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

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

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = \/ 3 *cos(x) - sin(x)
f(x)=−sin⁡(x)+3cos⁡(x)f{\left(x \right)} = - \sin{\left(x \right)} + \sqrt{3} \cos{\left(x \right)} 
f = -sin(x) + sqrt(3)*cos(x)
The graph of the function
0-80-70-60-50-40-30-20-10102030405060705-5
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:
−sin⁡(x)+3cos⁡(x)=0- \sin{\left(x \right)} + \sqrt{3} \cos{\left(x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=π3x_{1} = \frac{\pi}{3}
Numerical solution
x1=−42.9350995990605x_{1} = -42.9350995990605
x2=−80.634211442138x_{2} = -80.634211442138
x3=73.3038285837618x_{3} = 73.3038285837618
x4=−83.7758040957278x_{4} = -83.7758040957278
x5=−46.0766922526503x_{5} = -46.0766922526503
x6=67.0206432765823x_{6} = 67.0206432765823
x7=85.870199198121x_{7} = 85.870199198121
x8=13.6135681655558x_{8} = 13.6135681655558
x9=−93.2005820564972x_{9} = -93.2005820564972
x10=−14.6607657167524x_{10} = -14.6607657167524
x11=−99.4837673636768x_{11} = -99.4837673636768
x12=63.8790506229925x_{12} = 63.8790506229925
x13=76.4454212373516x_{13} = 76.4454212373516
x14=104.71975511966x_{14} = 104.71975511966
x15=51.3126800086333x_{15} = 51.3126800086333
x16=−58.6430628670095x_{16} = -58.6430628670095
x17=32.4631240870945x_{17} = 32.4631240870945
x18=−49.2182849062401x_{18} = -49.2182849062401
x19=−11.5191730631626x_{19} = -11.5191730631626
x20=82.7286065445312x_{20} = 82.7286065445312
x21=4.18879020478639x_{21} = 4.18879020478639
x22=19.8967534727354x_{22} = 19.8967534727354
x23=57.5958653158129x_{23} = 57.5958653158129
x24=−74.3510261349584x_{24} = -74.3510261349584
x25=23.0383461263252x_{25} = 23.0383461263252
x26=−61.7846555205993x_{26} = -61.7846555205993
x27=−39.7935069454707x_{27} = -39.7935069454707
x28=10.471975511966x_{28} = 10.471975511966
x29=−55.5014702134197x_{29} = -55.5014702134197
x30=16.7551608191456x_{30} = 16.7551608191456
x31=−64.9262481741891x_{31} = -64.9262481741891
x32=−71.2094334813686x_{32} = -71.2094334813686
x33=−8.37758040957278x_{33} = -8.37758040957278
x34=92.1533845053006x_{34} = 92.1533845053006
x35=−77.4926187885482x_{35} = -77.4926187885482
x36=−68.0678408277789x_{36} = -68.0678408277789
x37=−5.23598775598299x_{37} = -5.23598775598299
x38=48.1710873550435x_{38} = 48.1710873550435
x39=70.162235930172x_{39} = 70.162235930172
x40=−17.8023583703422x_{40} = -17.8023583703422
x41=45.0294947014537x_{41} = 45.0294947014537
x42=26.1799387799149x_{42} = 26.1799387799149
x43=−30.3687289847013x_{43} = -30.3687289847013
x44=29.3215314335047x_{44} = 29.3215314335047
x45=38.7463093942741x_{45} = 38.7463093942741
x46=−90.0589894029074x_{46} = -90.0589894029074
x47=60.7374579694027x_{47} = 60.7374579694027
x48=54.4542726622231x_{48} = 54.4542726622231
x49=−96.342174710087x_{49} = -96.342174710087
x50=35.6047167406843x_{50} = 35.6047167406843
x51=41.8879020478639x_{51} = 41.8879020478639
x52=7.33038285837618x_{52} = 7.33038285837618
x53=−24.0855436775217x_{53} = -24.0855436775217
x54=−20.943951023932x_{54} = -20.943951023932
x55=−2.0943951023932x_{55} = -2.0943951023932
x56=−27.2271363311115x_{56} = -27.2271363311115
x57=−86.9173967493176x_{57} = -86.9173967493176
x58=95.2949771588904x_{58} = 95.2949771588904
x59=98.4365698124802x_{59} = 98.4365698124802
x60=1.0471975511966x_{60} = 1.0471975511966
x61=−52.3598775598299x_{61} = -52.3598775598299
x62=−33.5103216382911x_{62} = -33.5103216382911
x63=−36.6519142918809x_{63} = -36.6519142918809
x64=79.5870138909414x_{64} = 79.5870138909414
x65=89.0117918517108x_{65} = 89.0117918517108
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sqrt(3)*cos(x) - sin(x).
−sin⁡(0)+3cos⁡(0)- \sin{\left(0 \right)} + \sqrt{3} \cos{\left(0 \right)}
The result:
f(0)=3f{\left(0 \right)} = \sqrt{3}
The point:
(0, sqrt(3))
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⁡(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}
The values of the extrema at the points:
 -pi     
(----, 2)
  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:
The function has no minima
Maxima of the function at points:
x1=−π6x_{1} = - \frac{\pi}{6}
Decreasing at intervals
(−∞,−π6]\left(-\infty, - \frac{\pi}{6}\right]
Increasing at intervals
[−π6,∞)\left[- \frac{\pi}{6}, \infty\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
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}

