Mister Exam

Other calculators

Plotting a function graph/ sin^2x-sinx

Graphing y = sin^2x-sinx

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          2            
f(x) = sin (x) - sin(x)
f(x)=sin⁡2(x)−sin⁡(x)f{\left(x \right)} = \sin^{2}{\left(x \right)} - \sin{\left(x \right)} 
f = sin(x)^2 - sin(x)
The graph of the function
02468-8-6-4-2-10102.5-2.5
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:
sin⁡2(x)−sin⁡(x)=0\sin^{2}{\left(x \right)} - \sin{\left(x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
x2=π2x_{2} = \frac{\pi}{2}
x3=πx_{3} = \pi
Numerical solution
x1=−92.6769832292373x_{1} = -92.6769832292373
x2=81.6814089933346x_{2} = 81.6814089933346
x3=32.986723044911x_{3} = 32.986723044911
x4=−67.5442421642546x_{4} = -67.5442421642546
x5=39.2699086388565x_{5} = 39.2699086388565
x6=59.6902604182061x_{6} = 59.6902604182061
x7=−29.8451300972765x_{7} = -29.8451300972765
x8=34.5575191894877x_{8} = 34.5575191894877
x9=−36.1283154212439x_{9} = -36.1283154212439
x10=−21.9911485751286x_{10} = -21.9911485751286
x11=15.707963267949x_{11} = 15.707963267949
x12=89.5353908137952x_{12} = 89.5353908137952
x13=120.951318648179x_{13} = 120.951318648179
x14=−81.6814089933346x_{14} = -81.6814089933346
x15=−54.9778717129156x_{15} = -54.9778717129156
x16=−40.8407044966673x_{16} = -40.8407044966673
x17=−80.1106125810393x_{17} = -80.1106125810393
x18=−94.2477796076938x_{18} = -94.2477796076938
x19=14.1371670985871x_{19} = 14.1371670985871
x20=−59.6902604182061x_{20} = -59.6902604182061
x21=−43.9822971502571x_{21} = -43.9822971502571
x22=7.85398173796495x_{22} = 7.85398173796495
x23=−31.4159265358979x_{23} = -31.4159265358979
x24=58.1194647431527x_{24} = 58.1194647431527
x25=12.5663706143592x_{25} = 12.5663706143592
x26=51.8362788966528x_{26} = 51.8362788966528
x27=43.9822971502571x_{27} = 43.9822971502571
x28=95.8185760548644x_{28} = 95.8185760548644
x29=−15.707963267949x_{29} = -15.707963267949
x30=9.42477796076938x_{30} = 9.42477796076938
x31=−84.8230016469244x_{31} = -84.8230016469244
x32=25.1327412287183x_{32} = 25.1327412287183
x33=0x_{33} = 0
x34=−65.9734457253857x_{34} = -65.9734457253857
x35=−62.8318530717959x_{35} = -62.8318530717959
x36=−28.2743338823081x_{36} = -28.2743338823081
x37=−86.3937977915432x_{37} = -86.3937977915432
x38=−69.1150383789755x_{38} = -69.1150383789755
x39=−17.2787597741434x_{39} = -17.2787597741434
x40=−50.2654824574367x_{40} = -50.2654824574367
x41=−18.8495559215388x_{41} = -18.8495559215388
x42=62.8318530717959x_{42} = 62.8318530717959
x43=−75.398223686155x_{43} = -75.398223686155
x44=91.106186954104x_{44} = 91.106186954104
x45=83.2522058001693x_{45} = 83.2522058001693
x46=64.4026493102586x_{46} = 64.4026493102586
x47=28.2743338823081x_{47} = 28.2743338823081
x48=32.9867223690379x_{48} = 32.9867223690379
