Mister Exam

Other calculators

3sin3x
Plotting a function graph/ 3sin3x

Graphing y = 3sin3x

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
f(x) = 3*sin(3*x)
f(x)=3sin⁡(3x)f{\left(x \right)} = 3 \sin{\left(3 x \right)} 
f = 3*sin(3*x)
The graph of the function
0-70-60-50-40-30-20-10102030405060705-5
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:
3sin⁡(3x)=03 \sin{\left(3 x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
x2=π3x_{2} = \frac{\pi}{3}
Numerical solution
x1=78.5398163397448x_{1} = 78.5398163397448
x2=−77.4926187885482x_{2} = -77.4926187885482
x3=−70.162235930172x_{3} = -70.162235930172
x4=70.162235930172x_{4} = 70.162235930172
x5=10.471975511966x_{5} = 10.471975511966
x6=−33.5103216382911x_{6} = -33.5103216382911
x7=−43.9822971502571x_{7} = -43.9822971502571
x8=−41.8879020478639x_{8} = -41.8879020478639
x9=−94.2477796076938x_{9} = -94.2477796076938
x10=8.37758040957278x_{10} = 8.37758040957278
x11=−26.1799387799149x_{11} = -26.1799387799149
x12=59.6902604182061x_{12} = 59.6902604182061
x13=−90.0589894029074x_{13} = -90.0589894029074
x14=96.342174710087x_{14} = 96.342174710087
x15=746.651854003174x_{15} = 746.651854003174
x16=19.8967534727354x_{16} = 19.8967534727354
x17=94.2477796076938x_{17} = 94.2477796076938
x18=83.7758040957278x_{18} = 83.7758040957278
x19=50.2654824574367x_{19} = 50.2654824574367
x20=15.707963267949x_{20} = 15.707963267949
x21=−4.18879020478639x_{21} = -4.18879020478639
x22=63.8790506229925x_{22} = 63.8790506229925
x23=−57.5958653158129x_{23} = -57.5958653158129
x24=−17.8023583703422x_{24} = -17.8023583703422
x25=24.0855436775217x_{25} = 24.0855436775217
x26=−55.5014702134197x_{26} = -55.5014702134197
x27=−83.7758040957278x_{27} = -83.7758040957278
x28=−99.4837673636768x_{28} = -99.4837673636768
x29=65.9734457253857x_{29} = 65.9734457253857
x30=61.7846555205993x_{30} = 61.7846555205993
x31=80.634211442138x_{31} = 80.634211442138
x32=81.6814089933346x_{32} = 81.6814089933346
x33=109.955742875643x_{33} = 109.955742875643
x34=39.7935069454707x_{34} = 39.7935069454707
x35=2491.28297429671x_{35} = 2491.28297429671
x36=−54.4542726622231x_{36} = -54.4542726622231
x37=4.18879020478639x_{37} = 4.18879020478639
x38=−24.0855436775217x_{38} = -24.0855436775217
x39=32.4631240870945x_{39} = 32.4631240870945
x40=−15.707963267949x_{40} = -15.707963267949
x41=−2.0943951023932x_{41} = -2.0943951023932
x42=−8.37758040957278x_{42} = -8.37758040957278
x43=−68.0678408277789x_{43} = -68.0678408277789
x44=74.3510261349584x_{44} = 74.3510261349584
x45=54.4542726622231x_{45} = 54.4542726622231
x46=68.0678408277789x_{46} = 68.0678408277789
x47=−81.6814089933346x_{47} = -81.6814089933346
x48=−21.9911485751286x_{48} = -21.9911485751286
x49=−50.2654824574367x_{49} = -50.2654824574367
x50=87.9645943005142x_{50} = 87.9645943005142
x51=76.4454212373516x_{51} = 76.4454212373516
x52=−28.2743338823081x_{52} = -28.2743338823081
x53=41.8879020478639x_{53} = 41.8879020478639
x54=98.4365698124802x_{54} = 98.4365698124802
x55=−19.8967534727354x_{55} = -19.8967534727354
x56=−46.0766922526503x_{56} = -46.0766922526503
x57=37.6991118430775x_{57} = 37.6991118430775
x58=−85.870199198121x_{58} = -85.870199198121
x59=−13.6135681655558x_{59} = -13.6135681655558
x60=30.3687289847013x_{60} = 30.3687289847013
x61=−37.6991118430775x_{61} = -37.6991118430775
x62=−87.9645943005142x_{62} = -87.9645943005142
x63=92.1533845053006x_{63} = 92.1533845053006
x64=72.2566310325652x_{64} = 72.2566310325652
x65=85.870199198121x_{65} = 85.870199198121
x66=690.103186238558x_{66} = 690.103186238558
x67=6.28318530717959x_{67} = 6.28318530717959
x68=−31.4159265358979x_{68} = -31.4159265358979
x69=46.0766922526503x_{69} = 46.0766922526503
x70=21.9911485751286x_{70} = 21.9911485751286
x71=34.5575191894877x_{71} = 34.5575191894877
x72=−63.8790506229925x_{72} = -63.8790506229925
x73=−98.4365698124802x_{73} = -98.4365698124802
x74=−79.5870138909414x_{74} = -79.5870138909414
x75=−61.7846555205993x_{75} = -61.7846555205993
x76=−35.6047167406843x_{76} = -35.6047167406843
x77=−29.3215314335047x_{77} = -29.3215314335047
x78=−39.7935069454707x_{78} = -39.7935069454707
x79=43.9822971502571x_{79} = 43.9822971502571
x80=52.3598775598299x_{80} = 52.3598775598299
x81=−92.1533845053006x_{81} = -92.1533845053006
x82=−10.471975511966x_{82} = -10.471975511966
x83=26.1799387799149x_{83} = 26.1799387799149
x84=56.5486677646163x_{84} = 56.5486677646163
x85=−11.5191730631626x_{85} = -11.5191730631626
x86=90.0589894029074x_{86} = 90.0589894029074
x87=28.2743338823081x_{87} = 28.2743338823081
x88=−48.1710873550435x_{88} = -48.1710873550435
x89=−72.2566310325652x_{89} = -72.2566310325652
x90=100.530964914873x_{90} = 100.530964914873
x91=0x_{91} = 0
x92=2.0943951023932x_{92} = 2.0943951023932
x93=−59.6902604182061x_{93} = -59.6902604182061
x94=17.8023583703422x_{94} = 17.8023583703422
x95=48.1710873550435x_{95} = 48.1710873550435
x96=−65.9734457253857x_{96} = -65.9734457253857
x97=−6.28318530717959x_{97} = -6.28318530717959
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 3*sin(3*x).
3sin⁡(3⋅0)3 \sin{\left(3 \cdot 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
9cos⁡(3x)=09 \cos{\left(3 x \right)} = 0
Solve this equation
The roots of this equation
x1=π6x_{1} = \frac{\pi}{6}
x2=π2x_{2} = \frac{\pi}{2}
The values of the extrema at the points:
 pi    
(--, 3)
 6     

