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

You entered:

-2*sin(x-pi/2)+2

What you mean?

-2*sin((x - pi)/2) + 2
 
-2*sin(x - pi/2) + 2
 
-2*sin(x - pi/2) + 2

Graphing y = -2*sin(x-pi/2)+2

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
              /    pi\    
f(x) = - 2*sin|x - --| + 2
              \    2 /
f(x)=2−2sin⁡(x−π2)f{\left(x \right)} = 2 - 2 \sin{\left(x - \frac{\pi}{2} \right)} 
f = 2 - 2*sin(x - pi/2)
The graph of the function
0-30-20-101020304050607005
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:
2−2sin⁡(x−π2)=02 - 2 \sin{\left(x - \frac{\pi}{2} \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=πx_{1} = \pi
Numerical solution
x1=−34.5575188899093x_{1} = -34.5575188899093
x2=59.6902599104079x_{2} = 59.6902599104079
x3=78.5398166181283x_{3} = 78.5398166181283
x4=53.4070746418597x_{4} = 53.4070746418597
x5=15.7079629803241x_{5} = 15.7079629803241
x6=9.42477826738203x_{6} = 9.42477826738203
x7=65.9734457529812x_{7} = 65.9734457529812
x8=97.3893717959212x_{8} = 97.3893717959212
x9=−28.2743337069329x_{9} = -28.2743337069329
x10=−78.5398168194507x_{10} = -78.5398168194507
x11=−53.4070745786761x_{11} = -53.4070745786761
x12=−65.9734453607004x_{12} = -65.9734453607004
x13=−47.1238901083229x_{13} = -47.1238901083229
x14=−65.9734461969855x_{14} = -65.9734461969855
x15=−3.14159295109225x_{15} = -3.14159295109225
x16=91.1061873718352x_{16} = 91.1061873718352
x17=3.1415922548952x_{17} = 3.1415922548952
x18=−15.7079635641079x_{18} = -15.7079635641079
x19=−72.2566308657983x_{19} = -72.2566308657983
x20=−21.991148226056x_{20} = -21.991148226056
x21=21.9911489072506x_{21} = 21.9911489072506
x22=34.5575195449229x_{22} = 34.5575195449229
x23=72.2566306985x_{23} = 72.2566306985
x24=−3.14159217367683x_{24} = -3.14159217367683
x25=−53.407075294995x_{25} = -53.407075294995
x26=−9.42477752082051x_{26} = -9.42477752082051
x27=−28.2743340989896x_{27} = -28.2743340989896
x28=−59.6902599212271x_{28} = -59.6902599212271
x29=78.5398152766482x_{29} = 78.5398152766482
x30=−21.9911490521325x_{30} = -21.9911490521325
x31=−9.4247781365785x_{31} = -9.4247781365785
x32=40.8407042062167x_{32} = 40.8407042062167
x33=−40.8407049290801x_{33} = -40.8407049290801
x34=21.9911480932338x_{34} = 21.9911480932338
x35=−59.6902606928653x_{35} = -59.6902606928653
x36=72.2566310277176x_{36} = 72.2566310277176
x37=15.707963957033x_{37} = 15.707963957033
x38=−78.5398160472843x_{38} = -78.5398160472843
x39=28.2743335663982x_{39} = 28.2743335663982
x40=28.2743338651796x_{40} = 28.2743338651796
x41=47.123889410773x_{41} = 47.123889410773
x42=65.9734452390837x_{42} = 65.9734452390837
x43=−97.3893716284562x_{43} = -97.3893716284562
x44=9.42477748794163x_{44} = 9.42477748794163
x45=−65.9734449870253x_{45} = -65.9734449870253
x46=−84.8230012511693x_{46} = -84.8230012511693
x47=−15.7079632965989x_{47} = -15.7079632965989
x48=28.2743343711514x_{48} = 28.2743343711514
x49=40.8407049800347x_{49} = 40.8407049800347
x50=−84.8230020565447x_{50} = -84.8230020565447
x51=59.6902600526626x_{51} = 59.6902600526626
x52=−72.2566311847166x_{52} = -72.2566311847166
x53=−47.1238893275319x_{53} = -47.1238893275319
x54=−59.6902604578012x_{54} = -59.6902604578012
x55=15.7079634518075x_{55} = 15.7079634518075
x56=59.6902606104322x_{56} = 59.6902606104322
x57=84.8230021335997x_{57} = 84.8230021335997
x58=−40.8407040952604x_{58} = -40.8407040952604
x59=−40.8407049008781x_{59} = -40.8407049008781
x60=21.9911485852059x_{60} = 21.9911485852059
x61=−53.4070745963886x_{61} = -53.4070745963886
x62=−28.2743343914215x_{62} = -28.2743343914215
x63=40.8407045792514x_{63} = 40.8407045792514
x64=−97.3893717476911x_{64} = -97.3893717476911
x65=−15.707962774825x_{65} = -15.707962774825
x66=−91.1061864815274x_{66} = -91.1061864815274
x67=91.1061865667532x_{67} = 91.1061865667532
x68=15.7079627593774x_{68} = 15.7079627593774
x69=47.1238902162437x_{69} = 47.1238902162437
x70=65.9734460390947x_{70} = 65.9734460390947
x71=−21.9911485864417x_{71} = -21.9911485864417
x72=53.407075424589x_{72} = 53.407075424589
x73=−72.2566315419804x_{73} = -72.2566315419804
x74=34.5575190219169x_{74} = 34.5575190219169
x75=3.14159306054457x_{75} = 3.14159306054457
x76=−1127.83176318906x_{76} = -1127.83176318906
x77=97.389372581711x_{77} = 97.389372581711
x78=78.5398168562347x_{78} = 78.5398168562347
x79=−9.42477744529557x_{79} = -9.42477744529557
x80=34.5575197055812x_{80} = 34.5575197055812
x81=72.2566315166773x_{81} = 72.2566315166773
x82=40.8407045848602x_{82} = 40.8407045848602
x83=−91.106187265474x_{83} = -91.106187265474
x84=78.5398161804942x_{84} = 78.5398161804942
x85=84.8230013636028x_{85} = 84.8230013636028
x86=−97.3893724533348x_{86} = -97.3893724533348
x87=−65.9734457649277x_{87} = -65.9734457649277
x88=−34.5575196658297x_{88} = -34.5575196658297
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 - pi/2) + 2.
2−2sin⁡(−π2+0)2 - 2 \sin{\left(- \frac{\pi}{2} + 0 \right)}
The result:
f(0)=4f{\left(0 \right)} = 4
The point:
(0, 4)
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−π2)=0- 2 \cos{\left(x - \frac{\pi}{2} \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=πx_{2} = \pi
The values of the extrema at the points:
(0, 4)

