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Plotting a function graph/ 2cos(2x)-sin(x)

Graphing y = 2cos(2x)-sin(x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = 2*cos(2*x) - sin(x)
f(x)=−sin⁡(x)+2cos⁡(2x)f{\left(x \right)} = - \sin{\left(x \right)} + 2 \cos{\left(2 x \right)} 
f = -sin(x) + 2*cos(2*x)
The graph of the function
-3.0-2.5-2.0-1.5-1.0-0.50.00.51.01.52.02.53.05-5
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⁡(x)+2cos⁡(2x)=0- \sin{\left(x \right)} + 2 \cos{\left(2 x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−ilog⁡(−215−338−i(1+33)8)x_{1} = - i \log{\left(- \frac{\sqrt{2} \sqrt{15 - \sqrt{33}}}{8} - \frac{i \left(1 + \sqrt{33}\right)}{8} \right)}
x2=−ilog⁡(−233+158−i(1−33)8)x_{2} = - i \log{\left(- \frac{\sqrt{2} \sqrt{\sqrt{33} + 15}}{8} - \frac{i \left(1 - \sqrt{33}\right)}{8} \right)}
x3=−ilog⁡(233+158−i(1−33)8)x_{3} = - i \log{\left(\frac{\sqrt{2} \sqrt{\sqrt{33} + 15}}{8} - \frac{i \left(1 - \sqrt{33}\right)}{8} \right)}
x4=−ilog⁡(215−338−33i8−i8)x_{4} = - i \log{\left(\frac{\sqrt{2} \sqrt{15 - \sqrt{33}}}{8} - \frac{\sqrt{33} i}{8} - \frac{i}{8} \right)}
Numerical solution
x1=0.634866871133571x_{1} = 0.634866871133571
x2=−16.3428301390825x_{2} = -16.3428301390825
x3=76.0330905572886x_{3} = 76.0330905572886
x4=71.6217641614317x_{4} = 71.6217641614317
x5=−57.5516347184825x_{5} = -57.5516347184825
x6=−51.2684494113029x_{6} = -51.2684494113029
x7=−18.2146890504052x_{7} = -18.2146890504052
x8=99.5279979610071x_{8} = 99.5279979610071
x9=60.6932273720723x_{9} = 60.6932273720723
x10=16.7109302218152x_{10} = 16.7109302218152
x11=46.4890229327133x_{11} = 46.4890229327133
x12=−2.13862569972354x_{12} = -2.13862569972354
x13=−47.7587566749805x_{13} = -47.7587566749805
x14=41.8436714505336x_{14} = 41.8436714505336
x15=40.2058376255337x_{15} = 40.2058376255337
x16=30.4129595820317x_{16} = 30.4129595820317
x17=38715.9848958867x_{17} = 38715.9848958867
x18=52.7722082398929x_{18} = 52.7722082398929
x19=−8561.47484790332x_{19} = -8561.47484790332
x20=10.4277449146356x_{20} = 10.4277449146356
x21=−5.64831843604602x_{21} = -5.64831843604602
x22=90.4713200829704x_{22} = 90.4713200829704
x23=19.4844227926723x_{23} = 19.4844227926723
x24=33.9226523183542x_{24} = 33.9226523183542
x25=24.1297742748521x_{25} = 24.1297742748521
x26=−3.77645952472336x_{26} = -3.77645952472336
x27=77.9049494686113x_{27} = 77.9049494686113
x28=−33.5545522356215x_{28} = -33.5545522356215
x29=−55.9138008934827x_{29} = -55.9138008934827
x30=−1.00296695386625x_{30} = -1.00296695386625
x31=−93.6129127365602x_{31} = -93.6129127365602
x32=63.4667199429294x_{32} = 63.4667199429294
x33=−49.6306155863031x_{33} = -49.6306155863031
x34=−99.8960980437398x_{34} = -99.8960980437398
x35=55.54570081075x_{35} = 55.54570081075
x36=−41.4755713678009x_{36} = -41.4755713678009
x37=−10.059644831903x_{37} = -10.059644831903
x38=−79.1746832108784x_{38} = -79.1746832108784
x39=48.1268567577131x_{39} = 48.1268567577131
x40=74.3952567322888x_{40} = 74.3952567322888
x41=−11.9315037432256x_{41} = -11.9315037432256
