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  • Graphing y =:
  • x^2+3x+3
  • x^2-3x-4
  • x^2-2x-2
  • (x^2+3)/(x-1) (x^2+3)/(x-1)

  • Identical expressions

  • two *sin(x- three / five)
  • 2 multiply by sinus of (x minus 3 divide by 5)
  • two multiply by sinus of (x minus three divide by five)
  • 2sin(x-3/5)
  • 2sinx-3/5
  • 2*sin(x-3 divide by 5)

  • Similar expressions

  • 2*sin(x+3/5)
Plotting a function graph/ 2*sin(x-3/5)

Graphing y = 2*sin(x-3/5)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
f(x) = 2*sin(x - 3/5)
f(x)=2sin⁡(x−35)f{\left(x \right)} = 2 \sin{\left(x - \frac{3}{5} \right)} 
f = 2*sin(x - 3/5)
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:
2sin⁡(x−35)=02 \sin{\left(x - \frac{3}{5} \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=35x_{1} = \frac{3}{5}
x2=35+πx_{2} = \frac{3}{5} + \pi
Numerical solution
x1=−65.3734457253857x_{1} = -65.3734457253857
x2=−40.2407044966673x_{2} = -40.2407044966673
x3=−93.6477796076938x_{3} = -93.6477796076938
x4=−612.01056745001x_{4} = -612.01056745001
x5=−59.0902604182061x_{5} = -59.0902604182061
x6=94.8477796076938x_{6} = 94.8477796076938
x7=−49.6654824574367x_{7} = -49.6654824574367
x8=−30.8159265358979x_{8} = -30.8159265358979
x9=69.7150383789754x_{9} = 69.7150383789754
x10=−46.5238898038469x_{10} = -46.5238898038469
x11=−33.9575191894877x_{11} = -33.9575191894877
x12=41.4407044966673x_{12} = 41.4407044966673
x13=97.9893722612836x_{13} = 97.9893722612836
x14=75.998223686155x_{14} = 75.998223686155
x15=79.1398163397448x_{15} = 79.1398163397448
x16=10.0247779607694x_{16} = 10.0247779607694
x17=25.7327412287183x_{17} = 25.7327412287183
x18=−77.9398163397448x_{18} = -77.9398163397448
x19=−81.0814089933346x_{19} = -81.0814089933346
x20=82.2814089933346x_{20} = 82.2814089933346
x21=60.2902604182061x_{21} = 60.2902604182061
x22=88.5645943005142x_{22} = 88.5645943005142
x23=57.1486677646163x_{23} = 57.1486677646163
x24=38.2991118430775x_{24} = 38.2991118430775
x25=13.1663706143592x_{25} = 13.1663706143592
x26=−1322.0105071613x_{26} = -1322.0105071613
x27=6.88318530717959x_{27} = 6.88318530717959
x28=72.8566310325652x_{28} = 72.8566310325652
x29=0.6x_{29} = 0.6
x30=44.5822971502571x_{30} = 44.5822971502571
x31=−71.6566310325652x_{31} = -71.6566310325652
x32=−90.506186954104x_{32} = -90.506186954104
x33=16.307963267949x_{33} = 16.307963267949
x34=−5.68318530717959x_{34} = -5.68318530717959
x35=−27.6743338823081x_{35} = -27.6743338823081
x36=3.74159265358979x_{36} = 3.74159265358979
x37=−21.3911485751286x_{37} = -21.3911485751286
x38=101.130964914873x_{38} = 101.130964914873
x39=32.0159265358979x_{39} = 32.0159265358979
x40=−8.82477796076938x_{40} = -8.82477796076938
x41=35.1575191894877x_{41} = 35.1575191894877
x42=22.5911485751286x_{42} = 22.5911485751286
x43=−74.798223686155x_{43} = -74.798223686155
x44=−11.9663706143592x_{44} = -11.9663706143592
x45=−55.9486677646163x_{45} = -55.9486677646163
x46=−87.3645943005142x_{46} = -87.3645943005142
x47=50.8654824574367x_{47} = 50.8654824574367
x48=−2.54159265358979x_{48} = -2.54159265358979
x49=28.8743338823081x_{49} = 28.8743338823081
x50=−18.2495559215388x_{50} = -18.2495559215388
x51=−43.3822971502571x_{51} = -43.3822971502571
x52=−68.5150383789754x_{52} = -68.5150383789754
x53=54.0070751110265x_{53} = 54.0070751110265
x54=−62.2318530717959x_{54} = -62.2318530717959
x55=−37.0991118430775x_{55} = -37.0991118430775
x56=91.706186954104x_{56} = 91.706186954104
x57=66.5734457253857x_{57} = 66.5734457253857
x58=−84.2230016469244x_{58} = -84.2230016469244
x59=−52.8070751110265x_{59} = -52.8070751110265
x60=−128.205298797182x_{60} = -128.205298797182
x61=−24.5327412287183x_{61} = -24.5327412287183
x62=−96.7893722612836x_{62} = -96.7893722612836
x63=−99.9309649148734x_{63} = -99.9309649148734
x64=85.4230016469244x_{64} = 85.4230016469244
x65=−15.107963267949x_{65} = -15.107963267949
x66=63.4318530717959x_{66} = 63.4318530717959
x67=47.7238898038469x_{67} = 47.7238898038469
x68=19.4495559215388x_{68} = 19.4495559215388
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 2*sin(x - 3/5).
2sin⁡(−35)2 \sin{\left(- \frac{3}{5} \right)}
The result:
f(0)=−2sin⁡(35)f{\left(0 \right)} = - 2 \sin{\left(\frac{3}{5} \right)}
The point:
(0, -2*sin(3/5))
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
2cos⁡(x−35)=02 \cos{\left(x - \frac{3}{5} \right)} = 0
Solve this equation
The roots of this equation
x1=35+π2x_{1} = \frac{3}{5} + \frac{\pi}{2}
x2=35+3π2x_{2} = \frac{3}{5} + \frac{3 \pi}{2}
The values of the extrema at the points:
 3   pi    
(- + --, 2)
 5   2     

