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Plotting a function graph/ cos(2*x)

Graphing y = cos(2*x)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = cos(2*x)
f(x)=cos⁡(2x)f{\left(x \right)} = \cos{\left(2 x \right)} 
f = cos(2*x)
The graph of the function
0.001.000.100.200.300.400.500.600.700.800.901.001.01
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:
cos⁡(2x)=0\cos{\left(2 x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=π4x_{1} = \frac{\pi}{4}
x2=3π4x_{2} = \frac{3 \pi}{4}
Numerical solution
x1=−2.35619449019234x_{1} = -2.35619449019234
x2=41.6261026600648x_{2} = 41.6261026600648
x3=99.7455667514759x_{3} = 99.7455667514759
x4=−25.9181393921158x_{4} = -25.9181393921158
x5=384.059701901352x_{5} = 384.059701901352
x6=−38.484510006475x_{6} = -38.484510006475
x7=−84.037603483527x_{7} = -84.037603483527
x8=96.6039740978861x_{8} = 96.6039740978861
x9=−55.7632696012188x_{9} = -55.7632696012188
x10=−71.4712328691678x_{10} = -71.4712328691678
x11=−79.3252145031423x_{11} = -79.3252145031423
x12=49.4800842940392x_{12} = 49.4800842940392
x13=5.49778714378214x_{13} = 5.49778714378214
x14=82.4668071567321x_{14} = 82.4668071567321
x15=−11.7809724509617x_{15} = -11.7809724509617
x16=60.4756585816035x_{16} = 60.4756585816035
x17=−35.3429173528852x_{17} = -35.3429173528852
x18=38.484510006475x_{18} = 38.484510006475
x19=44.7676953136546x_{19} = 44.7676953136546
x20=76.1836218495525x_{20} = 76.1836218495525
x21=−10.2101761241668x_{21} = -10.2101761241668
x22=−77.7544181763474x_{22} = -77.7544181763474
x23=16.4933614313464x_{23} = 16.4933614313464
x24=69.9004365423729x_{24} = 69.9004365423729
x25=32.2013246992954x_{25} = 32.2013246992954
x26=74.6128255227576x_{26} = 74.6128255227576
x27=68.329640215578x_{27} = 68.329640215578
x28=−16.4933614313464x_{28} = -16.4933614313464
x29=−85.6083998103219x_{29} = -85.6083998103219
x30=22.776546738526x_{30} = 22.776546738526
x31=11.7809724509617x_{31} = 11.7809724509617
x32=−3.92699081698724x_{32} = -3.92699081698724
x33=90.3207887907066x_{33} = 90.3207887907066
x34=−82.4668071567321x_{34} = -82.4668071567321
x35=−41.6261026600648x_{35} = -41.6261026600648
x36=63.6172512351933x_{36} = 63.6172512351933
x37=−27.4889357189107x_{37} = -27.4889357189107
x38=−18.0641577581413x_{38} = -18.0641577581413
x39=−62.0464549083984x_{39} = -62.0464549083984
x40=−19.6349540849362x_{40} = -19.6349540849362
x41=1973.70558461779x_{41} = 1973.70558461779
x42=−47.9092879672443x_{42} = -47.9092879672443
x43=−32.2013246992954x_{43} = -32.2013246992954
x44=24.3473430653209x_{44} = 24.3473430653209
x45=−49.4800842940392x_{45} = -49.4800842940392
x46=−93.4623814442964x_{46} = -93.4623814442964
x47=47.9092879672443x_{47} = 47.9092879672443
x48=8.63937979737193x_{48} = 8.63937979737193
x49=162.577419823272x_{49} = 162.577419823272
x50=77.7544181763474x_{50} = 77.7544181763474
x51=54.1924732744239x_{51} = 54.1924732744239
x52=−98.174770424681x_{52} = -98.174770424681
x53=27.4889357189107x_{53} = 27.4889357189107
x54=19.6349540849362x_{54} = 19.6349540849362
x55=85.6083998103219x_{55} = 85.6083998103219
x56=−13.3517687777566x_{56} = -13.3517687777566
x57=55.7632696012188x_{57} = 55.7632696012188
x58=−76.1836218495525x_{58} = -76.1836218495525
x59=−5.49778714378214x_{59} = -5.49778714378214
x60=−68.329640215578x_{60} = -68.329640215578
x61=−60.4756585816035x_{61} = -60.4756585816035
x62=84.037603483527x_{62} = 84.037603483527
x63=3.92699081698724x_{63} = 3.92699081698724
x64=10.2101761241668x_{64} = 10.2101761241668
x65=−69.9004365423729x_{65} = -69.9004365423729
x66=−12461.9126586273x_{66} = -12461.9126586273
x67=52.621676947629x_{67} = 52.621676947629
x68=−33.7721210260903x_{68} = -33.7721210260903
x69=66.7588438887831x_{69} = 66.7588438887831
x70=−54.1924732744239x_{70} = -54.1924732744239
x71=33.7721210260903x_{71} = 33.7721210260903
x72=−46.3384916404494x_{72} = -46.3384916404494
x73=46.3384916404494x_{73} = 46.3384916404494
x74=−63.6172512351933x_{74} = -63.6172512351933
x75=−91.8915851175014x_{75} = -91.8915851175014
x76=2.35619449019234x_{76} = 2.35619449019234
x77=88.7499924639117x_{77} = 88.7499924639117
x78=25.9181393921158x_{78} = 25.9181393921158
x79=−24.3473430653209x_{79} = -24.3473430653209
x80=40.0553063332699x_{80} = 40.0553063332699
x81=91.8915851175014x_{81} = 91.8915851175014
x82=−90.3207887907066x_{82} = -90.3207887907066
x83=30.6305283725005x_{83} = 30.6305283725005
x84=−40.0553063332699x_{84} = -40.0553063332699
x85=98.174770424681x_{85} = 98.174770424681
x86=18.0641577581413x_{86} = 18.0641577581413
x87=62.0464549083984x_{87} = 62.0464549083984
x88=87.1791961371168x_{88} = 87.1791961371168
x89=−99.7455667514759x_{89} = -99.7455667514759
x90=−57.3340659280137x_{90} = -57.3340659280137
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cos(2*x).
cos⁡(0⋅2)\cos{\left(0 \cdot 2 \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
−2sin⁡(2x)=0- 2 \sin{\left(2 x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=π2x_{2} = \frac{\pi}{2}
The values of the extrema at the points:
(0, 1)

