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Plotting a function graph/ 3sint

Graphing y = 3sint

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(t) = 3*sin(t)
f(t)=3sin⁡(t)f{\left(t \right)} = 3 \sin{\left(t \right)} 
f = 3*sin(t)
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 T at f = 0
so we need to solve the equation:
3sin⁡(t)=03 \sin{\left(t \right)} = 0
Solve this equation
The points of intersection with the axis T:

Analytical solution
t1=0t_{1} = 0
t2=πt_{2} = \pi
Numerical solution
t1=−43.9822971502571t_{1} = -43.9822971502571
t2=−31.4159265358979t_{2} = -31.4159265358979
t3=84.8230016469244t_{3} = 84.8230016469244
t4=−91.106186954104t_{4} = -91.106186954104
t5=−97.3893722612836t_{5} = -97.3893722612836
t6=91.106186954104t_{6} = 91.106186954104
t7=6.28318530717959t_{7} = 6.28318530717959
t8=−72.2566310325652t_{8} = -72.2566310325652
t9=−47.1238898038469t_{9} = -47.1238898038469
t10=−113.097335529233t_{10} = -113.097335529233
t11=94.2477796076938t_{11} = 94.2477796076938
t12=50.2654824574367t_{12} = 50.2654824574367
t13=56.5486677646163t_{13} = 56.5486677646163
t14=43.9822971502571t_{14} = 43.9822971502571
t15=47.1238898038469t_{15} = 47.1238898038469
t16=−50.2654824574367t_{16} = -50.2654824574367
t17=37.6991118430775t_{17} = 37.6991118430775
t18=−28.2743338823081t_{18} = -28.2743338823081
t19=65.9734457253857t_{19} = 65.9734457253857
t20=15.707963267949t_{20} = 15.707963267949
t21=28.2743338823081t_{21} = 28.2743338823081
t22=−62.8318530717959t_{22} = -62.8318530717959
t23=40.8407044966673t_{23} = 40.8407044966673
t24=−40.8407044966673t_{24} = -40.8407044966673
t25=−6.28318530717959t_{25} = -6.28318530717959
t26=−81.6814089933346t_{26} = -81.6814089933346
t27=−15.707963267949t_{27} = -15.707963267949
t28=−59.6902604182061t_{28} = -59.6902604182061
t29=72.2566310325652t_{29} = 72.2566310325652
t30=3.14159265358979t_{30} = 3.14159265358979
t31=−25.1327412287183t_{31} = -25.1327412287183
t32=21.9911485751286t_{32} = 21.9911485751286
t33=−75.398223686155t_{33} = -75.398223686155
t34=−56.5486677646163t_{34} = -56.5486677646163
t35=−267.035375555132t_{35} = -267.035375555132
t36=−69.1150383789755t_{36} = -69.1150383789755
t37=−84.8230016469244t_{37} = -84.8230016469244
t38=78.5398163397448t_{38} = 78.5398163397448
t39=9.42477796076938t_{39} = 9.42477796076938
t40=−232.477856365645t_{40} = -232.477856365645
t41=−53.4070751110265t_{41} = -53.4070751110265
t42=62.8318530717959t_{42} = 62.8318530717959
t43=−18.8495559215388t_{43} = -18.8495559215388
t44=25.1327412287183t_{44} = 25.1327412287183
t45=100.530964914873t_{45} = 100.530964914873
t46=−87.9645943005142t_{46} = -87.9645943005142
t47=−9.42477796076938t_{47} = -9.42477796076938
t48=75.398223686155t_{48} = 75.398223686155
t49=81.6814089933346t_{49} = 81.6814089933346
t50=87.9645943005142t_{50} = 87.9645943005142
t51=12.5663706143592t_{51} = 12.5663706143592
t52=−34.5575191894877t_{52} = -34.5575191894877
t53=69.1150383789755t_{53} = 69.1150383789755
t54=−3.14159265358979t_{54} = -3.14159265358979
t55=0t_{55} = 0
t56=−21.9911485751286t_{56} = -21.9911485751286
t57=−37.6991118430775t_{57} = -37.6991118430775
t58=31.4159265358979t_{58} = 31.4159265358979
t59=−78.5398163397448t_{59} = -78.5398163397448
t60=−12.5663706143592t_{60} = -12.5663706143592
t61=−94.2477796076938t_{61} = -94.2477796076938
t62=97.3893722612836t_{62} = 97.3893722612836
t63=−100.530964914873t_{63} = -100.530964914873
t64=59.6902604182061t_{64} = 59.6902604182061
t65=53.4070751110265t_{65} = 53.4070751110265
t66=34.5575191894877t_{66} = 34.5575191894877
t67=−65.9734457253857t_{67} = -65.9734457253857
t68=18.8495559215388t_{68} = 18.8495559215388
The points of intersection with the Y axis coordinate
The graph crosses Y axis when t equals 0:
substitute t = 0 to 3*sin(t).
3sin⁡(0)3 \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
ddtf(t)=0\frac{d}{d t} f{\left(t \right)} = 0
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
ddtf(t)=\frac{d}{d t} f{\left(t \right)} =
the first derivative
3cos⁡(t)=03 \cos{\left(t \right)} = 0
Solve this equation
The roots of this equation
t1=π2t_{1} = \frac{\pi}{2}
t2=3π2t_{2} = \frac{3 \pi}{2}
The values of the extrema at the points:
 pi    
(--, 3)
 2     

