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Plotting a function graph/ 2sinx-1

Graphing y = 2sinx-1

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = 2*sin(x) - 1
f(x)=2sin⁡(x)−1f{\left(x \right)} = 2 \sin{\left(x \right)} - 1 
f = 2*sin(x) - 1
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:
2sin⁡(x)−1=02 \sin{\left(x \right)} - 1 = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=π6x_{1} = \frac{\pi}{6}
x2=5π6x_{2} = \frac{5 \pi}{6}
Numerical solution
x1=69.6386371545737x_{1} = 69.6386371545737
x2=19.3731546971371x_{2} = 19.3731546971371
x3=−12.0427718387609x_{3} = -12.0427718387609
x4=−79.0634151153431x_{4} = -79.0634151153431
x5=6.80678408277789x_{5} = 6.80678408277789
x6=96.8657734856853x_{6} = 96.8657734856853
x7=−43.4586983746588x_{7} = -43.4586983746588
x8=−18.3259571459405x_{8} = -18.3259571459405
x9=2.61799387799149x_{9} = 2.61799387799149
x10=−4454.25478401473x_{10} = -4454.25478401473
x11=0.523598775598299x_{11} = 0.523598775598299
x12=−100.007366139275x_{12} = -100.007366139275
x13=27.7507351067098x_{13} = 27.7507351067098
x14=52.8834763354282x_{14} = 52.8834763354282
x15=44.5058959258554x_{15} = 44.5058959258554
x16=134.564885328763x_{16} = 134.564885328763
x17=57.0722665402146x_{17} = 57.0722665402146
x18=15.1843644923507x_{18} = 15.1843644923507
x19=−28.7979326579064x_{19} = -28.7979326579064
x20=94.7713783832921x_{20} = 94.7713783832921
x21=−91.6297857297023x_{21} = -91.6297857297023
x22=−93.7241808320955x_{22} = -93.7241808320955
x23=88.4881930761125x_{23} = 88.4881930761125
x24=75.9218224617533x_{24} = 75.9218224617533
x25=82.2050077689329x_{25} = 82.2050077689329
x26=65.4498469497874x_{26} = 65.4498469497874
x27=90.5825881785057x_{27} = 90.5825881785057
x28=71.733032256967x_{28} = 71.733032256967
x29=50.789081233035x_{29} = 50.789081233035
x30=78.0162175641465x_{30} = 78.0162175641465
x31=21.4675497995303x_{31} = 21.4675497995303
x32=138.753675533549x_{32} = 138.753675533549
x33=84.2994028713261x_{33} = 84.2994028713261
x34=63.3554518473942x_{34} = 63.3554518473942
x35=−87.4409955249159x_{35} = -87.4409955249159
x36=−97.9129710368819x_{36} = -97.9129710368819
x37=−16.2315620435473x_{37} = -16.2315620435473
x38=13.0899693899575x_{38} = 13.0899693899575
x39=8.90117918517108x_{39} = 8.90117918517108
x40=−9.94837673636768x_{40} = -9.94837673636768
x41=−56.025068989018x_{41} = -56.025068989018
x42=25.6563400043166x_{42} = 25.6563400043166
x43=−35.081117965086x_{43} = -35.081117965086
x44=−49.7418836818384x_{44} = -49.7418836818384
x45=−22.5147473507269x_{45} = -22.5147473507269
x46=−47.6474885794452x_{46} = -47.6474885794452
x47=−5.75958653158129x_{47} = -5.75958653158129
x48=−72.7802298081635x_{48} = -72.7802298081635
x49=38.2227106186758x_{49} = 38.2227106186758
x50=−3.66519142918809x_{50} = -3.66519142918809
x51=31.9395253114962x_{51} = 31.9395253114962
x52=−68.5914396033772x_{52} = -68.5914396033772
x53=−81.1578102177363x_{53} = -81.1578102177363
x54=−37.1755130674792x_{54} = -37.1755130674792
x55=−627.79493194236x_{55} = -627.79493194236
x56=−60.2138591938044x_{56} = -60.2138591938044
x57=−66.497044500984x_{57} = -66.497044500984
x58=101.054563690472x_{58} = 101.054563690472
x59=−85.3466004225227x_{59} = -85.3466004225227
x60=−53.9306738866248x_{60} = -53.9306738866248
x61=−41.3643032722656x_{61} = -41.3643032722656
x62=−62.3082542961976x_{62} = -62.3082542961976
x63=−24.60914245312x_{63} = -24.60914245312
x64=59.1666616426078x_{64} = 59.1666616426078
x65=40.317105721069x_{65} = 40.317105721069
x66=−2650.98060085419x_{66} = -2650.98060085419
x67=−74.8746249105567x_{67} = -74.8746249105567
x68=34.0339204138894x_{68} = 34.0339204138894
x69=46.6002910282486x_{69} = 46.6002910282486
x70=17438.4572213013x_{70} = 17438.4572213013
x71=−30.8923277602996x_{71} = -30.8923277602996
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 2*sin(x) - 1.
−1+2sin⁡(0)-1 + 2 \sin{\left(0 \right)}
The result:
f(0)=−1f{\left(0 \right)} = -1
The point:
(0, -1)
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⁡(x)=02 \cos{\left(x \right)} = 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    
(--, 1)
 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=3π2x_{1} = \frac{3 \pi}{2}
Maxima of the function at points:
x1=π2x_{1} = \frac{\pi}{2}
Decreasing at intervals
(−∞,π2]∪[3π2,∞)\left(-\infty, \frac{\pi}{2}\right] \cup \left[\frac{3 \pi}{2}, \infty\right)
Increasing at intervals
[π2,3π2]\left[\frac{\pi}{2}, \frac{3 \pi}{2}\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
−2sin⁡(x)=0- 2 \sin{\left(x \right)} = 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(-\infty, 0\right] \cup \left[\pi, \infty\right)
Convex at the intervals
[0,π]\left[0, \pi\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(2sin⁡(x)−1)=⟨−3,1⟩\lim_{x \to -\infty}\left(2 \sin{\left(x \right)} - 1\right) = \left\langle -3, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−3,1⟩y = \left\langle -3, 1\right\rangle
lim⁡x→∞(2sin⁡(x)−1)=⟨−3,1⟩\lim_{x \to \infty}\left(2 \sin{\left(x \right)} - 1\right) = \left\langle -3, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−3,1⟩y = \left\langle -3, 1\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 2*sin(x) - 1, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(2sin⁡(x)−1x)=0\lim_{x \to -\infty}\left(\frac{2 \sin{\left(x \right)} - 1}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(2sin⁡(x)−1x)=0\lim_{x \to \infty}\left(\frac{2 \sin{\left(x \right)} - 1}{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:
2sin⁡(x)−1=−2sin⁡(x)−12 \sin{\left(x \right)} - 1 = - 2 \sin{\left(x \right)} - 1
- No
2sin⁡(x)−1=2sin⁡(x)+12 \sin{\left(x \right)} - 1 = 2 \sin{\left(x \right)} + 1
- No
so, the function
not is
neither even, nor odd
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