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Plotting a function graph/ 8*cos(4*x)

Graphing y = 8*cos(4*x)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = 8*cos(4*x)
f(x)=8cos⁡(4x)f{\left(x \right)} = 8 \cos{\left(4 x \right)} 
f = 8*cos(4*x)
The graph of the function
02468-8-6-4-2-1010-2020
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:
8cos⁡(4x)=08 \cos{\left(4 x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=π8x_{1} = \frac{\pi}{8}
x2=3π8x_{2} = \frac{3 \pi}{8}
Numerical solution
x1=70.2931356240716x_{1} = 70.2931356240716
x2=40.4480054149686x_{2} = 40.4480054149686
x3=26.3108384738145x_{3} = 26.3108384738145
x4=93.8550805259951x_{4} = 93.8550805259951
x5=−65.5807466436869x_{5} = -65.5807466436869
x6=−82.0741080750334x_{6} = -82.0741080750334
x7=84.4303025652257x_{7} = 84.4303025652257
x8=−38.0918109247762x_{8} = -38.0918109247762
x9=82.0741080750334x_{9} = 82.0741080750334
x10=−93.8550805259951x_{10} = -93.8550805259951
x11=−67.9369411338793x_{11} = -67.9369411338793
x12=−68.7223392972767x_{12} = -68.7223392972767
x13=96.9966731795849x_{13} = 96.9966731795849
x14=−97.7820713429823x_{14} = -97.7820713429823
x15=−83.6449044018282x_{15} = -83.6449044018282
x16=86.0010988920206x_{16} = 86.0010988920206
x17=48.3019870489431x_{17} = 48.3019870489431
x18=66.3661448070844x_{18} = 66.3661448070844
x19=8.24668071567321x_{19} = 8.24668071567321
x20=−75.0055246044563x_{20} = -75.0055246044563
x21=27.8816348006094x_{21} = 27.8816348006094
x22=52.2289778659303x_{22} = 52.2289778659303
x23=12.1736715326604x_{23} = 12.1736715326604
x24=−96.9966731795849x_{24} = -96.9966731795849
x25=−31.8086256175967x_{25} = -31.8086256175967
x26=42.0188017417635x_{26} = 42.0188017417635
x27=−87.5718952188155x_{27} = -87.5718952188155
x28=−45.9457925587507x_{28} = -45.9457925587507
x29=−43.5895980685584x_{29} = -43.5895980685584
x30=92.2842841992002x_{30} = 92.2842841992002
x31=60.0829594999048x_{31} = 60.0829594999048
x32=−12.1736715326604x_{32} = -12.1736715326604
x33=74.2201264410589x_{33} = 74.2201264410589
x34=−100.138265833175x_{34} = -100.138265833175
x35=−39.6626072515711x_{35} = -39.6626072515711
x36=53.7997741927252x_{36} = 53.7997741927252
x37=−5.89048622548086x_{37} = -5.89048622548086
x38=−2.74889357189107x_{38} = -2.74889357189107
x39=−56.1559686829176x_{39} = -56.1559686829176
x40=5.89048622548086x_{40} = 5.89048622548086
x41=−61.6537558266997x_{41} = -61.6537558266997
x42=−57.7267650097125x_{42} = -57.7267650097125
x43=54.5851723561227x_{43} = 54.5851723561227
x44=−35.7356164345839x_{44} = -35.7356164345839
x45=31.8086256175967x_{45} = 31.8086256175967
x46=22.3838476568273x_{46} = 22.3838476568273
x47=34.164820107789x_{47} = 34.164820107789
x48=1.96349540849362x_{48} = 1.96349540849362
x49=4.31968989868597x_{49} = 4.31968989868597
x50=16.1006623496477x_{50} = 16.1006623496477
x51=64.009950316892x_{51} = 64.009950316892
x52=62.4391539900971x_{52} = 62.4391539900971
x53=104.850654813559x_{53} = 104.850654813559
x54=20.0276531666349x_{54} = 20.0276531666349
x55=44.3749962319558x_{55} = 44.3749962319558
x56=−9.8174770424681x_{56} = -9.8174770424681
x57=−75.7909227678538x_{57} = -75.7909227678538
x58=100.138265833175x_{58} = 100.138265833175
x59=−78.1471172580461x_{59} = -78.1471172580461
x60=−89.9280897090078x_{60} = -89.9280897090078
x61=38.0918109247762x_{61} = 38.0918109247762
x62=88.3572933822129x_{62} = 88.3572933822129
x63=−27.8816348006094x_{63} = -27.8816348006094
x64=23.9546439836222x_{64} = 23.9546439836222
x65=122.914812571701x_{65} = 122.914812571701
x66=−16.1006623496477x_{66} = -16.1006623496477
x67=−86.0010988920206x_{67} = -86.0010988920206
x68=9.8174770424681x_{68} = 9.8174770424681
x69=−21.5984494934298x_{69} = -21.5984494934298
x70=−42.0188017417635x_{70} = -42.0188017417635
x71=71.8639319508665x_{71} = 71.8639319508665
x72=45.9457925587507x_{72} = 45.9457925587507
x73=49.872783375738x_{73} = 49.872783375738
x74=96.2112750161874x_{74} = 96.2112750161874
x75=−10.6028752058656x_{75} = -10.6028752058656
x76=0.392699081698724x_{76} = 0.392699081698724
x77=−13.7444678594553x_{77} = -13.7444678594553
x78=56.1559686829176x_{78} = 56.1559686829176
x79=−53.7997741927252x_{79} = -53.7997741927252
x80=−34.164820107789x_{80} = -34.164820107789
x81=−49.872783375738x_{81} = -49.872783375738
x82=−17.6714586764426x_{82} = -17.6714586764426
x83=78.1471172580461x_{83} = 78.1471172580461
x84=−23.9546439836222x_{84} = -23.9546439836222
x85=−64.009950316892x_{85} = -64.009950316892
x86=−1.96349540849362x_{86} = -1.96349540849362
x87=−20.0276531666349x_{87} = -20.0276531666349
x88=−60.0829594999048x_{88} = -60.0829594999048
x89=89.9280897090078x_{89} = 89.9280897090078
x90=67.9369411338793x_{90} = 67.9369411338793
x91=−20.8130513300324x_{91} = -20.8130513300324
x92=30.2378292908018x_{92} = 30.2378292908018
x93=−71.8639319508665x_{93} = -71.8639319508665
x94=−79.717913584841x_{94} = -79.717913584841
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 8*cos(4*x).
8cos⁡(0⋅4)8 \cos{\left(0 \cdot 4 \right)}
The result:
f(0)=8f{\left(0 \right)} = 8
The point:
(0, 8)
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
−32sin⁡(4x)=0- 32 \sin{\left(4 x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=π4x_{2} = \frac{\pi}{4}
The values of the extrema at the points:
(0, 8)