 3*pi     
(----, -3)
  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:
t1=3π2t_{1} = \frac{3 \pi}{2}
Maxima of the function at points:
t1=π2t_{1} = \frac{\pi}{2}
Decreasing at intervals
(−∞,π2]∪[3π2,∞)\left(-\infty, \frac{\pi}{2}\right] \cup \left[\frac{3 \pi}{2}, \infty\right)
Increasing at intervals
[π2,3π2]\left[\frac{\pi}{2}, \frac{3 \pi}{2}\right]
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
d2dt2f(t)=0\frac{d^{2}}{d t^{2}} f{\left(t \right)} = 0
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
d2dt2f(t)=\frac{d^{2}}{d t^{2}} f{\left(t \right)} =
the second derivative
−3sin⁡(t)=0- 3 \sin{\left(t \right)} = 0
Solve this equation
The roots of this equation
t1=0t_{1} = 0
t2=πt_{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(-\infty, 0\right] \cup \left[\pi, \infty\right)
Convex at the intervals
[0,π]\left[0, \pi\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at t->+oo and t->-oo
lim⁡t→−∞(3sin⁡(t))=⟨−3,3⟩\lim_{t \to -\infty}\left(3 \sin{\left(t \right)}\right) = \left\langle -3, 3\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−3,3⟩y = \left\langle -3, 3\right\rangle
lim⁡t→∞(3sin⁡(t))=⟨−3,3⟩\lim_{t \to \infty}\left(3 \sin{\left(t \right)}\right) = \left\langle -3, 3\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−3,3⟩y = \left\langle -3, 3\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 3*sin(t), divided by t at t->+oo and t ->-oo
lim⁡t→−∞(3sin⁡(t)t)=0\lim_{t \to -\infty}\left(\frac{3 \sin{\left(t \right)}}{t}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡t→∞(3sin⁡(t)t)=0\lim_{t \to \infty}\left(\frac{3 \sin{\left(t \right)}}{t}\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(-t) и f = -f(-t).
So, check:
3sin⁡(t)=−3sin⁡(t)3 \sin{\left(t \right)} = - 3 \sin{\left(t \right)}
- No
3sin⁡(t)=3sin⁡(t)3 \sin{\left(t \right)} = 3 \sin{\left(t \right)}
- Yes
so, the function
is
odd
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