x42=−63.8348200256621x_{42} = -63.8348200256621
x43=17.8465889676725x_{43} = 17.8465889676725
x44=−39.8377375428011x_{44} = -39.8377375428011
x45=−98.0242391324172x_{45} = -98.0242391324172
x46=−116.873795053956x_{46} = -116.873795053956
x47=98.3923392151498x_{47} = 98.3923392151498
x48=−46.1209228499806x_{48} = -46.1209228499806
x49=29.2773008361744x_{49} = 29.2773008361744
x50=−32.4188934897642x_{50} = -32.4188934897642
x51=82.3162758644682x_{51} = 82.3162758644682
x52=121.887246618868x_{52} = 121.887246618868
x53=4.14455960745605x_{53} = 4.14455960745605
x54=8.78991108963581x_{54} = 8.78991108963581
x55=54.4100420648927x_{55} = 54.4100420648927
x56=−13.5693375682254x_{56} = -13.5693375682254
x57=61.8288861179296x_{57} = 61.8288861179296
x58=96.75450539015x_{58} = 96.75450539015
x59=−43.3474302791235x_{59} = -43.3474302791235
x60=2.50672578245622x_{60} = 2.50672578245622
x61=−90.1032200002378x_{61} = -90.1032200002378
x62=85.8259686007907x_{62} = 85.8259686007907
x63=−70.1180053328417x_{63} = -70.1180053328417
x64=−87.3297274293806x_{64} = -87.3297274293806
x65=−24.4978743575848x_{65} = -24.4978743575848
x66=−26.1357081825846x_{66} = -26.1357081825846
x67=25.7676080998519x_{67} = 25.7676080998519
x68=−71.253664078699x_{68} = -71.253664078699
x69=92.1091539079703x_{69} = 92.1091539079703
x70=69.749905250109x_{70} = 69.749905250109
x71=−91.7410538252376x_{71} = -91.7410538252376
x72=−366.56337351614x_{72} = -366.56337351614
x73=−27.2713669284419x_{73} = -27.2713669284419
x74=11.5634036604929x_{74} = 11.5634036604929
x75=38.3339787142111x_{75} = 38.3339787142111
x76=88.5994611716478x_{76} = 88.5994611716478
x77=−77.5368493858786x_{77} = -77.5368493858786
x78=27.6394670111746x_{78} = 27.6394670111746
x79=−76.4011906400213x_{79} = -76.4011906400213
x80=44.6171640213907x_{80} = 44.6171640213907
x81=32.0507934070315x_{81} = 32.0507934070315
x82=−54.0419419821601x_{82} = -54.0419419821601
x83=−68.4801715078419x_{83} = -68.4801715078419
x84=−85.457868518058x_{84} = -85.457868518058
x85=−62.1969862006623x_{85} = -62.1969862006623
x86=−19.852522875405x_{86} = -19.852522875405
x87=−60.3251272893396x_{87} = -60.3251272893396
x88=−35.1923860606213x_{88} = -35.1923860606213
x89=68.1120714251092x_{89} = 68.1120714251092
x90=−83.8200346930582x_{90} = -83.8200346930582
x91=84.1881347757908x_{91} = 84.1881347757908
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 2*cos(2*x) - sin(x).
−sin⁡(0)+2cos⁡(0⋅2)- \sin{\left(0 \right)} + 2 \cos{\left(0 \cdot 2 \right)}
The result:
f(0)=2f{\left(0 \right)} = 2
The point:
(0, 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
−4sin⁡(2x)−cos⁡(x)=0- 4 \sin{\left(2 x \right)} - \cos{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=−π2x_{1} = - \frac{\pi}{2}
x2=π2x_{2} = \frac{\pi}{2}
x3=−ilog⁡(−378−i8)x_{3} = - i \log{\left(- \frac{3 \sqrt{7}}{8} - \frac{i}{8} \right)}
x4=−ilog⁡(378−i8)x_{4} = - i \log{\left(\frac{3 \sqrt{7}}{8} - \frac{i}{8} \right)}
The values of the extrema at the points:
 -pi      
(----, -1)
  2       

