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Plotting a function graph/ cos3x

Graphing y = cos3x

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = cos(3*x)
f(x)=cos⁡(3x)f{\left(x \right)} = \cos{\left(3 x \right)} 
f = cos(3*x)
The graph of the function
02468-8-6-4-2-10102-2
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⁡(3x)=0\cos{\left(3 x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=π6x_{1} = \frac{\pi}{6}
x2=π2x_{2} = \frac{\pi}{2}
Numerical solution
x1=60.2138591938044x_{1} = 60.2138591938044
x2=73.8274273593601x_{2} = 73.8274273593601
x3=84.2994028713261x_{3} = 84.2994028713261
x4=−93.7241808320955x_{4} = -93.7241808320955
x5=14.1371669411541x_{5} = 14.1371669411541
x6=58.1194640914112x_{6} = 58.1194640914112
x7=9.94837673636768x_{7} = 9.94837673636768
x8=91.6297857297023x_{8} = 91.6297857297023
x9=40.317105721069x_{9} = 40.317105721069
x10=36.1283155162826x_{10} = 36.1283155162826
x11=−7.85398163397448x_{11} = -7.85398163397448
x12=−58.1194640914112x_{12} = -58.1194640914112
x13=86.3937979737193x_{13} = 86.3937979737193
x14=51.8362787842316x_{14} = 51.8362787842316
x15=−89.5353906273091x_{15} = -89.5353906273091
x16=1.5707963267949x_{16} = 1.5707963267949
x17=−82.2050077689329x_{17} = -82.2050077689329
x18=87.4409955249159x_{18} = 87.4409955249159
x19=−5.75958653158129x_{19} = -5.75958653158129
x20=−97.9129710368819x_{20} = -97.9129710368819
x21=56.025068989018x_{21} = 56.025068989018
x22=−75.9218224617533x_{22} = -75.9218224617533
x23=−3.66519142918809x_{23} = -3.66519142918809
x24=5.75958653158129x_{24} = 5.75958653158129
x25=−31.9395253114962x_{25} = -31.9395253114962
x26=−36.1283155162826x_{26} = -36.1283155162826
x27=−19.3731546971371x_{27} = -19.3731546971371
x28=−49.7418836818384x_{28} = -49.7418836818384
x29=0.523598775598299x_{29} = 0.523598775598299
x30=−16.2315620435473x_{30} = -16.2315620435473
x31=−14.1371669411541x_{31} = -14.1371669411541
x32=80.1106126665397x_{32} = 80.1106126665397
x33=95.8185759344887x_{33} = 95.8185759344887
x34=34.0339204138894x_{34} = 34.0339204138894
x35=12.0427718387609x_{35} = 12.0427718387609
x36=4.71238898038469x_{36} = 4.71238898038469
x37=49.7418836818384x_{37} = 49.7418836818384
x38=20.4203522483337x_{38} = 20.4203522483337
x39=−23.5619449019235x_{39} = -23.5619449019235
x40=−51.8362787842316x_{40} = -51.8362787842316
x41=−29.845130209103x_{41} = -29.845130209103
x42=27.7507351067098x_{42} = 27.7507351067098
x43=7.85398163397448x_{43} = 7.85398163397448
x44=16.2315620435473x_{44} = 16.2315620435473
x45=−62.3082542961976x_{45} = -62.3082542961976
x46=−95.8185759344887x_{46} = -95.8185759344887
x47=−9.94837673636768x_{47} = -9.94837673636768
x48=−69.6386371545737x_{48} = -69.6386371545737
x49=−27.7507351067098x_{49} = -27.7507351067098
x50=78.0162175641465x_{50} = 78.0162175641465
x51=−56.025068989018x_{51} = -56.025068989018
x52=−12.0427718387609x_{52} = -12.0427718387609
x53=88.4881930761125x_{53} = 88.4881930761125
x54=53.9306738866248x_{54} = 53.9306738866248
x55=−61.261056745001x_{55} = -61.261056745001
x56=−43.4586983746588x_{56} = -43.4586983746588
x57=−67.5442420521806x_{57} = -67.5442420521806
x58=−21.4675497995303x_{58} = -21.4675497995303
x59=31.9395253114962x_{59} = 31.9395253114962
x60=18.3259571459405x_{60} = 18.3259571459405
x61=−84.2994028713261x_{61} = -84.2994028713261
x62=66.497044500984x_{62} = 66.497044500984
x63=−45.553093477052x_{63} = -45.553093477052
x64=−71.733032256967x_{64} = -71.733032256967
x65=42.4115008234622x_{65} = 42.4115008234622
x66=−60.2138591938044x_{66} = -60.2138591938044
x67=68.5914396033772x_{67} = 68.5914396033772
x68=−73.8274273593601x_{68} = -73.8274273593601
x69=−25.6563400043166x_{69} = -25.6563400043166
x70=−1.5707963267949x_{70} = -1.5707963267949
x71=93.7241808320955x_{71} = 93.7241808320955
x72=1676.03968069015x_{72} = 1676.03968069015
x73=22.5147473507269x_{73} = 22.5147473507269
x74=−34.0339204138894x_{74} = -34.0339204138894
x75=100.007366139275x_{75} = 100.007366139275
x76=−91.6297857297023x_{76} = -91.6297857297023
x77=−47.6474885794452x_{77} = -47.6474885794452
x78=75.9218224617533x_{78} = 75.9218224617533
x79=82.2050077689329x_{79} = 82.2050077689329
x80=97.9129710368819x_{80} = 97.9129710368819
x81=−53.9306738866248x_{81} = -53.9306738866248
x82=−78.0162175641465x_{82} = -78.0162175641465
x83=26.7035375555132x_{83} = 26.7035375555132
x84=−98.9601685880785x_{84} = -98.9601685880785
x85=−38.2227106186758x_{85} = -38.2227106186758
x86=44.5058959258554x_{86} = 44.5058959258554
x87=15.1843644923507x_{87} = 15.1843644923507
x88=71.733032256967x_{88} = 71.733032256967
x89=−80.1106126665397x_{89} = -80.1106126665397
x90=−100.007366139275x_{90} = -100.007366139275
x91=64.4026493985908x_{91} = 64.4026493985908
x92=29.845130209103x_{92} = 29.845130209103
x93=62.3082542961976x_{93} = 62.3082542961976
x94=41.3643032722656x_{94} = 41.3643032722656
x95=−65.4498469497874x_{95} = -65.4498469497874
x96=38.2227106186758x_{96} = 38.2227106186758
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cos(3*x).
cos⁡(0⋅3)\cos{\left(0 \cdot 3 \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
−3sin⁡(3x)=0- 3 \sin{\left(3 x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=π3x_{2} = \frac{\pi}{3}
The values of the extrema at the points:
(0, 1)