x49=65.9734457253857x_{49} = 65.9734457253857
x50=70.6858345286456x_{50} = 70.6858345286456
x51=26.7035373768773x_{51} = 26.7035373768773
x52=−42.4115006392452x_{52} = -42.4115006392452
x53=39.2699080280542x_{53} = 39.2699080280542
x54=76.9690200976964x_{54} = 76.9690200976964
x55=−87.9645943005142x_{55} = -87.9645943005142
x56=−6.28318530717959x_{56} = -6.28318530717959
x57=47.1238898038469x_{57} = 47.1238898038469
x58=45.553093663481x_{58} = 45.553093663481
x59=−4.7123888305818x_{59} = -4.7123888305818
x60=1.57079651244662x_{60} = 1.57079651244662
x61=−97.3893722612836x_{61} = -97.3893722612836
x62=−48.6946861243056x_{62} = -48.6946861243056
x63=94.2477796076938x_{63} = 94.2477796076938
x64=−10.9955739732138x_{64} = -10.9955739732138
x65=72.2566310325652x_{65} = 72.2566310325652
x66=−54.9778709863297x_{66} = -54.9778709863297
x67=3.14159265358979x_{67} = 3.14159265358979
x68=−9.42477796076938x_{68} = -9.42477796076938
x69=56.5486677646163x_{69} = 56.5486677646163
x70=−61.2610569243204x_{70} = -61.2610569243204
x71=−4.71238903613963x_{71} = -4.71238903613963
x72=100.530964914873x_{72} = 100.530964914873
x73=76.969019673036x_{73} = 76.969019673036
x74=83.2522050600807x_{74} = 83.2522050600807
x75=−23.5619450064001x_{75} = -23.5619450064001
x76=6.28318530717959x_{76} = 6.28318530717959
x77=−53.4070751110265x_{77} = -53.4070751110265
x78=−10.9955747331165x_{78} = -10.9955747331165
x79=21.9911485751286x_{79} = 21.9911485751286
x80=−25.1327412287183x_{80} = -25.1327412287183
x81=18.8495559215388x_{81} = 18.8495559215388
x82=69.1150383789755x_{82} = 69.1150383789755
x83=−72.2566310325652x_{83} = -72.2566310325652
x84=87.9645943005142x_{84} = 87.9645943005142
x85=37.6991118430775x_{85} = 37.6991118430775
x86=−73.8274272802392x_{86} = -73.8274272802392
x87=50.2654824574367x_{87} = 50.2654824574367
x88=53.4070751110265x_{88} = 53.4070751110265
x89=97.3893722612836x_{89} = 97.3893722612836
x90=−1564.51314148772x_{90} = -1564.51314148772
x91=−37.6991118430775x_{91} = -37.6991118430775
x92=78.5398163397448x_{92} = 78.5398163397448
x93=−78.5398163397448x_{93} = -78.5398163397448
x94=20.42035215177x_{94} = 20.42035215177
x95=−34.5575191894877x_{95} = -34.5575191894877
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(x)^2 - sin(x).
sin⁡2(0)−sin⁡(0)\sin^{2}{\left(0 \right)} - \sin{\left(0 \right)}
The result:
f(0)=0f{\left(0 \right)} = 0
The point:
(0, 0)
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
2sin⁡(x)cos⁡(x)−cos⁡(x)=02 \sin{\left(x \right)} \cos{\left(x \right)} - \cos{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=−π2x_{1} = - \frac{\pi}{2}
x2=π6x_{2} = \frac{\pi}{6}
x3=π2x_{3} = \frac{\pi}{2}
x4=5π6x_{4} = \frac{5 \pi}{6}
The values of the extrema at the points:
 -pi     
(----, 2)
  2      