 3   3*pi     
(- + ----, -2)
 5    2


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=35+3π2x_{1} = \frac{3}{5} + \frac{3 \pi}{2}
Maxima of the function at points:
x1=35+π2x_{1} = \frac{3}{5} + \frac{\pi}{2}
Decreasing at intervals
(−∞,35+π2]∪[35+3π2,∞)\left(-\infty, \frac{3}{5} + \frac{\pi}{2}\right] \cup \left[\frac{3}{5} + \frac{3 \pi}{2}, \infty\right)
Increasing at intervals
[35+π2,35+3π2]\left[\frac{3}{5} + \frac{\pi}{2}, \frac{3}{5} + \frac{3 \pi}{2}\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
−2sin⁡(x−35)=0- 2 \sin{\left(x - \frac{3}{5} \right)} = 0
Solve this equation
The roots of this equation
x1=35x_{1} = \frac{3}{5}
x2=35+πx_{2} = \frac{3}{5} + \pi

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
(−∞,35]∪[35+π,∞)\left(-\infty, \frac{3}{5}\right] \cup \left[\frac{3}{5} + \pi, \infty\right)
Convex at the intervals
[35,35+π]\left[\frac{3}{5}, \frac{3}{5} + \pi\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(2sin⁡(x−35))=⟨−2,2⟩\lim_{x \to -\infty}\left(2 \sin{\left(x - \frac{3}{5} \right)}\right) = \left\langle -2, 2\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−2,2⟩y = \left\langle -2, 2\right\rangle
lim⁡x→∞(2sin⁡(x−35))=⟨−2,2⟩\lim_{x \to \infty}\left(2 \sin{\left(x - \frac{3}{5} \right)}\right) = \left\langle -2, 2\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−2,2⟩y = \left\langle -2, 2\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 2*sin(x - 3/5), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(2sin⁡(x−35)x)=0\lim_{x \to -\infty}\left(\frac{2 \sin{\left(x - \frac{3}{5} \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(2sin⁡(x−35)x)=0\lim_{x \to \infty}\left(\frac{2 \sin{\left(x - \frac{3}{5} \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:
2sin⁡(x−35)=−2sin⁡(x+35)2 \sin{\left(x - \frac{3}{5} \right)} = - 2 \sin{\left(x + \frac{3}{5} \right)}
- No
2sin⁡(x−35)=2sin⁡(x+35)2 \sin{\left(x - \frac{3}{5} \right)} = 2 \sin{\left(x + \frac{3}{5} \right)}
- No
so, the function
not is
neither even, nor odd
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