 pi     
(--, -1)
 3


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=π3x_{1} = \frac{\pi}{3}
Maxima of the function at points:
x1=0x_{1} = 0
Decreasing at intervals
(−∞,0]∪[π3,∞)\left(-\infty, 0\right] \cup \left[\frac{\pi}{3}, \infty\right)
Increasing at intervals
[0,π3]\left[0, \frac{\pi}{3}\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
−9cos⁡(3x)=0- 9 \cos{\left(3 x \right)} = 0
Solve this equation
The roots of this equation
x1=π6x_{1} = \frac{\pi}{6}
x2=π2x_{2} = \frac{\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
[π6,π2]\left[\frac{\pi}{6}, \frac{\pi}{2}\right]
Convex at the intervals
(−∞,π6]∪[π2,∞)\left(-\infty, \frac{\pi}{6}\right] \cup \left[\frac{\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→−∞cos⁡(3x)=⟨−1,1⟩\lim_{x \to -\infty} \cos{\left(3 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⁡(3x)=⟨−1,1⟩\lim_{x \to \infty} \cos{\left(3 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(3*x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(cos⁡(3x)x)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(3 x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(cos⁡(3x)x)=0\lim_{x \to \infty}\left(\frac{\cos{\left(3 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⁡(3x)=cos⁡(3x)\cos{\left(3 x \right)} = \cos{\left(3 x \right)}
- Yes
cos⁡(3x)=−cos⁡(3x)\cos{\left(3 x \right)} = - \cos{\left(3 x \right)}
- No
so, the function
is
even
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