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Plotting a function graph/ y=2sin3x+1

Graphing y = y=2sin3x+1

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = 2*sin(3*x) + 1
f(x)=2sin⁡(3x)+1f{\left(x \right)} = 2 \sin{\left(3 x \right)} + 1 
f = 2*sin(3*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⁡(3x)+1=02 \sin{\left(3 x \right)} + 1 = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−π18x_{1} = - \frac{\pi}{18}
x2=7π18x_{2} = \frac{7 \pi}{18}
Numerical solution
x1=70.3367688553715x_{1} = 70.3367688553715
x2=−44.1568300754565x_{2} = -44.1568300754565
x3=−61.6101225953998x_{3} = -61.6101225953998
x4=−35.7792496658838x_{4} = -35.7792496658838
x5=646.2954220135x_{5} = 646.2954220135
x6=91.9788515801012x_{6} = 91.9788515801012
x7=98.2620368872808x_{7} = 98.2620368872808
x8=68.2423737529783x_{8} = 68.2423737529783
x9=96.1676417848876x_{9} = 96.1676417848876
x10=85.6956662729216x_{10} = 85.6956662729216
x11=61.9591884457987x_{11} = 61.9591884457987
x12=72.4311639577647x_{12} = 72.4311639577647
x13=−31.5904594610974x_{13} = -31.5904594610974
x14=−19.7222205475359x_{14} = -19.7222205475359
x15=45.9021593274509x_{15} = 45.9021593274509
x16=−28.0998009571087x_{16} = -28.0998009571087
x17=−39.9680398706701x_{17} = -39.9680398706701
x18=−26.0054058547155x_{18} = -26.0054058547155
x19=−67.8933079025794x_{19} = -67.8933079025794
x20=41.7133691226645x_{20} = 41.7133691226645
x21=47.9965544298441x_{21} = 47.9965544298441
x22=−54.6288055874225x_{22} = -54.6288055874225
x23=12.3918376891597x_{23} = 12.3918376891597
x24=39.6189740202713x_{24} = 39.6189740202713
x25=74.5255590601579x_{25} = 74.5255590601579
x26=54.2797397370236x_{26} = 54.2797397370236
x27=8.20304748437335x_{27} = 8.20304748437335
x28=56.3741348394168x_{28} = 56.3741348394168
x29=−4.36332312998582x_{29} = -4.36332312998582
x30=−57.4213323906134x_{30} = -57.4213323906134
x31=76.6199541625511x_{31} = 76.6199541625511
x32=−86.0447321233204x_{32} = -86.0447321233204
x33=−92.3279174305x_{33} = -92.3279174305
x34=−46.2512251778497x_{34} = -46.2512251778497
x35=64.0535835481919x_{35} = 64.0535835481919
x36=−23.9110107523223x_{36} = -23.9110107523223
x37=1.91986217719376x_{37} = 1.91986217719376
x38=87.7900613753148x_{38} = 87.7900613753148
x39=−98.6111027376796x_{39} = -98.6111027376796
x40=−79.7615468161409x_{40} = -79.7615468161409
x41=22.165681500328x_{41} = 22.165681500328
x42=−37.873644768277x_{42} = -37.873644768277
x43=−42.0624349730633x_{43} = -42.0624349730633
x44=−51.1381470834339x_{44} = -51.1381470834339
x45=52.1853446346305x_{45} = 52.1853446346305
x46=−21.8166156499291x_{46} = -21.8166156499291
x47=43.8077642250577x_{47} = 43.8077642250577
x48=−11.3446401379631x_{48} = -11.3446401379631
x49=−16.929693744345x_{49} = -16.929693744345
x50=10.2974425867665x_{50} = 10.2974425867665
x51=−77.6671517137477x_{51} = -77.6671517137477
x52=20.0712863979348x_{52} = 20.0712863979348
x53=−0.174532925199433x_{53} = -0.174532925199433
x54=50.0909495322373x_{54} = 50.0909495322373
x55=94.0732466824944x_{55} = 94.0732466824944
x56=−48.3456202802429x_{56} = -48.3456202802429
