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Plotting a function graph/ sinx+1

Graphing y = sinx+1

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = sin(x) + 1
f(x)=sin⁡(x)+1f{\left(x \right)} = \sin{\left(x \right)} + 1 
f = sin(x) + 1
The graph of the function
02468-8-6-4-2-101004
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)+1=0\sin{\left(x \right)} + 1 = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−π2x_{1} = - \frac{\pi}{2}
x2=3π2x_{2} = \frac{3 \pi}{2}
Numerical solution
x1=67.54424230971x_{1} = 67.54424230971
x2=4.71239022926564x_{2} = 4.71239022926564
x3=−14.1371674455661x_{3} = -14.1371674455661
x4=23.5619437708833x_{4} = 23.5619437708833
x5=−1.57079643188553x_{5} = -1.57079643188553
x6=48.6946873020308x_{6} = 48.6946873020308
x7=86.3937978869933x_{7} = 86.3937978869933
x8=−102.101761026058x_{8} = -102.101761026058
x9=92.6769830592094x_{9} = 92.6769830592094
x10=−7.85398149665124x_{10} = -7.85398149665124
x11=−83.2522055723275x_{11} = -83.2522055723275
x12=−26.7035372004893x_{12} = -26.7035372004893
x13=−51.8362791922783x_{13} = -51.8362791922783
x14=−14.1371667858125x_{14} = -14.1371667858125
x15=29.845130330036x_{15} = 29.845130330036
x16=98.9601682515978x_{16} = 98.9601682515978
x17=42.4115007162407x_{17} = 42.4115007162407
x18=36.1283150875497x_{18} = 36.1283150875497
x19=538.783139388541x_{19} = 538.783139388541
x20=42.4115013353669x_{20} = 42.4115013353669
x21=−45.5530935025548x_{21} = -45.5530935025548
x22=−89.5353901118113x_{22} = -89.5353901118113
x23=80.1106130902139x_{23} = 80.1106130902139
x24=−32.9867232184024x_{24} = -32.9867232184024
x25=−39.2699076683741x_{25} = -39.2699076683741
x26=−39.2699069219675x_{26} = -39.2699069219675
x27=−7.85398205280014x_{27} = -7.85398205280014
x28=−70.6858343571487x_{28} = -70.6858343571487
x29=86.3937984838325x_{29} = 86.3937984838325
x30=−14.1371668370864x_{30} = -14.1371668370864
x31=−70.6858351534454x_{31} = -70.6858351534454
x32=−70.6858331259916x_{32} = -70.6858331259916
x33=4.71238874329685x_{33} = 4.71238874329685
x34=−20.420353265929x_{34} = -20.420353265929
x35=−64.4026491641039x_{35} = -64.4026491641039
x36=67.5442415586719x_{36} = 67.5442415586719
x37=4.7123894841958x_{37} = 4.7123894841958
x38=48.6946866365921x_{38} = 48.6946866365921
x39=−20.4203527465087x_{39} = -20.4203527465087
x40=23.5619444059921x_{40} = 23.5619444059921
x41=−95.818575476176x_{41} = -95.818575476176
x42=61.2610563112167x_{42} = 61.2610563112167
x43=−1.57079581340397x_{43} = -1.57079581340397
x44=−83.2522048211133x_{44} = -83.2522048211133
x45=−89.5353906059052x_{45} = -89.5353906059052
x46=98.9601692809083x_{46} = 98.9601692809083
x47=80.1106122287081x_{47} = 80.1106122287081
x48=10.9955747360645x_{48} = 10.9955747360645
x49=29.8451297031011x_{49} = 29.8451297031011
x50=−32.9867224188086x_{50} = -32.9867224188086
x51=−76.9690195738024x_{51} = -76.9690195738024
x52=92.6769843439965x_{52} = 92.6769843439965
x53=−95.8185763308148x_{53} = -95.8185763308148
x54=73.8274274830848x_{54} = 73.8274274830848
x55=−58.1194645939029x_{55} = -58.1194645939029
x56=29.8451303231501x_{56} = 29.8451303231501
x57=36.1283157235346x_{57} = 36.1283157235346
x58=−64.4026498988255x_{58} = -64.4026498988255
x59=67.5442408278864x_{59} = 67.5442408278864
x60=−45.5530929624673x_{60} = -45.5530929624673
x61=−58.1194639976905x_{61} = -58.1194639976905
x62=−7.85398119154045x_{62} = -7.85398119154045
x63=86.3937978309099x_{63} = 86.3937978309099
x64=−58.1194639046052x_{64} = -58.1194639046052
x65=54.9778710948428x_{65} = 54.9778710948428
x66=23.5619451518571x_{66} = 23.5619451518571
x67=98.9601690454399x_{67} = 98.9601690454399
x68=92.6769837888103x_{68} = 92.6769837888103
x69=−45.5530935911043x_{69} = -45.5530935911043
x70=17.2787591562062x_{70} = 17.2787591562062
x71=−20.4203520060805x_{71} = -20.4203520060805
x72=−95.8185758680502x_{72} = -95.8185758680502
x73=−89.535390750197x_{73} = -89.535390750197
x74=73.8274274426229x_{74} = 73.8274274426229
x75=61.2610571125526x_{75} = 61.2610571125526
x76=−76.9690203748894x_{76} = -76.9690203748894
x77=80.1106131368654x_{77} = 80.1106131368654
x78=10.9955739381756x_{78} = 10.9955739381756
x79=−83.2522042893833x_{79} = -83.2522042893833
x80=−1.57079639503667x_{80} = -1.57079639503667
x81=−39.2699084145515x_{81} = -39.2699084145515
x82=73.8274268520838x_{82} = 73.8274268520838
x83=54.9778718908148x_{83} = 54.9778718908148
x84=42.4115007274741x_{84} = 42.4115007274741
x85=−26.7035379986821x_{85} = -26.7035379986821
x86=48.6946859012172x_{86} = 48.6946859012172
x87=48.6946870830469x_{87} = 48.6946870830469
x88=17.2787599560783x_{88} = 17.2787599560783
x89=−51.8362783335234x_{89} = -51.8362783335234
x90=−51.8362786893284x_{90} = -51.8362786893284
x91=36.1283159497235x_{91} = 36.1283159497235
x92=−64.4026502975618x_{92} = -64.4026502975618
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(x) + 1.
sin⁡(0)+1\sin{\left(0 \right)} + 1
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
cos⁡(x)=0\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    
(--, 2)
 2     

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