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Plotting a function graph/ sin(5-2*x)

Graphing y = sin(5-2*x)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = sin(5 - 2*x)
f(x)=sin⁡(5−2x)f{\left(x \right)} = \sin{\left(5 - 2 x \right)} 
f = sin(5 - 2*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⁡(5−2x)=0\sin{\left(5 - 2 x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=52x_{1} = \frac{5}{2}
x2=π2+52x_{2} = \frac{\pi}{2} + \frac{5}{2}
Numerical solution
x1=92.0353906273091x_{1} = 92.0353906273091
x2=55.9070751110265x_{2} = 55.9070751110265
x3=4.0707963267949x_{3} = 4.0707963267949
x4=−3.78318530717959x_{4} = -3.78318530717959
x5=−24.2035375555132x_{5} = -24.2035375555132
x6=11.9247779607694x_{6} = 11.9247779607694
x7=27.6327412287183x_{7} = 27.6327412287183
x8=−91.7477796076938x_{8} = -91.7477796076938
x9=−28.9159265358979x_{9} = -28.9159265358979
x10=38.6283155162826x_{10} = 38.6283155162826
x11=−71.3274273593601x_{11} = -71.3274273593601
x12=40.1991118430775x_{12} = 40.1991118430775
x13=−99.6017612416683x_{13} = -99.6017612416683
x14=−16.3495559215388x_{14} = -16.3495559215388
x15=26.0619449019235x_{15} = 26.0619449019235
x16=5.64159265358979x_{16} = 5.64159265358979
x17=−39.9115008234622x_{17} = -39.9115008234622
x18=−69.7566310325652x_{18} = -69.7566310325652
x19=24.4911485751286x_{19} = 24.4911485751286
x20=77.898223686155x_{20} = 77.898223686155
x21=−98.0309649148734x_{21} = -98.0309649148734
x22=−38.3407044966673x_{22} = -38.3407044966673
x23=82.6106126665397x_{23} = 82.6106126665397
x24=10.3539816339745x_{24} = 10.3539816339745
x25=99.8893722612836x_{25} = 99.8893722612836
x26=−17.9203522483337x_{26} = -17.9203522483337
x27=49.6238898038469x_{27} = 49.6238898038469
x28=32.345130209103x_{28} = 32.345130209103
x29=13.4955742875643x_{29} = 13.4955742875643
x30=−11.6371669411541x_{30} = -11.6371669411541
x31=19.7787595947439x_{31} = 19.7787595947439
x32=2.5x_{32} = 2.5
x33=−46.1946861306418x_{33} = -46.1946861306418
x34=−54.0486677646163x_{34} = -54.0486677646163
x35=85.7522053201295x_{35} = 85.7522053201295
x36=−33.6283155162826x_{36} = -33.6283155162826
x37=−85.4645943005142x_{37} = -85.4645943005142
x38=62.1902604182061x_{38} = 62.1902604182061
x39=−63.4734457253857x_{39} = -63.4734457253857
x40=−5.35398163397448x_{40} = -5.35398163397448
x41=−90.1769832808989x_{41} = -90.1769832808989
x42=96.7477796076938x_{42} = 96.7477796076938
x43=−47.7654824574367x_{43} = -47.7654824574367
x44=−83.8937979737193x_{44} = -83.8937979737193
x45=63.761056745001x_{45} = 63.761056745001
x46=−49.3362787842316x_{46} = -49.3362787842316
x47=−60.3318530717959x_{47} = -60.3318530717959
x48=−76.0398163397448x_{48} = -76.0398163397448
x49=70.0442420521806x_{49} = 70.0442420521806
x50=−19.4911485751286x_{50} = -19.4911485751286
x51=−41.4822971502571x_{51} = -41.4822971502571
x52=−77.6106126665397x_{52} = -77.6106126665397
x53=57.4778714378214x_{53} = 57.4778714378214
x54=−52.4778714378214x_{54} = -52.4778714378214
x55=90.4645943005142x_{55} = 90.4645943005142
x56=−68.1858347057703x_{56} = -68.1858347057703
x57=−30.4867228626928x_{57} = -30.4867228626928
x58=−27.345130209103x_{58} = -27.345130209103
x59=104.601761241668x_{59} = 104.601761241668
x60=−55.6194640914112x_{60} = -55.6194640914112
x61=54.3362787842316x_{61} = 54.3362787842316
x62=41.7699081698724x_{62} = 41.7699081698724
x63=18.207963267949x_{63} = 18.207963267949
