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Plotting a function graph/ sin(2*x+pi/4)

Graphing y = sin(2*x+pi/4)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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          /      pi\
f(x) = sin|2*x + --|
          \      4 /
f(x)=sin⁡(2x+π4)f{\left(x \right)} = \sin{\left(2 x + \frac{\pi}{4} \right)} 
f = sin(2*x + pi/4)
The graph of the function
02468-8-6-4-2-10102-2
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+π4)=0\sin{\left(2 x + \frac{\pi}{4} \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−π8x_{1} = - \frac{\pi}{8}
x2=3π8x_{2} = \frac{3 \pi}{8}
Numerical solution
x1=−8.24668071567321x_{1} = -8.24668071567321
x2=−33.3794219443916x_{2} = -33.3794219443916
x3=−82.0741080750334x_{3} = -82.0741080750334
x4=−45.9457925587507x_{4} = -45.9457925587507
x5=−99.3528676697772x_{5} = -99.3528676697772
x6=−31.8086256175967x_{6} = -31.8086256175967
x7=13.7444678594553x_{7} = 13.7444678594553
x8=−1.96349540849362x_{8} = -1.96349540849362
x9=10.6028752058656x_{9} = 10.6028752058656
x10=76.5763209312512x_{10} = 76.5763209312512
x11=12.1736715326604x_{11} = 12.1736715326604
x12=−3.53429173528852x_{12} = -3.53429173528852
x13=−36.5210145979813x_{13} = -36.5210145979813
x14=−96.2112750161874x_{14} = -96.2112750161874
x15=40.4480054149686x_{15} = 40.4480054149686
x16=−66.3661448070844x_{16} = -66.3661448070844
x17=−11.388273369263x_{17} = -11.388273369263
x18=70.2931356240716x_{18} = 70.2931356240716
x19=18.45685683984x_{19} = 18.45685683984
x20=−44.3749962319558x_{20} = -44.3749962319558
x21=92.2842841992002x_{21} = 92.2842841992002
x22=5.89048622548086x_{22} = 5.89048622548086
x23=98.5674695063798x_{23} = 98.5674695063798
x24=93.8550805259951x_{24} = 93.8550805259951
x25=54.5851723561227x_{25} = 54.5851723561227
x26=−0.392699081698724x_{26} = -0.392699081698724
x27=65.5807466436869x_{27} = 65.5807466436869
x28=49.872783375738x_{28} = 49.872783375738
x29=−22.3838476568273x_{29} = -22.3838476568273
x30=−52.2289778659303x_{30} = -52.2289778659303
x31=56.1559686829176x_{31} = 56.1559686829176
x32=62.4391539900971x_{32} = 62.4391539900971
x33=20.0276531666349x_{33} = 20.0276531666349
x34=87.5718952188155x_{34} = 87.5718952188155
x35=−97.7820713429823x_{35} = -97.7820713429823
x36=−25.5254403104171x_{36} = -25.5254403104171
x37=−61.6537558266997x_{37} = -61.6537558266997
x38=68.7223392972767x_{38} = 68.7223392972767
x39=26.3108384738145x_{39} = 26.3108384738145
x40=71.8639319508665x_{40} = 71.8639319508665
x41=−39.6626072515711x_{41} = -39.6626072515711
x42=35.7356164345839x_{42} = 35.7356164345839
x43=32.5940237809941x_{43} = 32.5940237809941
x44=−47.5165888855456x_{44} = -47.5165888855456
x45=27.8816348006094x_{45} = 27.8816348006094
x46=−16.1006623496477x_{46} = -16.1006623496477
x47=−58.5121631731099x_{47} = -58.5121631731099
x48=−38.0918109247762x_{48} = -38.0918109247762
x49=−53.7997741927252x_{49} = -53.7997741927252
x50=−17.6714586764426x_{50} = -17.6714586764426
x51=64.009950316892x_{51} = 64.009950316892
x52=42.0188017417635x_{52} = 42.0188017417635
x53=−74.2201264410589x_{53} = -74.2201264410589
x54=−5389.79489631499x_{54} = -5389.79489631499
x55=−69.5077374606742x_{55} = -69.5077374606742
x56=−184.175869316702x_{56} = -184.175869316702
x57=−89.9280897090078x_{57} = -89.9280897090078
x58=−67.9369411338793x_{58} = -67.9369411338793
x59=4.31968989868597x_{59} = 4.31968989868597
x60=−77.3617190946487x_{60} = -77.3617190946487
x61=86.0010988920206x_{61} = 86.0010988920206
x62=48.3019870489431x_{62} = 48.3019870489431
x63=21.5984494934298x_{63} = 21.5984494934298
x64=90.7134878724053x_{64} = 90.7134878724053
x65=9.03207887907065x_{65} = 9.03207887907065
x66=−30.2378292908018x_{66} = -30.2378292908018
x67=−80.5033117482384x_{67} = -80.5033117482384
x68=−88.3572933822129x_{68} = -88.3572933822129
x69=−60.0829594999048x_{69} = -60.0829594999048
x70=84.4303025652257x_{70} = 84.4303025652257
x71=−91.4988860358027x_{71} = -91.4988860358027
x72=24.7400421470196x_{72} = 24.7400421470196
x73=2.74889357189107x_{73} = 2.74889357189107
x74=−75.7909227678538x_{74} = -75.7909227678538
x75=79.717913584841x_{75} = 79.717913584841
x76=−9.8174770424681x_{76} = -9.8174770424681
x77=43.5895980685584x_{77} = 43.5895980685584
x78=34.164820107789x_{78} = 34.164820107789
x79=−41.233403578366x_{79} = -41.233403578366
x80=100.138265833175x_{80} = 100.138265833175
x81=−19.2422550032375x_{81} = -19.2422550032375
x82=78.1471172580461x_{82} = 78.1471172580461
x83=−23.9546439836222x_{83} = -23.9546439836222
x84=46.7311907221482x_{84} = 46.7311907221482
x85=−55.3705705195201x_{85} = -55.3705705195201
x86=57.7267650097125x_{86} = 57.7267650097125
x87=−83.6449044018282x_{87} = -83.6449044018282
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 + pi/4).
sin⁡(0⋅2+π4)\sin{\left(0 \cdot 2 + \frac{\pi}{4} \right)}
The result:
f(0)=22f{\left(0 \right)} = \frac{\sqrt{2}}{2}
The point:
(0, sqrt(2)/2)
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+π4)=02 \cos{\left(2 x + \frac{\pi}{4} \right)} = 0
Solve this equation
The roots of this equation
x1=π8x_{1} = \frac{\pi}{8}
x2=5π8x_{2} = \frac{5 \pi}{8}
The values of the extrema at the points:
 pi     /pi   pi\ 
(--, sin|-- + --|)
 8      \4    4 / 