 pi     
(--, -8)
 4


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=π4x_{1} = \frac{\pi}{4}
Maxima of the function at points:
x1=0x_{1} = 0
Decreasing at intervals
(−∞,0]∪[π4,∞)\left(-\infty, 0\right] \cup \left[\frac{\pi}{4}, \infty\right)
Increasing at intervals
[0,π4]\left[0, \frac{\pi}{4}\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
−128cos⁡(4x)=0- 128 \cos{\left(4 x \right)} = 0
Solve this equation
The roots of this equation
x1=π8x_{1} = \frac{\pi}{8}
x2=3π8x_{2} = \frac{3 \pi}{8}

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
[π8,3π8]\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right]
Convex at the intervals
(−∞,π8]∪[3π8,∞)\left(-\infty, \frac{\pi}{8}\right] \cup \left[\frac{3 \pi}{8}, \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→−∞(8cos⁡(4x))=⟨−8,8⟩\lim_{x \to -\infty}\left(8 \cos{\left(4 x \right)}\right) = \left\langle -8, 8\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−8,8⟩y = \left\langle -8, 8\right\rangle
lim⁡x→∞(8cos⁡(4x))=⟨−8,8⟩\lim_{x \to \infty}\left(8 \cos{\left(4 x \right)}\right) = \left\langle -8, 8\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−8,8⟩y = \left\langle -8, 8\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 8*cos(4*x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(8cos⁡(4x)x)=0\lim_{x \to -\infty}\left(\frac{8 \cos{\left(4 x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(8cos⁡(4x)x)=0\lim_{x \to \infty}\left(\frac{8 \cos{\left(4 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:
8cos⁡(4x)=8cos⁡(4x)8 \cos{\left(4 x \right)} = 8 \cos{\left(4 x \right)}
- Yes
8cos⁡(4x)=−8cos⁡(4x)8 \cos{\left(4 x \right)} = - 8 \cos{\left(4 x \right)}
- No
so, the function
is
even
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