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Plotting a function graph/ 6cosx

Graphing y = 6cosx

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The graph:

from to

Intersection points:

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Piecewise:

The solution

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f(x) = 6*cos(x)
f(x)=6cos⁡(x)f{\left(x \right)} = 6 \cos{\left(x \right)} 
f = 6*cos(x)
The graph of the function
02468-8-6-4-2-1010-1010
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:
6cos⁡(x)=06 \cos{\left(x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=π2x_{1} = \frac{\pi}{2}
x2=3π2x_{2} = \frac{3 \pi}{2}
Numerical solution
x1=−1.5707963267949x_{1} = -1.5707963267949
x2=−64.4026493985908x_{2} = -64.4026493985908
x3=76.9690200129499x_{3} = 76.9690200129499
x4=−23.5619449019235x_{4} = -23.5619449019235
x5=−58.1194640914112x_{5} = -58.1194640914112
x6=61.261056745001x_{6} = 61.261056745001
x7=80.1106126665397x_{7} = 80.1106126665397
x8=−48.6946861306418x_{8} = -48.6946861306418
x9=−29.845130209103x_{9} = -29.845130209103
x10=−4.71238898038469x_{10} = -4.71238898038469
x11=−86.3937979737193x_{11} = -86.3937979737193
x12=−36.1283155162826x_{12} = -36.1283155162826
x13=−98.9601685880785x_{13} = -98.9601685880785
x14=1.5707963267949x_{14} = 1.5707963267949
x15=−39.2699081698724x_{15} = -39.2699081698724
x16=73.8274273593601x_{16} = 73.8274273593601
x17=−92.6769832808989x_{17} = -92.6769832808989
x18=42.4115008234622x_{18} = 42.4115008234622
x19=67.5442420521806x_{19} = 67.5442420521806
x20=−32.9867228626928x_{20} = -32.9867228626928
x21=14.1371669411541x_{21} = 14.1371669411541
x22=4.71238898038469x_{22} = 4.71238898038469
x23=32.9867228626928x_{23} = 32.9867228626928
x24=−10.9955742875643x_{24} = -10.9955742875643
x25=70.6858347057703x_{25} = 70.6858347057703
x26=36.1283155162826x_{26} = 36.1283155162826
x27=20.4203522483337x_{27} = 20.4203522483337
x28=−70.6858347057703x_{28} = -70.6858347057703
x29=−26.7035375555132x_{29} = -26.7035375555132
x30=10.9955742875643x_{30} = 10.9955742875643
x31=23.5619449019235x_{31} = 23.5619449019235
x32=45.553093477052x_{32} = 45.553093477052
x33=83.2522053201295x_{33} = 83.2522053201295
x34=−67.5442420521806x_{34} = -67.5442420521806
x35=−89.5353906273091x_{35} = -89.5353906273091
x36=−54.9778714378214x_{36} = -54.9778714378214
x37=95.8185759344887x_{37} = 95.8185759344887
x38=−17.2787595947439x_{38} = -17.2787595947439
x39=26.7035375555132x_{39} = 26.7035375555132
x40=17.2787595947439x_{40} = 17.2787595947439
x41=−42.4115008234622x_{41} = -42.4115008234622
x42=54.9778714378214x_{42} = 54.9778714378214
x43=−2266.65909956504x_{43} = -2266.65909956504
x44=−7.85398163397448x_{44} = -7.85398163397448
x45=48.6946861306418x_{45} = 48.6946861306418
x46=−51.8362787842316x_{46} = -51.8362787842316
x47=89.5353906273091x_{47} = 89.5353906273091
x48=92.6769832808989x_{48} = 92.6769832808989
x49=58.1194640914112x_{49} = 58.1194640914112
x50=−80.1106126665397x_{50} = -80.1106126665397
x51=−73.8274273593601x_{51} = -73.8274273593601
x52=86.3937979737193x_{52} = 86.3937979737193
x53=−76.9690200129499x_{53} = -76.9690200129499
x54=51.8362787842316x_{54} = 51.8362787842316
x55=39.2699081698724x_{55} = 39.2699081698724
x56=−20.4203522483337x_{56} = -20.4203522483337
x57=64.4026493985908x_{57} = 64.4026493985908
x58=−3626.96871856942x_{58} = -3626.96871856942
x59=−83.2522053201295x_{59} = -83.2522053201295
x60=98.9601685880785x_{60} = 98.9601685880785
x61=7.85398163397448x_{61} = 7.85398163397448
x62=−95.8185759344887x_{62} = -95.8185759344887
x63=−14.1371669411541x_{63} = -14.1371669411541
x64=−387.986692718339x_{64} = -387.986692718339
x65=29.845130209103x_{65} = 29.845130209103
x66=−45.553093477052x_{66} = -45.553093477052
x67=−61.261056745001x_{67} = -61.261056745001
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 6*cos(x).
6cos⁡(0)6 \cos{\left(0 \right)}
The result:
f(0)=6f{\left(0 \right)} = 6
The point:
(0, 6)
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
−6sin⁡(x)=0- 6 \sin{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=πx_{2} = \pi
The values of the extrema at the points:
(0, 6)

(pi, -6)


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