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Plotting a function graph/ -2,5sinx+0,5

Graphing y = -2,5sinx+0,5

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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         5*sin(x)   1
f(x) = - -------- + -
            2       2
f(x)=12−5sin⁡(x)2f{\left(x \right)} = \frac{1}{2} - \frac{5 \sin{\left(x \right)}}{2} 
f = 1/2 - 5*sin(x)/2
The graph of the function
02468-8-6-4-2-10105-5
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:
12−5sin⁡(x)2=0\frac{1}{2} - \frac{5 \sin{\left(x \right)}}{2} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=π−asin⁡(15)x_{1} = \pi - \operatorname{asin}{\left(\frac{1}{5} \right)}
x2=asin⁡(15)x_{2} = \operatorname{asin}{\left(\frac{1}{5} \right)}
Numerical solution
x1=31.6172844566883x_{1} = 31.6172844566883
x2=−56.3473098438259x_{2} = -56.3473098438259
x3=−43.7809392294668x_{3} = -43.7809392294668
x4=21.7897906543382x_{4} = 21.7897906543382
x5=90.9048290333137x_{5} = 90.9048290333137
x6=40.639346575877x_{6} = 40.639346575877
x7=−85.0243595677148x_{7} = -85.0243595677148
x8=−41.0420624174576x_{8} = -41.0420624174576
x9=−22.1925064959189x_{9} = -22.1925064959189
x10=−53.6084330318168x_{10} = -53.6084330318168
x11=12.7677285351495x_{11} = 12.7677285351495
x12=6.48454322796992x_{12} = 6.48454322796992
x13=97.1880143404933x_{13} = 97.1880143404933
x14=−72.4579889533556x_{14} = -72.4579889533556
x15=69.3163962997658x_{15} = 69.3163962997658
x16=−1683.69230440334x_{16} = -1683.69230440334
x17=−12.3650126935688x_{17} = -12.3650126935688
x18=−68.9136804581851x_{18} = -68.9136804581851
x19=94.4491375284841x_{19} = 94.4491375284841
x20=46.9225318830566x_{20} = 46.9225318830566
x21=−15.9093211887393x_{21} = -15.9093211887393
x22=−3.34295057438012x_{22} = -3.34295057438012
x23=65.7720878045953x_{23} = 65.7720878045953
x24=72.0552731117749x_{24} = 72.0552731117749
x25=2.94023473279946x_{25} = 2.94023473279946
x26=0.201357920790331x_{26} = 0.201357920790331
x27=−188.294201294597x_{27} = -188.294201294597
x28=−62.6304951510055x_{28} = -62.6304951510055
x29=−37.4977539222872x_{29} = -37.4977539222872
x30=81.882766914125x_{30} = 81.882766914125
x31=88.1659522213045x_{31} = 88.1659522213045
x32=−91.3075448748943x_{32} = -91.3075448748943
x33=15.5066053471586x_{33} = 15.5066053471586
x34=19.0509138423291x_{34} = 19.0509138423291
x35=−31.2145686151076x_{35} = -31.2145686151076
x36=63.0332109925862x_{36} = 63.0332109925862
x37=−59.8916183389964x_{37} = -59.8916183389964
x38=−24.931383307928x_{38} = -24.931383307928
x39=50.466840378227x_{39} = 50.466840378227
x40=−78.7411742605352x_{40} = -78.7411742605352
x41=75.5995816069454x_{41} = 75.5995816069454
x42=−66.174803646176x_{42} = -66.174803646176
x43=9.22342003997905x_{43} = 9.22342003997905
x44=100.732322835664x_{44} = 100.732322835664
x45=56.7500256854066x_{45} = 56.7500256854066
x46=−28.4756918030985x_{46} = -28.4756918030985
x47=78.3384584189545x_{47} = 78.3384584189545
x48=37.9004697638678x_{48} = 37.9004697638678
x49=−94.0464216869035x_{49} = -94.0464216869035
x50=84.6216437261341x_{50} = 84.6216437261341
x51=−9.62613588155971x_{51} = -9.62613588155971
x52=−97.5907301820739x_{52} = -97.5907301820739
x53=59.4889024974157x_{53} = 59.4889024974157
x54=−100.329606994083x_{54} = -100.329606994083
x55=−50.0641245366464x_{55} = -50.0641245366464
x56=−18.6481980007484x_{56} = -18.6481980007484
x57=−34.7588771102781x_{57} = -34.7588771102781
x58=−81.4800510725443x_{58} = -81.4800510725443
x59=53.2057171902362x_{59} = 53.2057171902362
x60=−87.7632363797239x_{60} = -87.7632363797239
x61=34.3561612686974x_{61} = 34.3561612686974
x62=28.0729759615178x_{62} = 28.0729759615178
x63=−6.08182738638926x_{63} = -6.08182738638926
x64=25.3340991495087x_{64} = 25.3340991495087
x65=44.1836550710474x_{65} = 44.1836550710474
x66=−47.3252477246372x_{66} = -47.3252477246372
x67=−75.1968657653647x_{67} = -75.1968657653647
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to -5*sin(x)/2 + 1/2.
12−5sin⁡(0)2\frac{1}{2} - \frac{5 \sin{\left(0 \right)}}{2}
The result:
f(0)=12f{\left(0 \right)} = \frac{1}{2}
The point:
(0, 1/2)
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
−5cos⁡(x)2=0- \frac{5 \cos{\left(x \right)}}{2} = 0
Solve this equation
The roots of this equation
x1=π2x_{1} = \frac{\pi}{2}
x2=3π2x_{2} = \frac{3 \pi}{2}
The values of the extrema at the points:
 pi     
(--, -2)
 2      

