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  • Graphing y =:
  • x^2-5x+6
  • 2/x
  • y=cos(1/2x+п/3)
  • cos(3*x)

  • Identical expressions

  • y=cos(one /2x+п/ three)
  • y equally co sinus of e of (1 divide by 2x plus п divide by 3)
  • y equally co sinus of e of (one divide by 2x plus п divide by three)
  • y=cos1/2x+п/3
  • y=cos(1 divide by 2x+п divide by 3)

  • Similar expressions

  • y=cos(1/2x-п/3)
Plotting a function graph/ y=cos(1/2x+п/3)

Graphing y = y=cos(1/2x+п/3)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          /x   pi\
f(x) = cos|- + --|
          \2   3 /
f(x)=cos⁡(x2+π3)f{\left(x \right)} = \cos{\left(\frac{x}{2} + \frac{\pi}{3} \right)} 
f = cos(x/2 + pi/3)
The graph of the function
02468-8-6-4-2-10102-2
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:
cos⁡(x2+π3)=0\cos{\left(\frac{x}{2} + \frac{\pi}{3} \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=π3x_{1} = \frac{\pi}{3}
x2=7π3x_{2} = \frac{7 \pi}{3}
Numerical solution
x1=−24.0855436775217x_{1} = -24.0855436775217
x2=−137.182879206754x_{2} = -137.182879206754
x3=32.4631240870945x_{3} = 32.4631240870945
x4=26.1799387799149x_{4} = 26.1799387799149
x5=−42.9350995990605x_{5} = -42.9350995990605
x6=−68.0678408277789x_{6} = -68.0678408277789
x7=19.8967534727354x_{7} = 19.8967534727354
x8=45.0294947014537x_{8} = 45.0294947014537
x9=−86.9173967493176x_{9} = -86.9173967493176
x10=−33714.5251607745x_{10} = -33714.5251607745
x11=57.5958653158129x_{11} = 57.5958653158129
x12=7.33038285837618x_{12} = 7.33038285837618
x13=95.2949771588904x_{13} = 95.2949771588904
x14=−30.3687289847013x_{14} = -30.3687289847013
x15=38.7463093942741x_{15} = 38.7463093942741
x16=82.7286065445312x_{16} = 82.7286065445312
x17=1.0471975511966x_{17} = 1.0471975511966
x18=−5.23598775598299x_{18} = -5.23598775598299
x19=−93.2005820564972x_{19} = -93.2005820564972
x20=−11.5191730631626x_{20} = -11.5191730631626
x21=−74.3510261349584x_{21} = -74.3510261349584
x22=−507.89081233035x_{22} = -507.89081233035
x23=76.4454212373516x_{23} = 76.4454212373516
x24=−80.634211442138x_{24} = -80.634211442138
x25=−17164.6150616634x_{25} = -17164.6150616634
x26=−17.8023583703422x_{26} = -17.8023583703422
x27=89.0117918517108x_{27} = 89.0117918517108
x28=−55.5014702134197x_{28} = -55.5014702134197
x29=63.8790506229925x_{29} = 63.8790506229925
x30=101.57816246607x_{30} = 101.57816246607
x31=−99.4837673636768x_{31} = -99.4837673636768
x32=51.3126800086333x_{32} = 51.3126800086333
x33=−49.2182849062401x_{33} = -49.2182849062401
x34=−36.6519142918809x_{34} = -36.6519142918809
x35=13.6135681655558x_{35} = 13.6135681655558
x36=70.162235930172x_{36} = 70.162235930172
x37=−61.7846555205993x_{37} = -61.7846555205993
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cos(x/2 + pi/3).
cos⁡(02+π3)\cos{\left(\frac{0}{2} + \frac{\pi}{3} \right)}
The result:
f(0)=12f{\left(0 \right)} = \frac{1}{2}
The point:
(0, 1/2)
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
−sin⁡(x2+π3)2=0- \frac{\sin{\left(\frac{x}{2} + \frac{\pi}{3} \right)}}{2} = 0
Solve this equation
The roots of this equation
x1=−2π3x_{1} = - \frac{2 \pi}{3}
x2=4π3x_{2} = \frac{4 \pi}{3}
The values of the extrema at the points:
 -2*pi     /pi   pi\ 
(-----, cos|-- - --|)
   3       \3    3 / 

 4*pi      /pi   pi\ 
(----, -sin|-- + --|)
  3        \6    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=4π3x_{1} = \frac{4 \pi}{3}
Maxima of the function at points:
x1=−2π3x_{1} = - \frac{2 \pi}{3}
Decreasing at intervals
(−∞,−2π3]∪[4π3,∞)\left(-\infty, - \frac{2 \pi}{3}\right] \cup \left[\frac{4 \pi}{3}, \infty\right)
Increasing at intervals
[−2π3,4π3]\left[- \frac{2 \pi}{3}, \frac{4 \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
−cos⁡(3x+2π6)4=0- \frac{\cos{\left(\frac{3 x + 2 \pi}{6} \right)}}{4} = 0
Solve this equation
The roots of this equation
x1=π3x_{1} = \frac{\pi}{3}
x2=7π3x_{2} = \frac{7 \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
[π3,7π3]\left[\frac{\pi}{3}, \frac{7 \pi}{3}\right]
Convex at the intervals
(−∞,π3]∪[7π3,∞)\left(-\infty, \frac{\pi}{3}\right] \cup \left[\frac{7 \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→−∞cos⁡(x2+π3)=⟨−1,1⟩\lim_{x \to -\infty} \cos{\left(\frac{x}{2} + \frac{\pi}{3} \right)} = \left\langle -1, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
lim⁡x→∞cos⁡(x2+π3)=⟨−1,1⟩\lim_{x \to \infty} \cos{\left(\frac{x}{2} + \frac{\pi}{3} \right)} = \left\langle -1, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cos(x/2 + pi/3), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(cos⁡(x2+π3)x)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(\frac{x}{2} + \frac{\pi}{3} \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(cos⁡(x2+π3)x)=0\lim_{x \to \infty}\left(\frac{\cos{\left(\frac{x}{2} + \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:
cos⁡(x2+π3)=cos⁡(x2−π3)\cos{\left(\frac{x}{2} + \frac{\pi}{3} \right)} = \cos{\left(\frac{x}{2} - \frac{\pi}{3} \right)}
- No
cos⁡(x2+π3)=−cos⁡(x2−π3)\cos{\left(\frac{x}{2} + \frac{\pi}{3} \right)} = - \cos{\left(\frac{x}{2} - \frac{\pi}{3} \right)}
- No
so, the function
not is
neither even, nor odd
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