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Plotting a function graph/ 2sinx+1

Graphing y = 2sinx+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
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⁡(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=−15.1843644923507x_{1} = -15.1843644923507
x2=−8.90117918517108x_{2} = -8.90117918517108
x3=−38.2227106186758x_{3} = -38.2227106186758
x4=−195.302343298165x_{4} = -195.302343298165
x5=74.8746249105567x_{5} = 74.8746249105567
x6=437.20497762458x_{6} = 437.20497762458
x7=−84.2994028713261x_{7} = -84.2994028713261
x8=−69.6386371545737x_{8} = -69.6386371545737
x9=49.7418836818384x_{9} = 49.7418836818384
x10=93.7241808320955x_{10} = 93.7241808320955
x11=−34.0339204138894x_{11} = -34.0339204138894
x12=−25.6563400043166x_{12} = -25.6563400043166
x13=−82.2050077689329x_{13} = -82.2050077689329
x14=−96.8657734856853x_{14} = -96.8657734856853
x15=−40.317105721069x_{15} = -40.317105721069
x16=37.1755130674792x_{16} = 37.1755130674792
x17=81.1578102177363x_{17} = 81.1578102177363
x18=−57.0722665402146x_{18} = -57.0722665402146
x19=9.94837673636768x_{19} = 9.94837673636768
x20=−63.3554518473942x_{20} = -63.3554518473942
x21=22.5147473507269x_{21} = 22.5147473507269
x22=91.6297857297023x_{22} = 91.6297857297023
x23=41.3643032722656x_{23} = 41.3643032722656
x24=5.75958653158129x_{24} = 5.75958653158129
x25=−31.9395253114962x_{25} = -31.9395253114962
x26=−101.054563690472x_{26} = -101.054563690472
x27=30.8923277602996x_{27} = 30.8923277602996
x28=56.025068989018x_{28} = 56.025068989018
x29=97.9129710368819x_{29} = 97.9129710368819
x30=−151.320046147908x_{30} = -151.320046147908
x31=100.007366139275x_{31} = 100.007366139275
x32=−94.7713783832921x_{32} = -94.7713783832921
x33=−59.1666616426078x_{33} = -59.1666616426078
x34=12.0427718387609x_{34} = 12.0427718387609
x35=−21.4675497995303x_{35} = -21.4675497995303
x36=−6.80678408277789x_{36} = -6.80678408277789
x37=66.497044500984x_{37} = 66.497044500984
x38=192.160750644576x_{38} = 192.160750644576
x39=85.3466004225227x_{39} = 85.3466004225227
x40=−78.0162175641465x_{40} = -78.0162175641465
x41=−46.6002910282486x_{41} = -46.6002910282486
x42=79.0634151153431x_{42} = 79.0634151153431
x43=−50.789081233035x_{43} = -50.789081233035
x44=−44.5058959258554x_{44} = -44.5058959258554
x45=−27.7507351067098x_{45} = -27.7507351067098
x46=−71.733032256967x_{46} = -71.733032256967
x47=68.5914396033772x_{47} = 68.5914396033772
x48=66400.1787274983x_{48} = 66400.1787274983
x49=62.3082542961976x_{49} = 62.3082542961976
x50=53.9306738866248x_{50} = 53.9306738866248
x51=24.60914245312x_{51} = 24.60914245312
x52=3.66519142918809x_{52} = 3.66519142918809
x53=72.7802298081635x_{53} = 72.7802298081635
x54=−2.61799387799149x_{54} = -2.61799387799149
x55=−75.9218224617533x_{55} = -75.9218224617533
x56=87.4409955249159x_{56} = 87.4409955249159
x57=−88.4881930761125x_{57} = -88.4881930761125
x58=43.4586983746588x_{58} = 43.4586983746588
x59=28.7979326579064x_{59} = 28.7979326579064
x60=60.2138591938044x_{60} = 60.2138591938044
x61=−52.8834763354282x_{61} = -52.8834763354282
x62=47.6474885794452x_{62} = 47.6474885794452
x63=16.2315620435473x_{63} = 16.2315620435473
x64=−13.0899693899575x_{64} = -13.0899693899575
x65=35.081117965086x_{65} = 35.081117965086
x66=18.3259571459405x_{66} = 18.3259571459405
x67=−90.5825881785057x_{67} = -90.5825881785057
x68=−19.3731546971371x_{68} = -19.3731546971371
x69=−0.523598775598299x_{69} = -0.523598775598299
x70=−65.4498469497874x_{70} = -65.4498469497874
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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