x64=−10.0663706143592x_{64} = -10.0663706143592
x65=−32.0575191894877x_{65} = -32.0575191894877
x66=16.6371669411541x_{66} = 16.6371669411541
x67=48.053093477052x_{67} = 48.053093477052
x68=33.9159265358979x_{68} = 33.9159265358979
x69=−2.21238898038469x_{69} = -2.21238898038469
x70=−61.9026493985908x_{70} = -61.9026493985908
x71=−82.3230016469244x_{71} = -82.3230016469244
x72=35.4867228626928x_{72} = 35.4867228626928
x73=74.7566310325652x_{73} = 74.7566310325652
x74=52.7654824574367x_{74} = 52.7654824574367
x75=−96.4601685880785x_{75} = -96.4601685880785
x76=−25.7743338823081x_{76} = -25.7743338823081
x77=−8.49557428756428x_{77} = -8.49557428756428
x78=76.3274273593601x_{78} = 76.3274273593601
x79=−74.4690200129499x_{79} = -74.4690200129499
x80=60.6194640914112x_{80} = 60.6194640914112
x81=98.3185759344887x_{81} = 98.3185759344887
x82=−93.3185759344887x_{82} = -93.3185759344887
x83=71.6150383789755x_{83} = 71.6150383789755
x84=84.1814089933346x_{84} = 84.1814089933346
x85=93.606186954104x_{85} = 93.606186954104
x86=46.4822971502571x_{86} = 46.4822971502571
x87=68.4734457253857x_{87} = 68.4734457253857
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(5 - 2*x).
sin⁡(5−0)\sin{\left(5 - 0 \right)}
The result:
f(0)=sin⁡(5)f{\left(0 \right)} = \sin{\left(5 \right)}
The point:
(0, sin(5))
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−5)=0- 2 \cos{\left(2 x - 5 \right)} = 0
Solve this equation
The roots of this equation
x1=π4+52x_{1} = \frac{\pi}{4} + \frac{5}{2}
x2=3π4+52x_{2} = \frac{3 \pi}{4} + \frac{5}{2}
The values of the extrema at the points:
 5   pi     
(- + --, -1)
 2   4      

 5   3*pi    
(- + ----, 1)
 2    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=π4+52x_{1} = \frac{\pi}{4} + \frac{5}{2}
Maxima of the function at points:
x1=3π4+52x_{1} = \frac{3 \pi}{4} + \frac{5}{2}
Decreasing at intervals
[π4+52,3π4+52]\left[\frac{\pi}{4} + \frac{5}{2}, \frac{3 \pi}{4} + \frac{5}{2}\right]
Increasing at intervals
(−∞,π4+52]∪[3π4+52,∞)\left(-\infty, \frac{\pi}{4} + \frac{5}{2}\right] \cup \left[\frac{3 \pi}{4} + \frac{5}{2}, \infty\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−5)=04 \sin{\left(2 x - 5 \right)} = 0
Solve this equation
The roots of this equation
x1=52x_{1} = \frac{5}{2}
x2=π2+52x_{2} = \frac{\pi}{2} + \frac{5}{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
[52,π2+52]\left[\frac{5}{2}, \frac{\pi}{2} + \frac{5}{2}\right]
Convex at the intervals
(−∞,52]∪[π2+52,∞)\left(-\infty, \frac{5}{2}\right] \cup \left[\frac{\pi}{2} + \frac{5}{2}, \infty\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⁡(5−2x)=⟨−1,1⟩\lim_{x \to -\infty} \sin{\left(5 - 2 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⁡(5−2x)=⟨−1,1⟩\lim_{x \to \infty} \sin{\left(5 - 2 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(5 - 2*x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡(5−2x)x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(5 - 2 x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(sin⁡(5−2x)x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(5 - 2 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⁡(5−2x)=sin⁡(2x+5)\sin{\left(5 - 2 x \right)} = \sin{\left(2 x + 5 \right)}
- No
sin⁡(5−2x)=−sin⁡(2x+5)\sin{\left(5 - 2 x \right)} = - \sin{\left(2 x + 5 \right)}
- No
so, the function
not is
neither even, nor odd
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