 pi       
(--, -1/4)
 6        

 pi    
(--, 0)
 2     

 5*pi       
(----, -1/4)
  6


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=π6x_{1} = \frac{\pi}{6}
x2=5π6x_{2} = \frac{5 \pi}{6}
Maxima of the function at points:
x2=−π2x_{2} = - \frac{\pi}{2}
x2=π2x_{2} = \frac{\pi}{2}
Decreasing at intervals
[5π6,∞)\left[\frac{5 \pi}{6}, \infty\right)
Increasing at intervals
(−∞,π6]∪[π2,5π6]\left(-\infty, \frac{\pi}{6}\right] \cup \left[\frac{\pi}{2}, \frac{5 \pi}{6}\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⁡2(x)+sin⁡(x)+2cos⁡2(x)=0- 2 \sin^{2}{\left(x \right)} + \sin{\left(x \right)} + 2 \cos^{2}{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=2atan⁡(−14+29−334+334)x_{1} = 2 \operatorname{atan}{\left(- \frac{1}{4} + \frac{\sqrt{2} \sqrt{9 - \sqrt{33}}}{4} + \frac{\sqrt{33}}{4} \right)}
x2=−2atan⁡(14+233+94+334)x_{2} = - 2 \operatorname{atan}{\left(\frac{1}{4} + \frac{\sqrt{2} \sqrt{\sqrt{33} + 9}}{4} + \frac{\sqrt{33}}{4} \right)}
x3=−2atan⁡(−334+14+29−334)x_{3} = - 2 \operatorname{atan}{\left(- \frac{\sqrt{33}}{4} + \frac{1}{4} + \frac{\sqrt{2} \sqrt{9 - \sqrt{33}}}{4} \right)}
x4=−2atan⁡(−233+94+14+334)x_{4} = - 2 \operatorname{atan}{\left(- \frac{\sqrt{2} \sqrt{\sqrt{33} + 9}}{4} + \frac{1}{4} + \frac{\sqrt{33}}{4} \right)}

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
[2atan⁡(−14+29−334+334),∞)\left[2 \operatorname{atan}{\left(- \frac{1}{4} + \frac{\sqrt{2} \sqrt{9 - \sqrt{33}}}{4} + \frac{\sqrt{33}}{4} \right)}, \infty\right)
Convex at the intervals
(−∞,−2atan⁡(−233+94+14+334)]∪[−2atan⁡(−334+14+29−334),2atan⁡(−14+29−334+334)]\left(-\infty, - 2 \operatorname{atan}{\left(- \frac{\sqrt{2} \sqrt{\sqrt{33} + 9}}{4} + \frac{1}{4} + \frac{\sqrt{33}}{4} \right)}\right] \cup \left[- 2 \operatorname{atan}{\left(- \frac{\sqrt{33}}{4} + \frac{1}{4} + \frac{\sqrt{2} \sqrt{9 - \sqrt{33}}}{4} \right)}, 2 \operatorname{atan}{\left(- \frac{1}{4} + \frac{\sqrt{2} \sqrt{9 - \sqrt{33}}}{4} + \frac{\sqrt{33}}{4} \right)}\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⁡2(x)−sin⁡(x))=⟨−1,2⟩\lim_{x \to -\infty}\left(\sin^{2}{\left(x \right)} - \sin{\left(x \right)}\right) = \left\langle -1, 2\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−1,2⟩y = \left\langle -1, 2\right\rangle
lim⁡x→∞(sin⁡2(x)−sin⁡(x))=⟨−1,2⟩\lim_{x \to \infty}\left(\sin^{2}{\left(x \right)} - \sin{\left(x \right)}\right) = \left\langle -1, 2\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−1,2⟩y = \left\langle -1, 2\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sin(x)^2 - sin(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡2(x)−sin⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{\sin^{2}{\left(x \right)} - \sin{\left(x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(sin⁡2(x)−sin⁡(x)x)=0\lim_{x \to \infty}\left(\frac{\sin^{2}{\left(x \right)} - \sin{\left(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⁡2(x)−sin⁡(x)=sin⁡2(x)+sin⁡(x)\sin^{2}{\left(x \right)} - \sin{\left(x \right)} = \sin^{2}{\left(x \right)} + \sin{\left(x \right)}
- No
sin⁡2(x)−sin⁡(x)=−sin⁡2(x)−sin⁡(x)\sin^{2}{\left(x \right)} - \sin{\left(x \right)} = - \sin^{2}{\left(x \right)} - \sin{\left(x \right)}
- No
so, the function
not is
neither even, nor odd
    © Mister Exam – Calculator