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
[π3,∞)\left[\frac{\pi}{3}, \infty\right)
Convex at the intervals
(−∞,π3]\left(-\infty, \frac{\pi}{3}\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(−sin⁡(x)+3cos⁡(x))=⟨3⟨−1,1⟩−1,3⟨−1,1⟩+1⟩\lim_{x \to -\infty}\left(- \sin{\left(x \right)} + \sqrt{3} \cos{\left(x \right)}\right) = \left\langle \sqrt{3} \left\langle -1, 1\right\rangle - 1, \sqrt{3} \left\langle -1, 1\right\rangle + 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨3⟨−1,1⟩−1,3⟨−1,1⟩+1⟩y = \left\langle \sqrt{3} \left\langle -1, 1\right\rangle - 1, \sqrt{3} \left\langle -1, 1\right\rangle + 1\right\rangle
lim⁡x→∞(−sin⁡(x)+3cos⁡(x))=⟨3⟨−1,1⟩−1,3⟨−1,1⟩+1⟩\lim_{x \to \infty}\left(- \sin{\left(x \right)} + \sqrt{3} \cos{\left(x \right)}\right) = \left\langle \sqrt{3} \left\langle -1, 1\right\rangle - 1, \sqrt{3} \left\langle -1, 1\right\rangle + 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨3⟨−1,1⟩−1,3⟨−1,1⟩+1⟩y = \left\langle \sqrt{3} \left\langle -1, 1\right\rangle - 1, \sqrt{3} \left\langle -1, 1\right\rangle + 1\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sqrt(3)*cos(x) - sin(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(−sin⁡(x)+3cos⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{- \sin{\left(x \right)} + \sqrt{3} \cos{\left(x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(−sin⁡(x)+3cos⁡(x)x)=0\lim_{x \to \infty}\left(\frac{- \sin{\left(x \right)} + \sqrt{3} \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:
−sin⁡(x)+3cos⁡(x)=sin⁡(x)+3cos⁡(x)- \sin{\left(x \right)} + \sqrt{3} \cos{\left(x \right)} = \sin{\left(x \right)} + \sqrt{3} \cos{\left(x \right)}
- No
−sin⁡(x)+3cos⁡(x)=−sin⁡(x)−3cos⁡(x)- \sin{\left(x \right)} + \sqrt{3} \cos{\left(x \right)} = - \sin{\left(x \right)} - \sqrt{3} \cos{\left(x \right)}
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = sqrt(3)*cos(x)-sin(x)
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