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sin(x+pi/3)*(-3)
Plotting a function graph/ sin(x+pi/3)*(-3)

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sin(x+pi/3)*(-3)

What you mean?

sin((x + pi)/3)*(-3)
 
sin(x + pi/3)*(-3)
 
sin(x + pi/3)*(-3)

Graphing y = sin(x+pi/3)*(-3)

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The graph:

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Intersection points:

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Piecewise:

The solution

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          /    pi\   
f(x) = sin|x + --|*-3
          \    3 /
f(x)=sin⁡(x+π3)(−3)f{\left(x \right)} = \sin{\left(x + \frac{\pi}{3} \right)} \left(-3\right) 
f = sin(x + pi/3)*(-3)
The graph of the function
0-70-60-50-40-30-20-10105-5
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:
sin⁡(x+π3)(−3)=0\sin{\left(x + \frac{\pi}{3} \right)} \left(-3\right) = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−π3x_{1} = - \frac{\pi}{3}
x2=2π3x_{2} = \frac{2 \pi}{3}
Numerical solution
x1=−19.8967534727354x_{1} = -19.8967534727354
x2=−35.6047167406843x_{2} = -35.6047167406843
x3=−70.162235930172x_{3} = -70.162235930172
x4=−4.18879020478639x_{4} = -4.18879020478639
x5=5.23598775598299x_{5} = 5.23598775598299
x6=−16.7551608191456x_{6} = -16.7551608191456
x7=42.9350995990605x_{7} = 42.9350995990605
x8=61.7846555205993x_{8} = 61.7846555205993
x9=102.625360017267x_{9} = 102.625360017267
x10=71.2094334813686x_{10} = 71.2094334813686
x11=−38.7463093942741x_{11} = -38.7463093942741
x12=30.3687289847013x_{12} = 30.3687289847013
x13=2.0943951023932x_{13} = 2.0943951023932
x14=86.9173967493176x_{14} = 86.9173967493176
x15=17.8023583703422x_{15} = 17.8023583703422
x16=52.3598775598299x_{16} = 52.3598775598299
x17=−82.7286065445312x_{17} = -82.7286065445312
x18=49.2182849062401x_{18} = 49.2182849062401
x19=24.0855436775217x_{19} = 24.0855436775217
x20=74.3510261349584x_{20} = 74.3510261349584
x21=−154.985237577096x_{21} = -154.985237577096
x22=−67.0206432765823x_{22} = -67.0206432765823
x23=−95.2949771588904x_{23} = -95.2949771588904
x24=−29.3215314335047x_{24} = -29.3215314335047
x25=14.6607657167524x_{25} = 14.6607657167524
x26=8.37758040957278x_{26} = 8.37758040957278
x27=39.7935069454707x_{27} = 39.7935069454707
x28=−54.4542726622231x_{28} = -54.4542726622231
x29=46.0766922526503x_{29} = 46.0766922526503
x30=−60.7374579694027x_{30} = -60.7374579694027
x31=−57.5958653158129x_{31} = -57.5958653158129
x32=20.943951023932x_{32} = 20.943951023932
x33=33.5103216382911x_{33} = 33.5103216382911
x34=36.6519142918809x_{34} = 36.6519142918809
x35=−73.3038285837618x_{35} = -73.3038285837618
x36=−45.0294947014537x_{36} = -45.0294947014537
x37=−7.33038285837618x_{37} = -7.33038285837618
x38=−89.0117918517108x_{38} = -89.0117918517108
x39=55.5014702134197x_{39} = 55.5014702134197
x40=−63.8790506229925x_{40} = -63.8790506229925
x41=−76.4454212373516x_{41} = -76.4454212373516
x42=−13.6135681655558x_{42} = -13.6135681655558
x43=−98.4365698124802x_{43} = -98.4365698124802
x44=64.9262481741891x_{44} = 64.9262481741891
x45=99.4837673636768x_{45} = 99.4837673636768
x46=−10.471975511966x_{46} = -10.471975511966
x47=−92.1533845053006x_{47} = -92.1533845053006
x48=−79.5870138909414x_{48} = -79.5870138909414
x49=80.634211442138x_{49} = 80.634211442138
x50=90.0589894029074x_{50} = 90.0589894029074
x51=−1.0471975511966x_{51} = -1.0471975511966
x52=−41.8879020478639x_{52} = -41.8879020478639
x53=−51.3126800086333x_{53} = -51.3126800086333
x54=−48.1710873550435x_{54} = -48.1710873550435
x55=−26.1799387799149x_{55} = -26.1799387799149
x56=−32.4631240870945x_{56} = -32.4631240870945
x57=4579.39489138272x_{57} = 4579.39489138272
x58=93.2005820564972x_{58} = 93.2005820564972
x59=58.6430628670095x_{59} = 58.6430628670095
x60=11.5191730631626x_{60} = 11.5191730631626
x61=83.7758040957278x_{61} = 83.7758040957278
x62=27.2271363311115x_{62} = 27.2271363311115
x63=−85.870199198121x_{63} = -85.870199198121
x64=68.0678408277789x_{64} = 68.0678408277789
x65=−23.0383461263252x_{65} = -23.0383461263252
x66=96.342174710087x_{66} = 96.342174710087
x67=77.4926187885482x_{67} = 77.4926187885482
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(x + pi/3)*(-3).
sin⁡(0+π3)(−3)\sin{\left(0 + \frac{\pi}{3} \right)} \left(-3\right)
The result:
f(0)=−332f{\left(0 \right)} = - \frac{3 \sqrt{3}}{2}
The point:
(0, -3*sqrt(3)/2)
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
−3cos⁡(x+π3)=0- 3 \cos{\left(x + \frac{\pi}{3} \right)} = 0
Solve this equation
The roots of this equation
x1=π6x_{1} = \frac{\pi}{6}
x2=7π6x_{2} = \frac{7 \pi}{6}
The values of the extrema at the points:
 pi     
(--, -3)
 6      

