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

Graphing y = sin(2x)+1

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = sin(2*x) + 1
f(x)=sin⁡(2x)+1f{\left(x \right)} = \sin{\left(2 x \right)} + 1 
f = sin(2*x) + 1
The graph of the function
010203040506070809004
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:
sin⁡(2x)+1=0\sin{\left(2 x \right)} + 1 = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−π4x_{1} = - \frac{\pi}{4}
x2=3π4x_{2} = \frac{3 \pi}{4}
Numerical solution
x1=40.0553062099627x_{1} = 40.0553062099627
x2=99.7455669089878x_{2} = 99.7455669089878
x3=14.9225649892472x_{3} = 14.9225649892472
x4=71.471233111112x_{4} = 71.471233111112
x5=87.1791963746667x_{5} = 87.1791963746667
x6=−38.4845103583538x_{6} = -38.4845103583538
x7=−66.7588439932285x_{7} = -66.7588439932285
x8=36.9137134309271x_{8} = 36.9137134309271
x9=33.7721211721652x_{9} = 33.7721211721652
x10=90.3207887447523x_{10} = 90.3207887447523
x11=14.9225648553046x_{11} = 14.9225648553046
x12=77.7544183301131x_{12} = 77.7544183301131
x13=−95.0331780209134x_{13} = -95.0331780209134
x14=−63.6172513149926x_{14} = -63.6172513149926
x15=−16.4933611966261x_{15} = -16.4933611966261
x16=−73.0420291965934x_{16} = -73.0420291965934
x17=−60.4756583509459x_{17} = -60.4756583509459
x18=−44.7676954314418x_{18} = -44.7676954314418
x19=−79.3252147038408x_{19} = -79.3252147038408
x20=93.462381687851x_{20} = 93.462381687851
x21=21.2057506565108x_{21} = 21.2057506565108
x22=−76.1836225361187x_{22} = -76.1836225361187
x23=−35.3429178934093x_{23} = -35.3429178934093
x24=96.6039739212089x_{24} = 96.6039739212089
x25=−19.6349541554673x_{25} = -19.6349541554673
x26=11.7809725930891x_{26} = 11.7809725930891
x27=−73.0420294459387x_{27} = -73.0420294459387
x28=−7.0685837201977x_{28} = -7.0685837201977
x29=43.1968992294597x_{29} = 43.1968992294597
x30=74.6128253428807x_{30} = 74.6128253428807
x31=27.4889359573966x_{31} = 27.4889359573966
x32=−10.2101759860574x_{32} = -10.2101759860574
x33=−29.0597320982225x_{33} = -29.0597320982225
x34=−44.7676950635734x_{34} = -44.7676950635734
x35=−38.4845097737495x_{35} = -38.4845097737495
x36=2.35619442440536x_{36} = 2.35619442440536
x37=49.4800845342934x_{37} = 49.4800845342934
x38=−51.0508808708243x_{38} = -51.0508808708243
x39=46.3384915843947x_{39} = 46.3384915843947
x40=52.6216767646073x_{40} = 52.6216767646073
x41=68.3296401645238x_{41} = 68.3296401645238
x42=−35.3429175479242x_{42} = -35.3429175479242
x43=−0.785397916651749x_{43} = -0.785397916651749
x44=−76.1836217239439x_{44} = -76.1836217239439
x45=−98.1747703033985x_{45} = -98.1747703033985
x46=55.7632697511726x_{46} = 55.7632697511726
x47=−57.3340661259123x_{47} = -57.3340661259123
x48=−51.0508806461775x_{48} = -51.0508806461775
x49=30.6305281863893x_{49} = 30.6305281863893
x50=65.1880473923958x_{50} = 65.1880473923958
x51=24.3473430043575x_{51} = 24.3473430043575
x52=18.0641575837788x_{52} = 18.0641575837788
x53=−0.785398304611266x_{53} = -0.785398304611266
x54=80.8960105951638x_{54} = 80.8960105951638
x55=−7.06858355298869x_{55} = -7.06858355298869
x56=84.0376034199549x_{56} = 84.0376034199549
x57=−101.316363786262x_{57} = -101.316363786262
x58=5.49778738042301x_{58} = 5.49778738042301
x59=−66.758843637313x_{59} = -66.758843637313
x60=−91.8915845717239x_{60} = -91.8915845717239
x61=36.9137135281996x_{61} = 36.9137135281996
x62=87.1791959595656x_{62} = 87.1791959595656
x63=80.896010582516x_{63} = 80.896010582516
x64=58.9048620066655x_{64} = 58.9048620066655
x65=96.6039739773235x_{65} = 96.6039739773235
x66=55.7632697006301x_{66} = 55.7632697006301
x67=58.9048620635881x_{67} = 58.9048620635881
x68=−3.92699107594367x_{68} = -3.92699107594367
x69=−82.4668069282144x_{69} = -82.4668069282144
x70=21.2057502602382x_{70} = 21.2057502602382
x71=62.0464547256144x_{71} = 62.0464547256144
x72=−88.7499922112266x_{72} = -88.7499922112266
x73=−54.1924731445711x_{73} = -54.1924731445711
x74=−32.2013245652766x_{74} = -32.2013245652766
x75=−22.7765464900166x_{75} = -22.7765464900166
x76=−13.3517689698773x_{76} = -13.3517689698773
x77=−85.608399894653x_{77} = -85.608399894653
x78=8.63937960822746x_{78} = 8.63937960822746
x79=65.1880478021829x_{79} = 65.1880478021829
x80=62.0464548195549x_{80} = 62.0464548195549
x81=−29.0597322955756x_{81} = -29.0597322955756
x82=−96953.4762811305x_{82} = -96953.4762811305
x83=−60.4756585174615x_{83} = -60.4756585174615
x84=−41.6261027352652x_{84} = -41.6261027352652
x85=−22.7765468685351x_{85} = -22.7765468685351
x86=−88.7499925537801x_{86} = -88.7499925537801
x87=−95.0331777492301x_{87} = -95.0331777492301
x88=43.1968988259316x_{88} = 43.1968988259316
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(2*x) + 1.
sin⁡(2⋅0)+1\sin{\left(2 \cdot 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⁡(2x)=02 \cos{\left(2 x \right)} = 0
Solve this equation
The roots of this equation
x1=π4x_{1} = \frac{\pi}{4}
x2=3π4x_{2} = \frac{3 \pi}{4}
The values of the extrema at the points:
 pi    
(--, 2)
 4     

