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(abs(cos(3x)))
Plotting a function graph/ (abs(cos(3x)))

Graphing y = (abs(cos(3x)))

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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)} = \left|{\cos{\left(3 x \right)}}\right| 
f = Abs(cos(3*x))
The graph of the function
05-35-30-25-20-15-10-51002
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\left|{\cos{\left(3 x \right)}}\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=−8.90117918517108x_{1} = -8.90117918517108
x2=−29.845130209103x_{2} = -29.845130209103
x3=−36.1283155162826x_{3} = -36.1283155162826
x4=84.2994028713261x_{4} = 84.2994028713261
x5=9.94837673636768x_{5} = 9.94837673636768
x6=−39.2699081698724x_{6} = -39.2699081698724
x7=62.3082542961976x_{7} = 62.3082542961976
x8=−5.75958653158129x_{8} = -5.75958653158129
x9=100.007366139275x_{9} = 100.007366139275
x10=−78.0162175641465x_{10} = -78.0162175641465
x11=75.9218224617533x_{11} = 75.9218224617533
x12=−18.3259571459405x_{12} = -18.3259571459405
x13=15.1843644923507x_{13} = 15.1843644923507
x14=38.2227106186758x_{14} = 38.2227106186758
x15=−100.007366139275x_{15} = -100.007366139275
x16=12.0427718387609x_{16} = 12.0427718387609
x17=−27.7507351067098x_{17} = -27.7507351067098
x18=−34.0339204138894x_{18} = -34.0339204138894
x19=91.6297857297023x_{19} = 91.6297857297023
x20=−1.5707963267949x_{20} = -1.5707963267949
x21=73.8274273593601x_{21} = 73.8274273593601
x22=51.8362787842316x_{22} = 51.8362787842316
x23=−58.1194640914112x_{23} = -58.1194640914112
x24=−3.66519142918809x_{24} = -3.66519142918809
x25=−91.6297857297023x_{25} = -91.6297857297023
x26=−80.1106126665397x_{26} = -80.1106126665397
x27=18.3259571459405x_{27} = 18.3259571459405
x28=78.0162175641465x_{28} = 78.0162175641465
x29=27.7507351067098x_{29} = 27.7507351067098
x30=−71.733032256967x_{30} = -71.733032256967
x31=40.317105721069x_{31} = 40.317105721069
x32=−51.8362787842316x_{32} = -51.8362787842316
x33=−12.0427718387609x_{33} = -12.0427718387609
x34=7.85398163397448x_{34} = 7.85398163397448
x35=−25.6563400043166x_{35} = -25.6563400043166
x36=−45.553093477052x_{36} = -45.553093477052
x37=86.3937979737193x_{37} = 86.3937979737193
x38=−73.8274273593601x_{38} = -73.8274273593601
x39=97.9129710368819x_{39} = 97.9129710368819
x40=2.61799387799149x_{40} = 2.61799387799149
x41=−47.6474885794452x_{41} = -47.6474885794452
x42=−95.8185759344887x_{42} = -95.8185759344887
x43=56.025068989018x_{43} = 56.025068989018
x44=20.4203522483337x_{44} = 20.4203522483337
x45=−56.025068989018x_{45} = -56.025068989018
x46=64.4026493985908x_{46} = 64.4026493985908
x47=34.0339204138894x_{47} = 34.0339204138894
x48=31.9395253114962x_{48} = 31.9395253114962
x49=53.9306738866248x_{49} = 53.9306738866248
x50=42.4115008234622x_{50} = 42.4115008234622
x51=16.2315620435473x_{51} = 16.2315620435473
x52=−947782.375272524x_{52} = -947782.375272524
x53=−14.1371669411541x_{53} = -14.1371669411541
x54=82.2050077689329x_{54} = 82.2050077689329
x55=−31.9395253114962x_{55} = -31.9395253114962
x56=95.8185759344887x_{56} = 95.8185759344887
x57=−93.7241808320955x_{57} = -93.7241808320955
x58=54.9778714378214x_{58} = 54.9778714378214
x59=−49.7418836818384x_{59} = -49.7418836818384
x60=−69.6386371545737x_{60} = -69.6386371545737
x61=60.2138591938044x_{61} = 60.2138591938044
x62=29.845130209103x_{62} = 29.845130209103
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to Abs(cos(3*x)).
∣cos⁡(3⋅0)∣\left|{\cos{\left(3 \cdot 0 \right)}}\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)sign⁡(cos⁡(3x))=0- 3 \sin{\left(3 x \right)} \operatorname{sign}{\left(\cos{\left(3 x \right)} \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:
The function has no minima
Maxima of the function at points:
x2=0x_{2} = 0
x2=π3x_{2} = \frac{\pi}{3}
Decreasing at intervals
(−∞,0]\left(-\infty, 0\right]
Increasing at intervals
[π3,∞)\left[\frac{\pi}{3}, \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} \left|{\cos{\left(3 x \right)}}\right| = \left|{\left\langle -1, 1\right\rangle}\right|
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=∣⟨−1,1⟩∣y = \left|{\left\langle -1, 1\right\rangle}\right|
lim⁡x→∞∣cos⁡(3x)∣=∣⟨−1,1⟩∣\lim_{x \to \infty} \left|{\cos{\left(3 x \right)}}\right| = \left|{\left\langle -1, 1\right\rangle}\right|
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=∣⟨−1,1⟩∣y = \left|{\left\langle -1, 1\right\rangle}\right|
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of Abs(cos(3*x)), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(∣cos⁡(3x)∣x)=0\lim_{x \to -\infty}\left(\frac{\left|{\cos{\left(3 x \right)}}\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{\left|{\cos{\left(3 x \right)}}\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)∣\left|{\cos{\left(3 x \right)}}\right| = \left|{\cos{\left(3 x \right)}}\right|
- Yes
∣cos⁡(3x)∣=−∣cos⁡(3x)∣\left|{\cos{\left(3 x \right)}}\right| = - \left|{\cos{\left(3 x \right)}}\right|
- No
so, the function
is
even
The graph
Graphing y = (abs(cos(3x)))
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