 pi     
(--, -1)
 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=π2x_{1} = \frac{\pi}{2}
Maxima of the function at points:
x1=0x_{1} = 0
Decreasing at intervals
(−∞,0]∪[π2,∞)\left(-\infty, 0\right] \cup \left[\frac{\pi}{2}, \infty\right)
Increasing at intervals
[0,π2]\left[0, \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
−4cos⁡(2x)=0- 4 \cos{\left(2 x \right)} = 0
Solve this equation
The roots of this equation
x1=π4x_{1} = \frac{\pi}{4}
x2=3π4x_{2} = \frac{3 \pi}{4}

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
[π4,3π4]\left[\frac{\pi}{4}, \frac{3 \pi}{4}\right]
Convex at the intervals
(−∞,π4]∪[3π4,∞)\left(-\infty, \frac{\pi}{4}\right] \cup \left[\frac{3 \pi}{4}, \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→−∞cos⁡(2x)=⟨−1,1⟩\lim_{x \to -\infty} \cos{\left(2 x \right)} = \left\langle -1, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
lim⁡x→∞cos⁡(2x)=⟨−1,1⟩\lim_{x \to \infty} \cos{\left(2 x \right)} = \left\langle -1, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cos(2*x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(cos⁡(2x)x)=0\lim_{x \to -\infty}\left(\frac{\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→∞(cos⁡(2x)x)=0\lim_{x \to \infty}\left(\frac{\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:
cos⁡(2x)=cos⁡(2x)\cos{\left(2 x \right)} = \cos{\left(2 x \right)}
- Yes
cos⁡(2x)=−cos⁡(2x)\cos{\left(2 x \right)} = - \cos{\left(2 x \right)}
- No
so, the function
is
even
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