 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:
x1=π2x_{1} = \frac{\pi}{2}
Maxima of the function at points:
x1=π6x_{1} = \frac{\pi}{6}
Decreasing at intervals
(−∞,π6]∪[π2,∞)\left(-\infty, \frac{\pi}{6}\right] \cup \left[\frac{\pi}{2}, \infty\right)
Increasing at intervals
[π6,π2]\left[\frac{\pi}{6}, \frac{\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
−27sin⁡(3x)=0- 27 \sin{\left(3 x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=π3x_{2} = \frac{\pi}{3}

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]∪[π3,∞)\left(-\infty, 0\right] \cup \left[\frac{\pi}{3}, \infty\right)
Convex at the intervals
[0,π3]\left[0, \frac{\pi}{3}\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(3sin⁡(3x))=⟨−3,3⟩\lim_{x \to -\infty}\left(3 \sin{\left(3 x \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⁡x→∞(3sin⁡(3x))=⟨−3,3⟩\lim_{x \to \infty}\left(3 \sin{\left(3 x \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(3*x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(3sin⁡(3x)x)=0\lim_{x \to -\infty}\left(\frac{3 \sin{\left(3 x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(3sin⁡(3x)x)=0\lim_{x \to \infty}\left(\frac{3 \sin{\left(3 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:
3sin⁡(3x)=−3sin⁡(3x)3 \sin{\left(3 x \right)} = - 3 \sin{\left(3 x \right)}
- No
3sin⁡(3x)=3sin⁡(3x)3 \sin{\left(3 x \right)} = 3 \sin{\left(3 x \right)}
- Yes
so, the function
is
odd
The graph
Graphing y = 3sin3x
    © Mister Exam – Calculator