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pi*cos((x/2)-(pi/8))

  • How to use it?

  • Graphing y =:
  • x^2+4/2x
  • (4x^2+1)/x
  • 2x^3-3x^2-12x+1
  • (x/(x+5))^2

  • Identical expressions

  • pi*cos((x/ two)-(pi/ eight))
  • Pi multiply by co sinus of e of ((x divide by 2) minus ( Pi divide by 8))
  • Pi multiply by co sinus of e of ((x divide by two) minus ( Pi divide by eight))
  • picos((x/2)-(pi/8))
  • picosx/2-pi/8
  • pi*cos((x divide by 2)-(pi divide by 8))

  • Similar expressions

  • pi*cos((x/2)+(pi/8))
Plotting a function graph/ pi*cos((x/2)-(pi/8))

Graphing y = pi*cos((x/2)-(pi/8))

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
             /x   pi\
f(x) = pi*cos|- - --|
             \2   8 /
f(x)=πcos⁡(x2−π8)f{\left(x \right)} = \pi \cos{\left(\frac{x}{2} - \frac{\pi}{8} \right)} 
f = pi*cos(x/2 - pi/8)
The graph of the function
05-45-40-35-30-25-20-15-10-51015205-5
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:
πcos⁡(x2−π8)=0\pi \cos{\left(\frac{x}{2} - \frac{\pi}{8} \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−3π4x_{1} = - \frac{3 \pi}{4}
x2=5π4x_{2} = \frac{5 \pi}{4}
Numerical solution
x1=−46.3384916404494x_{1} = -46.3384916404494
x2=22.776546738526x_{2} = 22.776546738526
x3=−40.0553063332699x_{3} = -40.0553063332699
x4=−96.6039740978861x_{4} = -96.6039740978861
x5=16.4933614313464x_{5} = 16.4933614313464
x6=−2.35619449019234x_{6} = -2.35619449019234
x7=73.0420291959627x_{7} = 73.0420291959627
x8=−52.621676947629x_{8} = -52.621676947629
x9=85.6083998103219x_{9} = 85.6083998103219
x10=−77.7544181763474x_{10} = -77.7544181763474
x11=−153.152641862502x_{11} = -153.152641862502
x12=−115.453530019425x_{12} = -115.453530019425
x13=−84.037603483527x_{13} = -84.037603483527
x14=−27.4889357189107x_{14} = -27.4889357189107
x15=−58.9048622548086x_{15} = -58.9048622548086
x16=35.3429173528852x_{16} = 35.3429173528852
x17=3.92699081698724x_{17} = 3.92699081698724
x18=60.4756585816035x_{18} = 60.4756585816035
x19=47.9092879672443x_{19} = 47.9092879672443
x20=−8.63937979737193x_{20} = -8.63937979737193
x21=−90.3207887907066x_{21} = -90.3207887907066
x22=−33.7721210260903x_{22} = -33.7721210260903
x23=−65.1880475619882x_{23} = -65.1880475619882
x24=41.6261026600648x_{24} = 41.6261026600648
x25=−21.2057504117311x_{25} = -21.2057504117311
x26=91.8915851175014x_{26} = 91.8915851175014
x27=66.7588438887831x_{27} = 66.7588438887831
x28=98.174770424681x_{28} = 98.174770424681
x29=79.3252145031423x_{29} = 79.3252145031423
x30=10.2101761241668x_{30} = 10.2101761241668
x31=−102.887159405066x_{31} = -102.887159405066
x32=29.0597320457056x_{32} = 29.0597320457056
x33=−14.9225651045515x_{33} = -14.9225651045515
x34=54.1924732744239x_{34} = 54.1924732744239
x35=−71.4712328691678x_{35} = -71.4712328691678
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to pi*cos(x/2 - pi/8).
πcos⁡(−π8+02)\pi \cos{\left(- \frac{\pi}{8} + \frac{0}{2} \right)}
The result:
f(0)=π24+12f{\left(0 \right)} = \pi \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}
The point:
(0, pi*sqrt(1/2 + sqrt(2)/4))
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
−πsin⁡(x2−π8)2=0- \frac{\pi \sin{\left(\frac{x}{2} - \frac{\pi}{8} \right)}}{2} = 0
Solve this equation
The roots of this equation
x1=π4x_{1} = \frac{\pi}{4}
x2=9π4x_{2} = \frac{9 \pi}{4}
The values of the extrema at the points:
 pi        /pi   pi\ 
(--, pi*cos|-- - --|)
 4         \8    8 / 

 9*pi      
(----, -pi)
  4


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=9π4x_{1} = \frac{9 \pi}{4}
Maxima of the function at points:
x1=π4x_{1} = \frac{\pi}{4}
Decreasing at intervals
(−∞,π4]∪[9π4,∞)\left(-\infty, \frac{\pi}{4}\right] \cup \left[\frac{9 \pi}{4}, \infty\right)
Increasing at intervals
[π4,9π4]\left[\frac{\pi}{4}, \frac{9 \pi}{4}\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
−πsin⁡(4x+3π8)4=0- \frac{\pi \sin{\left(\frac{4 x + 3 \pi}{8} \right)}}{4} = 0
Solve this equation
The roots of this equation
x1=−3π4x_{1} = - \frac{3 \pi}{4}
x2=5π4x_{2} = \frac{5 \pi}{4}

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
(−∞,−3π4]∪[5π4,∞)\left(-\infty, - \frac{3 \pi}{4}\right] \cup \left[\frac{5 \pi}{4}, \infty\right)
Convex at the intervals
[−3π4,5π4]\left[- \frac{3 \pi}{4}, \frac{5 \pi}{4}\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(πcos⁡(x2−π8))=π⟨−1,1⟩\lim_{x \to -\infty}\left(\pi \cos{\left(\frac{x}{2} - \frac{\pi}{8} \right)}\right) = \pi \left\langle -1, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=π⟨−1,1⟩y = \pi \left\langle -1, 1\right\rangle
lim⁡x→∞(πcos⁡(x2−π8))=π⟨−1,1⟩\lim_{x \to \infty}\left(\pi \cos{\left(\frac{x}{2} - \frac{\pi}{8} \right)}\right) = \pi \left\langle -1, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=π⟨−1,1⟩y = \pi \left\langle -1, 1\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of pi*cos(x/2 - pi/8), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(πcos⁡(x2−π8)x)=0\lim_{x \to -\infty}\left(\frac{\pi \cos{\left(\frac{x}{2} - \frac{\pi}{8} \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(πcos⁡(x2−π8)x)=0\lim_{x \to \infty}\left(\frac{\pi \cos{\left(\frac{x}{2} - \frac{\pi}{8} \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:
πcos⁡(x2−π8)=πcos⁡(x2+π8)\pi \cos{\left(\frac{x}{2} - \frac{\pi}{8} \right)} = \pi \cos{\left(\frac{x}{2} + \frac{\pi}{8} \right)}
- No
πcos⁡(x2−π8)=−πcos⁡(x2+π8)\pi \cos{\left(\frac{x}{2} - \frac{\pi}{8} \right)} = - \pi \cos{\left(\frac{x}{2} + \frac{\pi}{8} \right)}
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = pi*cos((x/2)-(pi/8))
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