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Plotting a function graph/ 3sin(2x)-16cos(x)

Graphing y = 3sin(2x)-16cos(x)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = 3*sin(2*x) - 16*cos(x)
f(x)=3sin⁡(2x)−16cos⁡(x)f{\left(x \right)} = 3 \sin{\left(2 x \right)} - 16 \cos{\left(x \right)} 
f = 3*sin(2*x) - 16*cos(x)
The graph of the function
02468-8-6-4-2-1010-5050
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:
3sin⁡(2x)−16cos⁡(x)=03 \sin{\left(2 x \right)} - 16 \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=π2x_{2} = \frac{\pi}{2}
Numerical solution
x1=−139.800873084746x_{1} = -139.800873084746
x2=−1.5707963267949x_{2} = -1.5707963267949
x3=−64.4026493985908x_{3} = -64.4026493985908
x4=76.9690200129499x_{4} = 76.9690200129499
x5=−23.5619449019235x_{5} = -23.5619449019235
x6=−58.1194640914112x_{6} = -58.1194640914112
x7=61.261056745001x_{7} = 61.261056745001
x8=80.1106126665397x_{8} = 80.1106126665397
x9=−48.6946861306418x_{9} = -48.6946861306418
x10=−29.845130209103x_{10} = -29.845130209103
x11=−4.71238898038469x_{11} = -4.71238898038469
x12=−86.3937979737193x_{12} = -86.3937979737193
x13=−36.1283155162826x_{13} = -36.1283155162826
x14=−98.9601685880785x_{14} = -98.9601685880785
x15=1.5707963267949x_{15} = 1.5707963267949
x16=−39.2699081698724x_{16} = -39.2699081698724
x17=73.8274273593601x_{17} = 73.8274273593601
x18=−92.6769832808989x_{18} = -92.6769832808989
x19=42.4115008234622x_{19} = 42.4115008234622
x20=67.5442420521806x_{20} = 67.5442420521806
x21=−32.9867228626928x_{21} = -32.9867228626928
x22=14.1371669411541x_{22} = 14.1371669411541
x23=4.71238898038469x_{23} = 4.71238898038469
x24=32.9867228626928x_{24} = 32.9867228626928
x25=−10.9955742875643x_{25} = -10.9955742875643
x26=70.6858347057703x_{26} = 70.6858347057703
x27=36.1283155162826x_{27} = 36.1283155162826
x28=20.4203522483337x_{28} = 20.4203522483337
x29=−70.6858347057703x_{29} = -70.6858347057703
x30=−26.7035375555132x_{30} = -26.7035375555132
x31=10.9955742875643x_{31} = 10.9955742875643
x32=23.5619449019235x_{32} = 23.5619449019235
x33=45.553093477052x_{33} = 45.553093477052
x34=83.2522053201295x_{34} = 83.2522053201295
x35=−67.5442420521806x_{35} = -67.5442420521806
x36=−89.5353906273091x_{36} = -89.5353906273091
x37=−54.9778714378214x_{37} = -54.9778714378214
x38=95.8185759344887x_{38} = 95.8185759344887
x39=−17.2787595947439x_{39} = -17.2787595947439
x40=26.7035375555132x_{40} = 26.7035375555132
x41=17.2787595947439x_{41} = 17.2787595947439
x42=177.499984927823x_{42} = 177.499984927823
x43=−42.4115008234622x_{43} = -42.4115008234622
x44=54.9778714378214x_{44} = 54.9778714378214
x45=−7.85398163397448x_{45} = -7.85398163397448
x46=48.6946861306418x_{46} = 48.6946861306418
x47=−51.8362787842316x_{47} = -51.8362787842316
x48=89.5353906273091x_{48} = 89.5353906273091
x49=92.6769832808989x_{49} = 92.6769832808989
x50=58.1194640914112x_{50} = 58.1194640914112
x51=−80.1106126665397x_{51} = -80.1106126665397
x52=−73.8274273593601x_{52} = -73.8274273593601
x53=86.3937979737193x_{53} = 86.3937979737193
x54=−76.9690200129499x_{54} = -76.9690200129499
x55=365.995544143211x_{55} = 365.995544143211
x56=51.8362787842316x_{56} = 51.8362787842316
x57=350.287580875262x_{57} = 350.287580875262
x58=39.2699081698724x_{58} = 39.2699081698724
x59=−20.4203522483337x_{59} = -20.4203522483337
x60=64.4026493985908x_{60} = 64.4026493985908
x61=−83.2522053201295x_{61} = -83.2522053201295
x62=98.9601685880785x_{62} = 98.9601685880785
x63=7.85398163397448x_{63} = 7.85398163397448
x64=−95.8185759344887x_{64} = -95.8185759344887
x65=−14.1371669411541x_{65} = -14.1371669411541
x66=29.845130209103x_{66} = 29.845130209103
x67=−45.553093477052x_{67} = -45.553093477052
x68=−61.261056745001x_{68} = -61.261056745001
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 3*sin(2*x) - 16*cos(x).
