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Plotting a function graph/ cos(3*x)

Graphing y = cos(3*x)

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
2.003.002.102.202.302.402.502.602.702.802.902-2
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=82.2050077689329x_{1} = 82.2050077689329
x2=1.5707963267949x_{2} = 1.5707963267949
x3=62.3082542961976x_{3} = 62.3082542961976
x4=97.9129710368819x_{4} = 97.9129710368819
x5=−69.6386371545737x_{5} = -69.6386371545737
x6=14.1371669411541x_{6} = 14.1371669411541
x7=−36.1283155162826x_{7} = -36.1283155162826
x8=44.5058959258554x_{8} = 44.5058959258554
x9=7.85398163397448x_{9} = 7.85398163397448
x10=42.4115008234622x_{10} = 42.4115008234622
x11=9.94837673636768x_{11} = 9.94837673636768
x12=12.0427718387609x_{12} = 12.0427718387609
x13=91.6297857297023x_{13} = 91.6297857297023
x14=−38.2227106186758x_{14} = -38.2227106186758
x15=−89.5353906273091x_{15} = -89.5353906273091
x16=−49.7418836818384x_{16} = -49.7418836818384
x17=−75.9218224617533x_{17} = -75.9218224617533
x18=−67.5442420521806x_{18} = -67.5442420521806
x19=−51.8362787842316x_{19} = -51.8362787842316
x20=−19.3731546971371x_{20} = -19.3731546971371
x21=−14.1371669411541x_{21} = -14.1371669411541
x22=64.4026493985908x_{22} = 64.4026493985908
x23=−47.6474885794452x_{23} = -47.6474885794452
x24=71.733032256967x_{24} = 71.733032256967
x25=93.7241808320955x_{25} = 93.7241808320955
x26=38.2227106186758x_{26} = 38.2227106186758
x27=88.4881930761125x_{27} = 88.4881930761125
x28=34.0339204138894x_{28} = 34.0339204138894
x29=−31.9395253114962x_{29} = -31.9395253114962
x30=−62.3082542961976x_{30} = -62.3082542961976
x31=60.2138591938044x_{31} = 60.2138591938044
x32=−80.1106126665397x_{32} = -80.1106126665397
x33=−29.845130209103x_{33} = -29.845130209103
x34=−78.0162175641465x_{34} = -78.0162175641465
x35=4.71238898038469x_{35} = 4.71238898038469
x36=−16.2315620435473x_{36} = -16.2315620435473
x37=−93.7241808320955x_{37} = -93.7241808320955
x38=20.4203522483337x_{38} = 20.4203522483337
x39=5.75958653158129x_{39} = 5.75958653158129
x40=87.4409955249159x_{40} = 87.4409955249159
x41=75.9218224617533x_{41} = 75.9218224617533
x42=−1.5707963267949x_{42} = -1.5707963267949
x43=−82.2050077689329x_{43} = -82.2050077689329
x44=−21.4675497995303x_{44} = -21.4675497995303
x45=−25.6563400043166x_{45} = -25.6563400043166
x46=86.3937979737193x_{46} = 86.3937979737193
x47=−60.2138591938044x_{47} = -60.2138591938044
x48=−43.4586983746588x_{48} = -43.4586983746588
x49=68.5914396033772x_{49} = 68.5914396033772
x50=−91.6297857297023x_{50} = -91.6297857297023
x51=15.1843644923507x_{51} = 15.1843644923507
x52=36.1283155162826x_{52} = 36.1283155162826
x53=22.5147473507269x_{53} = 22.5147473507269
x54=27.7507351067098x_{54} = 27.7507351067098
x55=51.8362787842316x_{55} = 51.8362787842316
x56=−12.0427718387609x_{56} = -12.0427718387609
x57=80.1106126665397x_{57} = 80.1106126665397
x58=56.025068989018x_{58} = 56.025068989018
x59=−56.025068989018x_{59} = -56.025068989018
x60=41.3643032722656x_{60} = 41.3643032722656
x61=78.0162175641465x_{61} = 78.0162175641465
x62=1676.03968069015x_{62} = 1676.03968069015
x63=18.3259571459405x_{63} = 18.3259571459405
x64=58.1194640914112x_{64} = 58.1194640914112
x65=73.8274273593601x_{65} = 73.8274273593601
x66=−58.1194640914112x_{66} = -58.1194640914112
x67=−65.4498469497874x_{67} = -65.4498469497874
x68=−9.94837673636768x_{68} = -9.94837673636768
x69=−34.0339204138894x_{69} = -34.0339204138894
x70=−100.007366139275x_{70} = -100.007366139275
x71=−95.8185759344887x_{71} = -95.8185759344887
x72=29.845130209103x_{72} = 29.845130209103
x73=26.7035375555132x_{73} = 26.7035375555132
x74=49.7418836818384x_{74} = 49.7418836818384
x75=−23.5619449019235x_{75} = -23.5619449019235
x76=−84.2994028713261x_{76} = -84.2994028713261
x77=40.317105721069x_{77} = 40.317105721069
x78=−97.9129710368819x_{78} = -97.9129710368819
x79=−61.261056745001x_{79} = -61.261056745001
x80=100.007366139275x_{80} = 100.007366139275
x81=−53.9306738866248x_{81} = -53.9306738866248
x82=−98.9601685880785x_{82} = -98.9601685880785
x83=31.9395253114962x_{83} = 31.9395253114962
x84=−3.66519142918809x_{84} = -3.66519142918809
x85=0.523598775598299x_{85} = 0.523598775598299
x86=84.2994028713261x_{86} = 84.2994028713261
x87=66.497044500984x_{87} = 66.497044500984
x88=53.9306738866248x_{88} = 53.9306738866248
x89=−73.8274273593601x_{89} = -73.8274273593601
x90=16.2315620435473x_{90} = 16.2315620435473
x91=−71.733032256967x_{91} = -71.733032256967
x92=−45.553093477052x_{92} = -45.553093477052
x93=−7.85398163397448x_{93} = -7.85398163397448
x94=−5.75958653158129x_{94} = -5.75958653158129
x95=95.8185759344887x_{95} = 95.8185759344887
x96=−27.7507351067098x_{96} = -27.7507351067098
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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