 3*pi    
(----, 3)
  2


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=π2x_{1} = \frac{\pi}{2}
Maxima of the function at points:
x1=3π2x_{1} = \frac{3 \pi}{2}
Decreasing at intervals
[π2,3π2]\left[\frac{\pi}{2}, \frac{3 \pi}{2}\right]
Increasing at intervals
(−∞,π2]∪[3π2,∞)\left(-\infty, \frac{\pi}{2}\right] \cup \left[\frac{3 \pi}{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
5sin⁡(x)2=0\frac{5 \sin{\left(x \right)}}{2} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=πx_{2} = \pi

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,π]\left[0, \pi\right]
Convex at the intervals
(−∞,0]∪[π,∞)\left(-\infty, 0\right] \cup \left[\pi, \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→−∞(12−5sin⁡(x)2)=⟨−2,3⟩\lim_{x \to -\infty}\left(\frac{1}{2} - \frac{5 \sin{\left(x \right)}}{2}\right) = \left\langle -2, 3\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−2,3⟩y = \left\langle -2, 3\right\rangle
lim⁡x→∞(12−5sin⁡(x)2)=⟨−2,3⟩\lim_{x \to \infty}\left(\frac{1}{2} - \frac{5 \sin{\left(x \right)}}{2}\right) = \left\langle -2, 3\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−2,3⟩y = \left\langle -2, 3\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of -5*sin(x)/2 + 1/2, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(12−5sin⁡(x)2x)=0\lim_{x \to -\infty}\left(\frac{\frac{1}{2} - \frac{5 \sin{\left(x \right)}}{2}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(12−5sin⁡(x)2x)=0\lim_{x \to \infty}\left(\frac{\frac{1}{2} - \frac{5 \sin{\left(x \right)}}{2}}{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:
12−5sin⁡(x)2=5sin⁡(x)2+12\frac{1}{2} - \frac{5 \sin{\left(x \right)}}{2} = \frac{5 \sin{\left(x \right)}}{2} + \frac{1}{2}
- No
12−5sin⁡(x)2=−5sin⁡(x)2−12\frac{1}{2} - \frac{5 \sin{\left(x \right)}}{2} = - \frac{5 \sin{\left(x \right)}}{2} - \frac{1}{2}
- No
so, the function
not is
neither even, nor odd
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