−16cos⁡(0)+3sin⁡(0⋅2)- 16 \cos{\left(0 \right)} + 3 \sin{\left(0 \cdot 2 \right)}
The result:
f(0)=−16f{\left(0 \right)} = -16
The point:
(0, -16)
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
16sin⁡(x)+6cos⁡(2x)=016 \sin{\left(x \right)} + 6 \cos{\left(2 x \right)} = 0
Solve this equation
The roots of this equation
x1=−ilog⁡(−2−7+4346+i(4−34)6)x_{1} = - i \log{\left(- \frac{\sqrt{2} \sqrt{-7 + 4 \sqrt{34}}}{6} + \frac{i \left(4 - \sqrt{34}\right)}{6} \right)}
x2=−ilog⁡(2−7+4346+i(4−34)6)x_{2} = - i \log{\left(\frac{\sqrt{2} \sqrt{-7 + 4 \sqrt{34}}}{6} + \frac{i \left(4 - \sqrt{34}\right)}{6} \right)}
The values of the extrema at the points:
       /           _______________                 \          /     /           _______________                 \\        /       /           _______________                 \\ 
       |    ___   /          ____      /      ____\|          |     |    ___   /          ____      /      ____\||        |       |    ___   /          ____      /      ____\|| 
       |  \/ 2 *\/  -7 + 4*\/ 34     I*\4 - \/ 34 /|          |     |  \/ 2 *\/  -7 + 4*\/ 34     I*\4 - \/ 34 /||        |       |  \/ 2 *\/  -7 + 4*\/ 34     I*\4 - \/ 34 /|| 
(-I*log|- ------------------------ + --------------|, - 16*cos|I*log|- ------------------------ + --------------|| - 3*sin|2*I*log|- ------------------------ + --------------||)
       \             6                     6       /          \     \             6                     6       //        \       \             6                     6       // 

       /                          _______________\          /     /                          _______________\\        /       /                          _______________\\ 
       |  /      ____\     ___   /          ____ |          |     |  /      ____\     ___   /          ____ ||        |       |  /      ____\     ___   /          ____ || 
       |I*\4 - \/ 34 /   \/ 2 *\/  -7 + 4*\/ 34  |          |     |I*\4 - \/ 34 /   \/ 2 *\/  -7 + 4*\/ 34  ||        |       |I*\4 - \/ 34 /   \/ 2 *\/  -7 + 4*\/ 34  || 
(-I*log|-------------- + ------------------------|, - 16*cos|I*log|-------------- + ------------------------|| - 3*sin|2*I*log|-------------- + ------------------------||)
       \      6                     6            /          \     \      6                     6            //        \       \      6                     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=atan⁡(2(4−34)2−7+434)x_{1} = \operatorname{atan}{\left(\frac{\sqrt{2} \left(4 - \sqrt{34}\right)}{2 \sqrt{-7 + 4 \sqrt{34}}} \right)}
Maxima of the function at points:
x1=−π−atan⁡(2(4−34)2−7+434)x_{1} = - \pi - \operatorname{atan}{\left(\frac{\sqrt{2} \left(4 - \sqrt{34}\right)}{2 \sqrt{-7 + 4 \sqrt{34}}} \right)}