 pi     
(--, -3)
 2      

       /      ___    \       /       /      ___    \\      /     /      ___    \\ 
       |  3*\/ 7    I|       |       |  3*\/ 7    I||      |     |  3*\/ 7    I|| 
(-I*log|- ------- - -|, 2*cos|2*I*log|- ------- - -|| + sin|I*log|- ------- - -||)
       \     8      8/       \       \     8      8//      \     \     8      8// 

       /          ___\       /       /          ___\\      /     /          ___\\ 
       |  I   3*\/ 7 |       |       |  I   3*\/ 7 ||      |     |  I   3*\/ 7 || 
(-I*log|- - + -------|, 2*cos|2*I*log|- - + -------|| + sin|I*log|- - + -------||)
       \  8      8   /       \       \  8      8   //      \     \  8      8   //


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}
x2=π2x_{2} = \frac{\pi}{2}
Maxima of the function at points:
x2=−π+atan⁡(721)x_{2} = - \pi + \operatorname{atan}{\left(\frac{\sqrt{7}}{21} \right)}
x2=−atan⁡(721)x_{2} = - \operatorname{atan}{\left(\frac{\sqrt{7}}{21} \right)}
Decreasing at intervals
[π2,∞)\left[\frac{\pi}{2}, \infty\right)
Increasing at intervals
(−∞,−π2]∪[−atan⁡(721),π2]\left(-\infty, - \frac{\pi}{2}\right] \cup \left[- \operatorname{atan}{\left(\frac{\sqrt{7}}{21} \right)}, \frac{\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)−8cos⁡(2x)=0\sin{\left(x \right)} - 8 \cos{\left(2 x \right)} = 0
Solve this equation
The roots of this equation
x1=−ilog⁡(−685−5732−i(1+357)32)x_{1} = - i \log{\left(- \frac{\sqrt{6} \sqrt{85 - \sqrt{57}}}{32} - \frac{i \left(1 + 3 \sqrt{57}\right)}{32} \right)}
x2=−ilog⁡(−657+8532−i(1−357)32)x_{2} = - i \log{\left(- \frac{\sqrt{6} \sqrt{\sqrt{57} + 85}}{32} - \frac{i \left(1 - 3 \sqrt{57}\right)}{32} \right)}
x3=−ilog⁡(657+8532−i(1−357)32)x_{3} = - i \log{\left(\frac{\sqrt{6} \sqrt{\sqrt{57} + 85}}{32} - \frac{i \left(1 - 3 \sqrt{57}\right)}{32} \right)}
x4=−ilog⁡(685−5732−357i32−i32)x_{4} = - i \log{\left(\frac{\sqrt{6} \sqrt{85 - \sqrt{57}}}{32} - \frac{3 \sqrt{57} i}{32} - \frac{i}{32} \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
[−π+atan⁡(−6(−357−1)685−57),−atan⁡(−6(−357−1)685−57)]∪[atan⁡(6(−1+357)657+85),∞)\left[- \pi + \operatorname{atan}{\left(- \frac{\sqrt{6} \left(- 3 \sqrt{57} - 1\right)}{6 \sqrt{85 - \sqrt{57}}} \right)}, - \operatorname{atan}{\left(- \frac{\sqrt{6} \left(- 3 \sqrt{57} - 1\right)}{6 \sqrt{85 - \sqrt{57}}} \right)}\right] \cup \left[\operatorname{atan}{\left(\frac{\sqrt{6} \left(-1 + 3 \sqrt{57}\right)}{6 \sqrt{\sqrt{57} + 85}} \right)}, \infty\right)
Convex at the intervals
(−∞,−π+atan⁡(−6(−357−1)685−57)]\left(-\infty, - \pi + \operatorname{atan}{\left(- \frac{\sqrt{6} \left(- 3 \sqrt{57} - 1\right)}{6 \sqrt{85 - \sqrt{57}}} \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⁡(x)+2cos⁡(2x))=⟨−3,3⟩\lim_{x \to -\infty}\left(- \sin{\left(x \right)} + 2 \cos{\left(2 x \right)}\right) = \left\langle -3, 3\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−3,3⟩y = \left\langle -3, 3\right\rangle
lim⁡x→∞(−sin⁡(x)+2cos⁡(2x))=⟨−3,3⟩\lim_{x \to \infty}\left(- \sin{\left(x \right)} + 2 \cos{\left(2 x \right)}\right) = \left\langle -3, 3\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−3,3⟩y = \left\langle -3, 3\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 2*cos(2*x) - sin(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(−sin⁡(x)+2cos⁡(2x)x)=0\lim_{x \to -\infty}\left(\frac{- \sin{\left(x \right)} + 2 \cos{\left(2 x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(−sin⁡(x)+2cos⁡(2x)x)=0\lim_{x \to \infty}\left(\frac{- \sin{\left(x \right)} + 2 \cos{\left(2 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⁡(x)+2cos⁡(2x)=sin⁡(x)+2cos⁡(2x)- \sin{\left(x \right)} + 2 \cos{\left(2 x \right)} = \sin{\left(x \right)} + 2 \cos{\left(2 x \right)}
- No
−sin⁡(x)+2cos⁡(2x)=−sin⁡(x)−2cos⁡(2x)- \sin{\left(x \right)} + 2 \cos{\left(2 x \right)} = - \sin{\left(x \right)} - 2 \cos{\left(2 x \right)}
- No
so, the function
not is
neither even, nor odd
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