 5*pi      /pi   pi\ 
(----, -sin|-- + --|)
  8        \4    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=5π8x_{1} = \frac{5 \pi}{8}
Maxima of the function at points:
x1=π8x_{1} = \frac{\pi}{8}
Decreasing at intervals
(−∞,π8]∪[5π8,∞)\left(-\infty, \frac{\pi}{8}\right] \cup \left[\frac{5 \pi}{8}, \infty\right)
Increasing at intervals
[π8,5π8]\left[\frac{\pi}{8}, \frac{5 \pi}{8}\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+π4)=0- 4 \sin{\left(2 x + \frac{\pi}{4} \right)} = 0
Solve this equation
The roots of this equation
x1=−π8x_{1} = - \frac{\pi}{8}
x2=3π8x_{2} = \frac{3 \pi}{8}

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
(−∞,−π8]∪[3π8,∞)\left(-\infty, - \frac{\pi}{8}\right] \cup \left[\frac{3 \pi}{8}, \infty\right)
Convex at the intervals
[−π8,3π8]\left[- \frac{\pi}{8}, \frac{3 \pi}{8}\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+π4)=⟨−1,1⟩\lim_{x \to -\infty} \sin{\left(2 x + \frac{\pi}{4} \right)} = \left\langle -1, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
lim⁡x→∞sin⁡(2x+π4)=⟨−1,1⟩\lim_{x \to \infty} \sin{\left(2 x + \frac{\pi}{4} \right)} = \left\langle -1, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sin(2*x + pi/4), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡(2x+π4)x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(2 x + \frac{\pi}{4} \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(sin⁡(2x+π4)x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(2 x + \frac{\pi}{4} \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:
sin⁡(2x+π4)=−sin⁡(2x−π4)\sin{\left(2 x + \frac{\pi}{4} \right)} = - \sin{\left(2 x - \frac{\pi}{4} \right)}
- No
sin⁡(2x+π4)=sin⁡(2x−π4)\sin{\left(2 x + \frac{\pi}{4} \right)} = \sin{\left(2 x - \frac{\pi}{4} \right)}
- No
so, the function
not is
neither even, nor odd
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