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Plotting a function graph/ sin(7x)

Graphing y = sin(7x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = sin(7*x)
f(x)=sin⁡(7x)f{\left(x \right)} = \sin{\left(7 x \right)} 
f = sin(7*x)
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⁡(7x)=0\sin{\left(7 x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
x2=π7x_{2} = \frac{\pi}{7}
Numerical solution
x1=−4.03919055461545x_{1} = -4.03919055461545
x2=17.9519580205131x_{2} = 17.9519580205131
x3=−89.7597901025655x_{3} = -89.7597901025655
x4=24.2351433276927x_{4} = 24.2351433276927
x5=48.0214877048726x_{5} = 48.0214877048726
x6=−19.7471538225644x_{6} = -19.7471538225644
x7=−43.9822971502571x_{7} = -43.9822971502571
x8=28.2743338823081x_{8} = 28.2743338823081
x9=−5.83438635666676x_{9} = -5.83438635666676
x10=−17.9519580205131x_{10} = -17.9519580205131
x11=4.03919055461545x_{11} = 4.03919055461545
x12=92.0037848551297x_{12} = 92.0037848551297
x13=−23.7863443771799x_{13} = -23.7863443771799
x14=−15.707963267949x_{14} = -15.707963267949
x15=−37.6991118430775x_{15} = -37.6991118430775
x16=74.0518268346165x_{16} = 74.0518268346165
x17=−41.738302397693x_{17} = -41.738302397693
x18=52.060678259488x_{18} = 52.060678259488
x19=56.0998688141035x_{19} = 56.0998688141035
x20=−72.7054299830781x_{20} = -72.7054299830781
x21=−92.0037848551297x_{21} = -92.0037848551297
x22=82.1302079438475x_{22} = 82.1302079438475
x23=−96.0429754097451x_{23} = -96.0429754097451
x24=38.1479107935903x_{24} = 38.1479107935903
x25=63.2806520223087x_{25} = 63.2806520223087
x26=−8.97597901025655x_{26} = -8.97597901025655
x27=−39.9431065956417x_{27} = -39.9431065956417
x28=65.9734457253857x_{28} = 65.9734457253857
x29=−70.0126362800011x_{29} = -70.0126362800011
x30=−63.7294509728215x_{30} = -63.7294509728215
x31=−53.8558740615393x_{31} = -53.8558740615393
x32=16.1567622184618x_{32} = 16.1567622184618
x33=21.9911485751286x_{33} = 21.9911485751286
x34=−71.8078320820524x_{34} = -71.8078320820524
x35=86.1693984984629x_{35} = 86.1693984984629
x36=−32.3135244369236x_{36} = -32.3135244369236
x37=−65.9734457253857x_{37} = -65.9734457253857
x38=8.0783811092309x_{38} = 8.0783811092309
x39=70.0126362800011x_{39} = 70.0126362800011
x40=−35.9039160410262x_{40} = -35.9039160410262
x41=−52.060678259488x_{41} = -52.060678259488
x42=39.9431065956417x_{42} = 39.9431065956417
x43=−93.798980657181x_{43} = -93.798980657181
x44=32.3135244369236x_{44} = 32.3135244369236
x45=83.4766047953859x_{45} = 83.4766047953859
x46=30.0695296843594x_{46} = 30.0695296843594
x47=98.2869701623092x_{47} = 98.2869701623092
x48=26.030339129744x_{48} = 26.030339129744
x49=46.2262919028212x_{49} = 46.2262919028212
x50=−21.9911485751286x_{50} = -21.9911485751286
x51=−101.877361766412x_{51} = -101.877361766412
x52=20.1959527730772x_{52} = 20.1959527730772
x53=−79.8862131912833x_{53} = -79.8862131912833
x54=96.0429754097451x_{54} = 96.0429754097451
x55=−75.8470226366679x_{55} = -75.8470226366679
x56=−83.9254037458988x_{56} = -83.9254037458988
x57=−85.7205995479501x_{57} = -85.7205995479501
x58=−31.8647254864108x_{58} = -31.8647254864108
x59=−59.6902604182061x_{59} = -59.6902604182061
x60=68.2174404779498x_{60} = 68.2174404779498
x61=−87.9645943005142x_{61} = -87.9645943005142
x62=64.1782499233343x_{62} = 64.1782499233343
x63=42.1871013482058x_{63} = 42.1871013482058
x64=−61.9342551707702x_{64} = -61.9342551707702
x65=−45.7774929523084x_{65} = -45.7774929523084
x66=−74.0518268346165x_{66} = -74.0518268346165
x67=34.1087202389749x_{67} = 34.1087202389749
x68=78.091017389232x_{68} = 78.091017389232
x69=6.28318530717959x_{69} = 6.28318530717959
x70=−1.79519580205131x_{70} = -1.79519580205131
x71=12.1175716638463x_{71} = 12.1175716638463
x72=0x_{72} = 0
x73=61.9342551707702x_{73} = 61.9342551707702
x74=2.24399475256414x_{74} = 2.24399475256414
x75=87.9645943005142x_{75} = 87.9645943005142
x76=43.9822971502571x_{76} = 43.9822971502571
x77=54.3046730120521x_{77} = 54.3046730120521
x78=−26.030339129744x_{78} = -26.030339129744
x79=−57.8950646161548x_{79} = -57.8950646161548
x80=90.2085890530783x_{80} = 90.2085890530783
x81=100.082165964361x_{81} = 100.082165964361
x82=−9.87357691128221x_{82} = -9.87357691128221
x83=−13.9127674658977x_{83} = -13.9127674658977
x84=72.2566310325652x_{84} = 72.2566310325652
x85=−97.8381712117964x_{85} = -97.8381712117964
x86=94.2477796076938x_{86} = 94.2477796076938
x87=−67.768641527437x_{87} = -67.768641527437
x88=−48.0214877048726x_{88} = -48.0214877048726
x89=−30.0695296843594x_{89} = -30.0695296843594
x90=−27.8255349317953x_{90} = -27.8255349317953
x91=−81.6814089933346x_{91} = -81.6814089933346
x92=60.1390593687189x_{92} = 60.1390593687189
x93=76.2958215871807x_{93} = 76.2958215871807
x94=−49.8166835069239x_{94} = -49.8166835069239
x95=50.2654824574367x_{95} = 50.2654824574367
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(7*x).
sin⁡(0⋅7)\sin{\left(0 \cdot 7 \right)}
The result:
f(0)=0f{\left(0 \right)} = 0
The point:
(0, 0)
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
7cos⁡(7x)=07 \cos{\left(7 x \right)} = 0
Solve this equation
The roots of this equation
x1=π14x_{1} = \frac{\pi}{14}
x2=3π14x_{2} = \frac{3 \pi}{14}
The values of the extrema at the points:
 pi    
(--, 1)
 14    

