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

Graphing y = sqrt(3)sin(x)-cos(x)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = \/ 3 *sin(x) - cos(x)
f(x)=3sin⁡(x)−cos⁡(x)f{\left(x \right)} = \sqrt{3} \sin{\left(x \right)} - \cos{\left(x \right)} 
f = sqrt(3)*sin(x) - cos(x)
The graph of the function
02468-8-6-4-2-10105-5
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⁡(x)−cos⁡(x)=0\sqrt{3} \sin{\left(x \right)} - \cos{\left(x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=π6x_{1} = \frac{\pi}{6}
Numerical solution
x1=−100.007366139275x_{1} = -100.007366139275
x2=69.6386371545737x_{2} = 69.6386371545737
x3=−5.75958653158129x_{3} = -5.75958653158129
x4=−15.1843644923507x_{4} = -15.1843644923507
x5=−8.90117918517108x_{5} = -8.90117918517108
x6=−74.8746249105567x_{6} = -74.8746249105567
x7=13.0899693899575x_{7} = 13.0899693899575
x8=−2509.60893144265x_{8} = -2509.60893144265
x9=−68.5914396033772x_{9} = -68.5914396033772
x10=101.054563690472x_{10} = 101.054563690472
x11=−84.2994028713261x_{11} = -84.2994028713261
x12=−34.0339204138894x_{12} = -34.0339204138894
x13=−96.8657734856853x_{13} = -96.8657734856853
x14=31.9395253114962x_{14} = 31.9395253114962
x15=−40.317105721069x_{15} = -40.317105721069
x16=25.6563400043166x_{16} = 25.6563400043166
x17=−24.60914245312x_{17} = -24.60914245312
x18=9.94837673636768x_{18} = 9.94837673636768
x19=−56.025068989018x_{19} = -56.025068989018
x20=−37.1755130674792x_{20} = -37.1755130674792
x21=22.5147473507269x_{21} = 22.5147473507269
x22=75.9218224617533x_{22} = 75.9218224617533
x23=91.6297857297023x_{23} = 91.6297857297023
x24=41.3643032722656x_{24} = 41.3643032722656
x25=63.3554518473942x_{25} = 63.3554518473942
x26=−62.3082542961976x_{26} = -62.3082542961976
x27=−30.8923277602996x_{27} = -30.8923277602996
x28=97.9129710368819x_{28} = 97.9129710368819
x29=19.3731546971371x_{29} = 19.3731546971371
x30=−59.1666616426078x_{30} = -59.1666616426078
x31=−21.4675497995303x_{31} = -21.4675497995303
x32=−90.5825881785057x_{32} = -90.5825881785057
x33=−81.1578102177363x_{33} = -81.1578102177363
x34=66.497044500984x_{34} = 66.497044500984
x35=82.2050077689329x_{35} = 82.2050077689329
x36=85.3466004225227x_{36} = 85.3466004225227
x37=−78.0162175641465x_{37} = -78.0162175641465
x38=−46.6002910282486x_{38} = -46.6002910282486
x39=−12.0427718387609x_{39} = -12.0427718387609
x40=0.523598775598299x_{40} = 0.523598775598299
x41=79.0634151153431x_{41} = 79.0634151153431
x42=−87.4409955249159x_{42} = -87.4409955249159
x43=−43.4586983746588x_{43} = -43.4586983746588
x44=−27.7507351067098x_{44} = -27.7507351067098
x45=−71.733032256967x_{45} = -71.733032256967
x46=44.5058959258554x_{46} = 44.5058959258554
x47=53.9306738866248x_{47} = 53.9306738866248
x48=−93.7241808320955x_{48} = -93.7241808320955
x49=3.66519142918809x_{49} = 3.66519142918809
x50=50.789081233035x_{50} = 50.789081233035
x51=88.4881930761125x_{51} = 88.4881930761125
x52=72.7802298081635x_{52} = 72.7802298081635
x53=−2.61799387799149x_{53} = -2.61799387799149
x54=28.7979326579064x_{54} = 28.7979326579064
x55=94.7713783832921x_{55} = 94.7713783832921
x56=60.2138591938044x_{56} = 60.2138591938044
x57=−52.8834763354282x_{57} = -52.8834763354282
x58=47.6474885794452x_{58} = 47.6474885794452
x59=38.2227106186758x_{59} = 38.2227106186758
x60=−18.3259571459405x_{60} = -18.3259571459405
x61=16.2315620435473x_{61} = 16.2315620435473
x62=927.293431584587x_{62} = 927.293431584587
x63=35.081117965086x_{63} = 35.081117965086
x64=57.0722665402146x_{64} = 57.0722665402146
x65=6.80678408277789x_{65} = 6.80678408277789
x66=119.90411961201x_{66} = 119.90411961201
x67=−49.7418836818384x_{67} = -49.7418836818384
x68=−65.4498469497874x_{68} = -65.4498469497874
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sqrt(3)*sin(x) - cos(x).
−cos⁡(0)+3sin⁡(0)- \cos{\left(0 \right)} + \sqrt{3} \sin{\left(0 \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
sin⁡(x)+3cos⁡(x)=0\sin{\left(x \right)} + \sqrt{3} \cos{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=−π3x_{1} = - \frac{\pi}{3}
The values of the extrema at the points:
 -pi      
(----, -2)
  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}
The function has no maxima
Decreasing at intervals
[−π3,∞)\left[- \frac{\pi}{3}, \infty\right)
Increasing at intervals
(−∞,−π3]\left(-\infty, - \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
−3sin⁡(x)+cos⁡(x)=0- \sqrt{3} \sin{\left(x \right)} + \cos{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=π6x_{1} = \frac{\pi}{6}

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]\left(-\infty, \frac{\pi}{6}\right]
Convex at the intervals
[π6,∞)\left[\frac{\pi}{6}, \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→−∞(3sin⁡(x)−cos⁡(x))=⟨−1,1⟩+3⟨−1,1⟩\lim_{x \to -\infty}\left(\sqrt{3} \sin{\left(x \right)} - \cos{\left(x \right)}\right) = \left\langle -1, 1\right\rangle + \sqrt{3} \left\langle -1, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−1,1⟩+3⟨−1,1⟩y = \left\langle -1, 1\right\rangle + \sqrt{3} \left\langle -1, 1\right\rangle
lim⁡x→∞(3sin⁡(x)−cos⁡(x))=⟨−1,1⟩+3⟨−1,1⟩\lim_{x \to \infty}\left(\sqrt{3} \sin{\left(x \right)} - \cos{\left(x \right)}\right) = \left\langle -1, 1\right\rangle + \sqrt{3} \left\langle -1, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−1,1⟩+3⟨−1,1⟩y = \left\langle -1, 1\right\rangle + \sqrt{3} \left\langle -1, 1\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sqrt(3)*sin(x) - cos(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(3sin⁡(x)−cos⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{\sqrt{3} \sin{\left(x \right)} - \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⁡(x)−cos⁡(x)x)=0\lim_{x \to \infty}\left(\frac{\sqrt{3} \sin{\left(x \right)} - \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⁡(x)−cos⁡(x)=−3sin⁡(x)−cos⁡(x)\sqrt{3} \sin{\left(x \right)} - \cos{\left(x \right)} = - \sqrt{3} \sin{\left(x \right)} - \cos{\left(x \right)}
- No
3sin⁡(x)−cos⁡(x)=3sin⁡(x)+cos⁡(x)\sqrt{3} \sin{\left(x \right)} - \cos{\left(x \right)} = \sqrt{3} \sin{\left(x \right)} + \cos{\left(x \right)}
- No
so, the function
not is
neither even, nor odd
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