954104x_{28} = 91.106186954104
x29=97.3893722612836x_{29} = 97.3893722612836
x30=78.5398163397448x_{30} = 78.5398163397448
x31=9.42477796076938x_{31} = 9.42477796076938
x32=−6.28318530717959x_{32} = -6.28318530717959
x33=75.398223686155x_{33} = 75.398223686155
x34=12.5663706143592x_{34} = 12.5663706143592
x35=56.5486677646163x_{35} = 56.5486677646163
x36=−12.5663706143592x_{36} = -12.5663706143592
x37=−34.5575191894877x_{37} = -34.5575191894877
x38=−28.2743338823081x_{38} = -28.2743338823081
x39=18.8495559215388x_{39} = 18.8495559215388
x40=−78.5398163397448x_{40} = -78.5398163397448
x41=−94.2477796076938x_{41} = -94.2477796076938
x42=−25.1327412287183x_{42} = -25.1327412287183
x43=59.6902604182061x_{43} = 59.6902604182061
x44=−21.9911485751286x_{44} = -21.9911485751286
x45=−72.2566310325652x_{45} = -72.2566310325652
x46=−75.398223686155x_{46} = -75.398223686155
x47=31.4159265358979x_{47} = 31.4159265358979
x48=43.9822971502571x_{48} = 43.9822971502571
x49=28.2743338823081x_{49} = 28.2743338823081
x50=−40.8407044966673x_{50} = -40.8407044966673
x51=−50.2654824574367x_{51} = -50.2654824574367
x52=47.1238898038469x_{52} = 47.1238898038469
x53=−84.8230016469244x_{53} = -84.8230016469244
x54=−97.3893722612836x_{54} = -97.3893722612836
x55=−43.9822971502571x_{55} = -43.9822971502571
x56=69.1150383789755x_{56} = 69.1150383789755
x57=−18.8495559215388x_{57} = -18.8495559215388
x58=65.9734457253857x_{58} = 65.9734457253857
x59=−65.9734457253857x_{59} = -65.9734457253857
x60=34.5575191894877x_{60} = 34.5575191894877
x61=0x_{61} = 0
x62=−59.6902604182061x_{62} = -59.6902604182061
x63=−62.8318530717959x_{63} = -62.8318530717959
x64=−2642.07942166902x_{64} = -2642.07942166902
x65=21.9911485751286x_{65} = 21.9911485751286
x66=87.9645943005142x_{66} = 87.9645943005142
x67=37.6991118430775x_{67} = 37.6991118430775
x68=−100.530964914873x_{68} = -100.530964914873
x69=−69.1150383789755x_{69} = -69.1150383789755
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to -sin(x).
−sin⁡(0)- \sin{\left(0 \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)=0- \cos{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=π2x_{1} = \frac{\pi}{2}
x2=3π2x_{2} = \frac{3 \pi}{2}
The values of the extrema at the points:
 pi     
(--, -1)
 2      

 3*pi    
(----, 1)
  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=π2x_{1} = \frac{\pi}{2}
Maxima of the function at points:
x1=3π2x_{1} = \frac{3 \pi}{2}
Decreasing at intervals
[π2,3π2]\left[\frac{\pi}{2}, \frac{3 \pi}{2}\right]
Increasing at intervals
(−∞,π2]∪[3π2,∞)\left(-\infty, \frac{\pi}{2}\right] \cup \left[\frac{3 \pi}{2}, \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)=0\sin{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=πx_{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,π]\left[0, \pi\right]
Convex at the intervals
(−∞,0]∪[π,∞)\left(-\infty, 0\right] \cup \left[\pi, \infty\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⁡(x))=⟨−1,1⟩\lim_{x \to -\infty}\left(- \sin{\left(x \right)}\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⁡(x))=⟨−1,1⟩\lim_{x \to \infty}\left(- \sin{\left(x \right)}\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), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(−sin⁡(x)x)=0\lim_{x \to -\infty}\left(- \frac{\sin{\left(x \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{\sin{\left(x \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)- \sin{\left(x \right)} = \sin{\left(x \right)}
- No
−sin⁡(x)=−sin⁡(x)- \sin{\left(x \right)} = - \sin{\left(x \right)}
- Yes
so, the function
is
odd
The graph
Graphing y = -sin(x)
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