 7*pi    
(----, 3)
  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=π6x_{1} = \frac{\pi}{6}
Maxima of the function at points:
x1=7π6x_{1} = \frac{7 \pi}{6}
Decreasing at intervals
[π6,7π6]\left[\frac{\pi}{6}, \frac{7 \pi}{6}\right]
Increasing at intervals
(−∞,π6]∪[7π6,∞)\left(-\infty, \frac{\pi}{6}\right] \cup \left[\frac{7 \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
3sin⁡(x+π3)=03 \sin{\left(x + \frac{\pi}{3} \right)} = 0
Solve this equation
The roots of this equation
x1=−π3x_{1} = - \frac{\pi}{3}
x2=2π3x_{2} = \frac{2 \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
[−π3,2π3]\left[- \frac{\pi}{3}, \frac{2 \pi}{3}\right]
Convex at the intervals
(−∞,−π3]∪[2π3,∞)\left(-\infty, - \frac{\pi}{3}\right] \cup \left[\frac{2 \pi}{3}, \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→−∞(sin⁡(x+π3)(−3))=⟨−3,3⟩\lim_{x \to -\infty}\left(\sin{\left(x + \frac{\pi}{3} \right)} \left(-3\right)\right) = \left\langle -3, 3\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−3,3⟩y = \left\langle -3, 3\right\rangle
lim⁡x→∞(sin⁡(x+π3)(−3))=⟨−3,3⟩\lim_{x \to \infty}\left(\sin{\left(x + \frac{\pi}{3} \right)} \left(-3\right)\right) = \left\langle -3, 3\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−3,3⟩y = \left\langle -3, 3\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sin(x + pi/3)*(-3), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(−3sin⁡(x+π3)x)=0\lim_{x \to -\infty}\left(- \frac{3 \sin{\left(x + \frac{\pi}{3} \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(−3sin⁡(x+π3)x)=0\lim_{x \to \infty}\left(- \frac{3 \sin{\left(x + \frac{\pi}{3} \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+π3)(−3)=−3cos⁡(x+π6)\sin{\left(x + \frac{\pi}{3} \right)} \left(-3\right) = - 3 \cos{\left(x + \frac{\pi}{6} \right)}
- No
sin⁡(x+π3)(−3)=3cos⁡(x+π6)\sin{\left(x + \frac{\pi}{3} \right)} \left(-3\right) = 3 \cos{\left(x + \frac{\pi}{6} \right)}
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = sin(x+pi/3)*(-3)
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