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

Graphing y = sin(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|x + --|
          \    4 /
f(x)=sin⁡(x+π4)f{\left(x \right)} = \sin{\left(x + \frac{\pi}{4} \right)} 
f = sin(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⁡(x+π4)=0\sin{\left(x + \frac{\pi}{4} \right)} = 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=21.2057504117311x_{1} = 21.2057504117311
x2=−35.3429173528852x_{2} = -35.3429173528852
x3=−51.0508806208341x_{3} = -51.0508806208341
x4=−107.59954838545x_{4} = -107.59954838545
x5=43.1968989868597x_{5} = 43.1968989868597
x6=2.35619449019234x_{6} = 2.35619449019234
x7=−98.174770424681x_{7} = -98.174770424681
x8=391.913683535327x_{8} = 391.913683535327
x9=1745.94011723253x_{9} = 1745.94011723253
x10=11.7809724509617x_{10} = 11.7809724509617
x11=−38.484510006475x_{11} = -38.484510006475
x12=−79.3252145031423x_{12} = -79.3252145031423
x13=90.3207887907066x_{13} = 90.3207887907066
x14=30.6305283725005x_{14} = 30.6305283725005
x15=80.8960108299372x_{15} = 80.8960108299372
x16=46.3384916404494x_{16} = 46.3384916404494
x17=−88.7499924639117x_{17} = -88.7499924639117
x18=−7.06858347057703x_{18} = -7.06858347057703
x19=62.0464549083984x_{19} = 62.0464549083984
x20=36.9137136796801x_{20} = 36.9137136796801
x21=40.0553063332699x_{21} = 40.0553063332699
x22=52.621676947629x_{22} = 52.621676947629
x23=−66.7588438887831x_{23} = -66.7588438887831
x24=−63.6172512351933x_{24} = -63.6172512351933
x25=49.4800842940392x_{25} = 49.4800842940392
x26=65.1880475619882x_{26} = 65.1880475619882
x27=−16.4933614313464x_{27} = -16.4933614313464
x28=−95.0331777710912x_{28} = -95.0331777710912
x29=−10.2101761241668x_{29} = -10.2101761241668
x30=−32.2013246992954x_{30} = -32.2013246992954
x31=58.9048622548086x_{31} = 58.9048622548086
x32=−25.9181393921158x_{32} = -25.9181393921158
x33=−47.9092879672443x_{33} = -47.9092879672443
x34=−0.785398163397448x_{34} = -0.785398163397448
x35=96.6039740978861x_{35} = 96.6039740978861
x36=33.7721210260903x_{36} = 33.7721210260903
x37=27.4889357189107x_{37} = 27.4889357189107
x38=55.7632696012188x_{38} = 55.7632696012188
x39=87.1791961371168x_{39} = 87.1791961371168
x40=−73.0420291959627x_{40} = -73.0420291959627
x41=−41.6261026600648x_{41} = -41.6261026600648
x42=99.7455667514759x_{42} = 99.7455667514759
x43=−91.8915851175014x_{43} = -91.8915851175014
x44=84.037603483527x_{44} = 84.037603483527
x45=71.4712328691678x_{45} = 71.4712328691678
x46=77.7544181763474x_{46} = 77.7544181763474
x47=14.9225651045515x_{47} = 14.9225651045515
x48=24.3473430653209x_{48} = 24.3473430653209
x49=−82.4668071567321x_{49} = -82.4668071567321
x50=93.4623814442964x_{50} = 93.4623814442964
x51=−76.1836218495525x_{51} = -76.1836218495525
x52=68.329640215578x_{52} = 68.329640215578
x53=−60.4756585816035x_{53} = -60.4756585816035
x54=74.6128255227576x_{54} = 74.6128255227576
x55=−19.6349540849362x_{55} = -19.6349540849362
x56=−13.3517687777566x_{56} = -13.3517687777566
x57=−29.0597320457056x_{57} = -29.0597320457056
x58=5.49778714378214x_{58} = 5.49778714378214
x59=−69.9004365423729x_{59} = -69.9004365423729
x60=−22.776546738526x_{60} = -22.776546738526
x61=−57.3340659280137x_{61} = -57.3340659280137
x62=−44.7676953136546x_{62} = -44.7676953136546
x63=18.0641577581413x_{63} = 18.0641577581413
x64=8.63937979737193x_{64} = 8.63937979737193
x65=−3.92699081698724x_{65} = -3.92699081698724
x66=−54.1924732744239x_{66} = -54.1924732744239
x67=−85.6083998103219x_{67} = -85.6083998103219
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(x + pi/4).
sin⁡(π4)\sin{\left(\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
cos⁡(x+π4)=0\cos{\left(x + \frac{\pi}{4} \right)} = 0
Solve this equation
The roots of this equation
x1=π4x_{1} = \frac{\pi}{4}
x2=5π4x_{2} = \frac{5 \pi}{4}
The values of the extrema at the points:
 pi     /pi   pi\ 
(--, sin|-- + --|)
 4      \4    4 / 

 5*pi      /pi   pi\ 
(----, -sin|-- + --|)
  4        \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=5π4x_{1} = \frac{5 \pi}{4}
Maxima of the function at points:
x1=π4x_{1} = \frac{\pi}{4}
Decreasing at intervals
(−∞,π4]∪[5π4,∞)\left(-\infty, \frac{\pi}{4}\right] \cup \left[\frac{5 \pi}{4}, \infty\right)
Increasing at intervals
[π4,5π4]\left[\frac{\pi}{4}, \frac{5 \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
−sin⁡(x+π4)=0- \sin{\left(x + \frac{\pi}{4} \right)} = 0
Solve this equation
The roots of this equation
x1=−π4x_{1} = - \frac{\pi}{4}
x2=3π4x_{2} = \frac{3 \pi}{4}

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
(−∞,−π4]∪[3π4,∞)\left(-\infty, - \frac{\pi}{4}\right] \cup \left[\frac{3 \pi}{4}, \infty\right)
Convex at the intervals
[−π4,3π4]\left[- \frac{\pi}{4}, \frac{3 \pi}{4}\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⁡(x+π4)=⟨−1,1⟩\lim_{x \to -\infty} \sin{\left(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⁡(x+π4)=⟨−1,1⟩\lim_{x \to \infty} \sin{\left(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(x + pi/4), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡(x+π4)x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(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⁡(x+π4)x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(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⁡(x+π4)=−sin⁡(x−π4)\sin{\left(x + \frac{\pi}{4} \right)} = - \sin{\left(x - \frac{\pi}{4} \right)}
- No
sin⁡(x+π4)=sin⁡(x−π4)\sin{\left(x + \frac{\pi}{4} \right)} = \sin{\left(x - \frac{\pi}{4} \right)}
- No
so, the function
not is
neither even, nor odd
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