 3*pi    
(----, 0)
  4


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=3π4x_{1} = \frac{3 \pi}{4}
Maxima of the function at points:
x1=π4x_{1} = \frac{\pi}{4}
Decreasing at intervals
(−∞,π4]∪[3π4,∞)\left(-\infty, \frac{\pi}{4}\right] \cup \left[\frac{3 \pi}{4}, \infty\right)
Increasing at intervals
[π4,3π4]\left[\frac{\pi}{4}, \frac{3 \pi}{4}\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
−4sin⁡(2x)=0- 4 \sin{\left(2 x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=π2x_{2} = \frac{\pi}{2}

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]∪[π2,∞)\left(-\infty, 0\right] \cup \left[\frac{\pi}{2}, \infty\right)
Convex at the intervals
[0,π2]\left[0, \frac{\pi}{2}\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(sin⁡(2x)+1)=⟨0,2⟩\lim_{x \to -\infty}\left(\sin{\left(2 x \right)} + 1\right) = \left\langle 0, 2\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨0,2⟩y = \left\langle 0, 2\right\rangle
lim⁡x→∞(sin⁡(2x)+1)=⟨0,2⟩\lim_{x \to \infty}\left(\sin{\left(2 x \right)} + 1\right) = \left\langle 0, 2\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨0,2⟩y = \left\langle 0, 2\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sin(2*x) + 1, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡(2x)+1x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(2 x \right)} + 1}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(sin⁡(2x)+1x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(2 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:
sin⁡(2x)+1=1−sin⁡(2x)\sin{\left(2 x \right)} + 1 = 1 - \sin{\left(2 x \right)}
- No
sin⁡(2x)+1=sin⁡(2x)−1\sin{\left(2 x \right)} + 1 = \sin{\left(2 x \right)} - 1
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = sin(2x)+1
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