(pi, 0)


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=πx_{1} = \pi
Maxima of the function at points:
x1=0x_{1} = 0
Decreasing at intervals
(−∞,0]∪[π,∞)\left(-\infty, 0\right] \cup \left[\pi, \infty\right)
Increasing at intervals
[0,π]\left[0, \pi\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
−2cos⁡(x)=0- 2 \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}

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
[π2,3π2]\left[\frac{\pi}{2}, \frac{3 \pi}{2}\right]
Convex at the intervals
(−∞,π2]∪[3π2,∞)\left(-\infty, \frac{\pi}{2}\right] \cup \left[\frac{3 \pi}{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→−∞(2−2sin⁡(x−π2))=⟨0,4⟩\lim_{x \to -\infty}\left(2 - 2 \sin{\left(x - \frac{\pi}{2} \right)}\right) = \left\langle 0, 4\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨0,4⟩y = \left\langle 0, 4\right\rangle
lim⁡x→∞(2−2sin⁡(x−π2))=⟨0,4⟩\lim_{x \to \infty}\left(2 - 2 \sin{\left(x - \frac{\pi}{2} \right)}\right) = \left\langle 0, 4\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨0,4⟩y = \left\langle 0, 4\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of -2*sin(x - pi/2) + 2, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(2−2sin⁡(x−π2)x)=0\lim_{x \to -\infty}\left(\frac{2 - 2 \sin{\left(x - \frac{\pi}{2} \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(2−2sin⁡(x−π2)x)=0\lim_{x \to \infty}\left(\frac{2 - 2 \sin{\left(x - \frac{\pi}{2} \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:
2−2sin⁡(x−π2)=2cos⁡(x)+22 - 2 \sin{\left(x - \frac{\pi}{2} \right)} = 2 \cos{\left(x \right)} + 2
- No
2−2sin⁡(x−π2)=−2cos⁡(x)−22 - 2 \sin{\left(x - \frac{\pi}{2} \right)} = - 2 \cos{\left(x \right)} - 2
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = -2*sin(x-pi/2)+2
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