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Plotting a function graph/ 2*sin(x)+1

Graphing y = 2*sin(x)+1

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = 2*sin(x) + 1
f(x)=2sin⁡(x)+1f{\left(x \right)} = 2 \sin{\left(x \right)} + 1 
f = 2*sin(x) + 1
The graph of the function
-1.0-0.8-0.6-0.4-0.21.00.00.20.40.60.85-5
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⁡(x)+1=02 \sin{\left(x \right)} + 1 = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−π6x_{1} = - \frac{\pi}{6}
x2=7π6x_{2} = \frac{7 \pi}{6}
Numerical solution
x1=9.94837673636768x_{1} = 9.94837673636768
x2=−75.9218224617533x_{2} = -75.9218224617533
x3=37.1755130674792x_{3} = 37.1755130674792
x4=74.8746249105567x_{4} = 74.8746249105567
x5=24.60914245312x_{5} = 24.60914245312
x6=60.2138591938044x_{6} = 60.2138591938044
x7=−15.1843644923507x_{7} = -15.1843644923507
x8=−101.054563690472x_{8} = -101.054563690472
x9=−31.9395253114962x_{9} = -31.9395253114962
x10=−25.6563400043166x_{10} = -25.6563400043166
x11=93.7241808320955x_{11} = 93.7241808320955
x12=−195.302343298165x_{12} = -195.302343298165
x13=−52.8834763354282x_{13} = -52.8834763354282
x14=85.3466004225227x_{14} = 85.3466004225227
x15=−0.523598775598299x_{15} = -0.523598775598299
x16=−38.2227106186758x_{16} = -38.2227106186758
x17=−57.0722665402146x_{17} = -57.0722665402146
x18=−84.2994028713261x_{18} = -84.2994028713261
x19=−90.5825881785057x_{19} = -90.5825881785057
x20=62.3082542961976x_{20} = 62.3082542961976
x21=66.497044500984x_{21} = 66.497044500984
x22=16.2315620435473x_{22} = 16.2315620435473
x23=22.5147473507269x_{23} = 22.5147473507269
x24=28.7979326579064x_{24} = 28.7979326579064
x25=12.0427718387609x_{25} = 12.0427718387609
x26=−46.6002910282486x_{26} = -46.6002910282486
x27=−78.0162175641465x_{27} = -78.0162175641465
x28=−82.2050077689329x_{28} = -82.2050077689329
x29=100.007366139275x_{29} = 100.007366139275
x30=5.75958653158129x_{30} = 5.75958653158129
x31=−8.90117918517108x_{31} = -8.90117918517108
x32=−13.0899693899575x_{32} = -13.0899693899575
x33=−88.4881930761125x_{33} = -88.4881930761125
x34=437.20497762458x_{34} = 437.20497762458
x35=−94.7713783832921x_{35} = -94.7713783832921
x36=−65.4498469497874x_{36} = -65.4498469497874
x37=79.0634151153431x_{37} = 79.0634151153431
x38=−19.3731546971371x_{38} = -19.3731546971371
x39=56.025068989018x_{39} = 56.025068989018
x40=−34.0339204138894x_{40} = -34.0339204138894
x41=−63.3554518473942x_{41} = -63.3554518473942
x42=91.6297857297023x_{42} = 91.6297857297023
x43=−44.5058959258554x_{43} = -44.5058959258554
x44=−6.80678408277789x_{44} = -6.80678408277789
x45=−2.61799387799149x_{45} = -2.61799387799149
x46=72.7802298081635x_{46} = 72.7802298081635
x47=192.160750644576x_{47} = 192.160750644576
x48=35.081117965086x_{48} = 35.081117965086
x49=−40.317105721069x_{49} = -40.317105721069
x50=97.9129710368819x_{50} = 97.9129710368819
x51=−96.8657734856853x_{51} = -96.8657734856853
x52=−151.320046147908x_{52} = -151.320046147908
x53=87.4409955249159x_{53} = 87.4409955249159
x54=81.1578102177363x_{54} = 81.1578102177363
x55=66400.1787274983x_{55} = 66400.1787274983
x56=53.9306738866248x_{56} = 53.9306738866248
x57=68.5914396033772x_{57} = 68.5914396033772
x58=3.66519142918809x_{58} = 3.66519142918809
x59=−71.733032256967x_{59} = -71.733032256967
x60=−50.789081233035x_{60} = -50.789081233035
x61=−59.1666616426078x_{61} = -59.1666616426078
x62=30.8923277602996x_{62} = 30.8923277602996
x63=47.6474885794452x_{63} = 47.6474885794452
x64=−27.7507351067098x_{64} = -27.7507351067098
x65=18.3259571459405x_{65} = 18.3259571459405
x66=43.4586983746588x_{66} = 43.4586983746588
x67=41.3643032722656x_{67} = 41.3643032722656
x68=−21.4675497995303x_{68} = -21.4675497995303
x69=49.7418836818384x_{69} = 49.7418836818384
x70=−69.6386371545737x_{70} = -69.6386371545737
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 2*sin(x) + 1.
2sin⁡(0)+12 \sin{\left(0 \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
2cos⁡(x)=02 \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    
(--, 3)
 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=3π2x_{1} = \frac{3 \pi}{2}
Maxima of the function at points:
x1=π2x_{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
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
−2sin⁡(x)=0- 2 \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(-\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 x->+oo and x->-oo
lim⁡x→−∞(2sin⁡(x)+1)=⟨−1,3⟩\lim_{x \to -\infty}\left(2 \sin{\left(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⁡(x)+1)=⟨−1,3⟩\lim_{x \to \infty}\left(2 \sin{\left(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(x) + 1, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(2sin⁡(x)+1x)=0\lim_{x \to -\infty}\left(\frac{2 \sin{\left(x \right)} + 1}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(2sin⁡(x)+1x)=0\lim_{x \to \infty}\left(\frac{2 \sin{\left(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⁡(x)+1=1−2sin⁡(x)2 \sin{\left(x \right)} + 1 = 1 - 2 \sin{\left(x \right)}
- No
2sin⁡(x)+1=2sin⁡(x)−12 \sin{\left(x \right)} + 1 = 2 \sin{\left(x \right)} - 1
- No
so, the function
not is
neither even, nor odd
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