Decreasing at intervals
(−∞,−π−atan⁡(2(4−34)2−7+434)]∪[atan⁡(2(4−34)2−7+434),∞)\left(-\infty, - \pi - \operatorname{atan}{\left(\frac{\sqrt{2} \left(4 - \sqrt{34}\right)}{2 \sqrt{-7 + 4 \sqrt{34}}} \right)}\right] \cup \left[\operatorname{atan}{\left(\frac{\sqrt{2} \left(4 - \sqrt{34}\right)}{2 \sqrt{-7 + 4 \sqrt{34}}} \right)}, \infty\right)
Increasing at intervals
[−π−atan⁡(2(4−34)2−7+434),atan⁡(2(4−34)2−7+434)]\left[- \pi - \operatorname{atan}{\left(\frac{\sqrt{2} \left(4 - \sqrt{34}\right)}{2 \sqrt{-7 + 4 \sqrt{34}}} \right)}, \operatorname{atan}{\left(\frac{\sqrt{2} \left(4 - \sqrt{34}\right)}{2 \sqrt{-7 + 4 \sqrt{34}}} \right)}\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
4(−3sin⁡(2x)+4cos⁡(x))=04 \left(- 3 \sin{\left(2 x \right)} + 4 \cos{\left(x \right)}\right) = 0
Solve this equation
The roots of this equation
x1=−π2x_{1} = - \frac{\pi}{2}
x2=π2x_{2} = \frac{\pi}{2}
x3=i(log⁡(3)−log⁡(−5+2i))x_{3} = i \left(\log{\left(3 \right)} - \log{\left(- \sqrt{5} + 2 i \right)}\right)
x4=i(log⁡(3)−log⁡(5+2i))x_{4} = i \left(\log{\left(3 \right)} - \log{\left(\sqrt{5} + 2 i \right)}\right)

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,atan⁡(255)]∪[π2,∞)\left[- \frac{\pi}{2}, \operatorname{atan}{\left(\frac{2 \sqrt{5}}{5} \right)}\right] \cup \left[\frac{\pi}{2}, \infty\right)
Convex at the intervals
(−∞,−π2]\left(-\infty, - \frac{\pi}{2}\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(3sin⁡(2x)−16cos⁡(x))=⟨−19,19⟩\lim_{x \to -\infty}\left(3 \sin{\left(2 x \right)} - 16 \cos{\left(x \right)}\right) = \left\langle -19, 19\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−19,19⟩y = \left\langle -19, 19\right\rangle
lim⁡x→∞(3sin⁡(2x)−16cos⁡(x))=⟨−19,19⟩\lim_{x \to \infty}\left(3 \sin{\left(2 x \right)} - 16 \cos{\left(x \right)}\right) = \left\langle -19, 19\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−19,19⟩y = \left\langle -19, 19\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 3*sin(2*x) - 16*cos(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(3sin⁡(2x)−16cos⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{3 \sin{\left(2 x \right)} - 16 \cos{\left(x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(3sin⁡(2x)−16cos⁡(x)x)=0\lim_{x \to \infty}\left(\frac{3 \sin{\left(2 x \right)} - 16 \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:
3sin⁡(2x)−16cos⁡(x)=−3sin⁡(2x)−16cos⁡(x)3 \sin{\left(2 x \right)} - 16 \cos{\left(x \right)} = - 3 \sin{\left(2 x \right)} - 16 \cos{\left(x \right)}
- No
3sin⁡(2x)−16cos⁡(x)=3sin⁡(2x)+16cos⁡(x)3 \sin{\left(2 x \right)} - 16 \cos{\left(x \right)} = 3 \sin{\left(2 x \right)} + 16 \cos{\left(x \right)}
- No
so, the function
not is
neither even, nor odd
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