x57=15.8824961931484x_{57} = 15.8824961931484
x58=−72.0820981073658x_{58} = -72.0820981073658
x59=−63.704517697793x_{59} = -63.704517697793
x60=−69.9877030049726x_{60} = -69.9877030049726
x61=24.2600766027212x_{61} = 24.2600766027212
x62=−15.5334303427495x_{62} = -15.5334303427495
x63=−65.7989128001862x_{63} = -65.7989128001862
x64=−7.15584993317675x_{64} = -7.15584993317675
x65=89.884456477708x_{65} = 89.884456477708
x66=−81.8559419185341x_{66} = -81.8559419185341
x67=28.4488668075076x_{67} = 28.4488668075076
x68=17.9768912955416x_{68} = 17.9768912955416
x69=26.3544717051144x_{69} = 26.3544717051144
x70=4.01425727958696x_{70} = 4.01425727958696
x71=100.356431989674x_{71} = 100.356431989674
x72=6.10865238198015x_{72} = 6.10865238198015
x73=33.3357887130917x_{73} = 33.3357887130917
x74=−90.2335223281068x_{74} = -90.2335223281068
x75=−75.5727566113545x_{75} = -75.5727566113545
x76=30.5432619099008x_{76} = 30.5432619099008
x77=32.637657012294x_{77} = 32.637657012294
x78=−83.9503370209273x_{78} = -83.9503370209273
x79=77.3180858633488x_{79} = 77.3180858633488
x80=−88.1391272257137x_{80} = -88.1391272257137
x81=−13.4390352403563x_{81} = -13.4390352403563
x82=−2.26892802759263x_{82} = -2.26892802759263
x83=−74.176493209759x_{83} = -74.176493209759
x84=66.1479786505851x_{84} = 66.1479786505851
x85=83.6012711705284x_{85} = 83.6012711705284
x86=−33.6848545634906x_{86} = -33.6848545634906
x87=59.8647933434055x_{87} = 59.8647933434055
x88=−59.5157274930066x_{88} = -59.5157274930066
x89=−17.6278254451427x_{89} = -17.6278254451427
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 2*sin(3*x) + 1.
2sin⁡(0⋅3)+12 \sin{\left(0 \cdot 3 \right)} + 1
The result:
f(0)=1f{\left(0 \right)} = 1
The point:
(0, 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
6cos⁡(3x)=06 \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     
(--, -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=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
−18sin⁡(3x)=0- 18 \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→−∞(2sin⁡(3x)+1)=⟨−1,3⟩\lim_{x \to -\infty}\left(2 \sin{\left(3 x \right)} + 1\right) = \left\langle -1, 3\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−1,3⟩y = \left\langle -1, 3\right\rangle
lim⁡x→∞(2sin⁡(3x)+1)=⟨−1,3⟩\lim_{x \to \infty}\left(2 \sin{\left(3 x \right)} + 1\right) = \left\langle -1, 3\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−1,3⟩y = \left\langle -1, 3\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 2*sin(3*x) + 1, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(2sin⁡(3x)+1x)=0\lim_{x \to -\infty}\left(\frac{2 \sin{\left(3 x \right)} + 1}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(2sin⁡(3x)+1x)=0\lim_{x \to \infty}\left(\frac{2 \sin{\left(3 x \right)} + 1}{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⁡(3x)+1=1−2sin⁡(3x)2 \sin{\left(3 x \right)} + 1 = 1 - 2 \sin{\left(3 x \right)}
- No
2sin⁡(3x)+1=2sin⁡(3x)−12 \sin{\left(3 x \right)} + 1 = 2 \sin{\left(3 x \right)} - 1
- No
so, the function
not is
neither even, nor odd
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