 3*pi     
(----, -1)
  14


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=3π14x_{1} = \frac{3 \pi}{14}
Maxima of the function at points:
x1=π14x_{1} = \frac{\pi}{14}
Decreasing at intervals
(−∞,π14]∪[3π14,∞)\left(-\infty, \frac{\pi}{14}\right] \cup \left[\frac{3 \pi}{14}, \infty\right)
Increasing at intervals
[π14,3π14]\left[\frac{\pi}{14}, \frac{3 \pi}{14}\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
−49sin⁡(7x)=0- 49 \sin{\left(7 x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=π7x_{2} = \frac{\pi}{7}

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]∪[π7,∞)\left(-\infty, 0\right] \cup \left[\frac{\pi}{7}, \infty\right)
Convex at the intervals
[0,π7]\left[0, \frac{\pi}{7}\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⁡(7x)=⟨−1,1⟩\lim_{x \to -\infty} \sin{\left(7 x \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⁡(7x)=⟨−1,1⟩\lim_{x \to \infty} \sin{\left(7 x \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(7*x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡(7x)x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(7 x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(sin⁡(7x)x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(7 x \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⁡(7x)=−sin⁡(7x)\sin{\left(7 x \right)} = - \sin{\left(7 x \right)}
- No
sin⁡(7x)=sin⁡(7x)\sin{\left(7 x \right)} = \sin{\left(7 x \right)}
- Yes
so, the function
is
odd
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