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Plotting a function graph/ -2cos(x-pi/4)

Graphing y = -2cos(x-pi/4)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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             /    pi\
f(x) = -2*cos|x - --|
             \    4 /
f(x)=−2cos⁡(x−π4)f{\left(x \right)} = - 2 \cos{\left(x - \frac{\pi}{4} \right)} 
f = -2*cos(x - pi/4)
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:
−2cos⁡(x−π4)=0- 2 \cos{\left(x - \frac{\pi}{4} \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−π4x_{1} = - \frac{\pi}{4}
x2=3π4x_{2} = \frac{3 \pi}{4}
Numerical solution
x1=−38.484510006475x_{1} = -38.484510006475
x2=−51.0508806208341x_{2} = -51.0508806208341
x3=18.0641577581413x_{3} = 18.0641577581413
x4=−7.06858347057703x_{4} = -7.06858347057703
x5=−57.3340659280137x_{5} = -57.3340659280137
x6=30.6305283725005x_{6} = 30.6305283725005
x7=87.1791961371168x_{7} = 87.1791961371168
x8=−73.0420291959627x_{8} = -73.0420291959627
x9=−60.4756585816035x_{9} = -60.4756585816035
x10=−47.9092879672443x_{10} = -47.9092879672443
x11=96.6039740978861x_{11} = 96.6039740978861
x12=52.621676947629x_{12} = 52.621676947629
x13=68.329640215578x_{13} = 68.329640215578
x14=8.63937979737193x_{14} = 8.63937979737193
x15=55.7632696012188x_{15} = 55.7632696012188
x16=62.0464549083984x_{16} = 62.0464549083984
x17=43.1968989868597x_{17} = 43.1968989868597
x18=−91.8915851175014x_{18} = -91.8915851175014
x19=−98.174770424681x_{19} = -98.174770424681
x20=−44.7676953136546x_{20} = -44.7676953136546
x21=27.4889357189107x_{21} = 27.4889357189107
x22=71.4712328691678x_{22} = 71.4712328691678
x23=49.4800842940392x_{23} = 49.4800842940392
x24=99.7455667514759x_{24} = 99.7455667514759
x25=391.913683535327x_{25} = 391.913683535327
x26=−3.92699081698724x_{26} = -3.92699081698724
x27=93.4623814442964x_{27} = 93.4623814442964
x28=−69.9004365423729x_{28} = -69.9004365423729
x29=−66.7588438887831x_{29} = -66.7588438887831
x30=36.9137136796801x_{30} = 36.9137136796801
x31=58.9048622548086x_{31} = 58.9048622548086
x32=−0.785398163397448x_{32} = -0.785398163397448
x33=−82.4668071567321x_{33} = -82.4668071567321
x34=−54.1924732744239x_{34} = -54.1924732744239
x35=84.037603483527x_{35} = 84.037603483527
x36=−19.6349540849362x_{36} = -19.6349540849362
x37=2.35619449019234x_{37} = 2.35619449019234
x38=80.8960108299372x_{38} = 80.8960108299372
x39=−88.7499924639117x_{39} = -88.7499924639117
x40=−63.6172512351933x_{40} = -63.6172512351933
x41=−22.776546738526x_{41} = -22.776546738526
x42=24.3473430653209x_{42} = 24.3473430653209
x43=−95.0331777710912x_{43} = -95.0331777710912
x44=−10.2101761241668x_{44} = -10.2101761241668
x45=−79.3252145031423x_{45} = -79.3252145031423
x46=−29.0597320457056x_{46} = -29.0597320457056
x47=−76.1836218495525x_{47} = -76.1836218495525
x48=−13.3517687777566x_{48} = -13.3517687777566
x49=−25.9181393921158x_{49} = -25.9181393921158
x50=65.1880475619882x_{50} = 65.1880475619882
x51=−16.4933614313464x_{51} = -16.4933614313464
x52=−35.3429173528852x_{52} = -35.3429173528852
x53=14.9225651045515x_{53} = 14.9225651045515
x54=33.7721210260903x_{54} = 33.7721210260903
x55=5.49778714378214x_{55} = 5.49778714378214
x56=46.3384916404494x_{56} = 46.3384916404494
x57=21.2057504117311x_{57} = 21.2057504117311
x58=−32.2013246992954x_{58} = -32.2013246992954
x59=11.7809724509617x_{59} = 11.7809724509617
x60=−85.6083998103219x_{60} = -85.6083998103219
x61=90.3207887907066x_{61} = 90.3207887907066
x62=−41.6261026600648x_{62} = -41.6261026600648
x63=1745.94011723253x_{63} = 1745.94011723253
x64=77.7544181763474x_{64} = 77.7544181763474
x65=−107.59954838545x_{65} = -107.59954838545
x66=74.6128255227576x_{66} = 74.6128255227576
x67=40.0553063332699x_{67} = 40.0553063332699
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to -2*cos(x - pi/4).
−2cos⁡(−π4)- 2 \cos{\left(- \frac{\pi}{4} \right)}
The result:
f(0)=−2f{\left(0 \right)} = - \sqrt{2}
The point:
(0, -sqrt(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
2sin⁡(x−π4)=02 \sin{\left(x - \frac{\pi}{4} \right)} = 0
Solve this equation
The roots of this equation
x1=π4x_{1} = \frac{\pi}{4}
x2=5π4x_{2} = \frac{5 \pi}{4}
The values of the extrema at the points:
 pi        /pi   pi\ 
(--, -2*cos|-- - --|)
 4         \4    4 / 

 5*pi       /pi   pi\ 
(----, 2*cos|-- - --|)
  4         \4    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=π4x_{1} = \frac{\pi}{4}
Maxima of the function at points:
x1=5π4x_{1} = \frac{5 \pi}{4}
Decreasing at intervals
[π4,5π4]\left[\frac{\pi}{4}, \frac{5 \pi}{4}\right]
Increasing at intervals
(−∞,π4]∪[5π4,∞)\left(-\infty, \frac{\pi}{4}\right] \cup \left[\frac{5 \pi}{4}, \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
2sin⁡(x+π4)=02 \sin{\left(x + \frac{\pi}{4} \right)} = 0
Solve this equation
The roots of this equation
x1=−π4x_{1} = - \frac{\pi}{4}
x2=3π4x_{2} = \frac{3 \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
[−π4,3π4]\left[- \frac{\pi}{4}, \frac{3 \pi}{4}\right]
Convex at the intervals
(−∞,−π4]∪[3π4,∞)\left(-\infty, - \frac{\pi}{4}\right] \cup \left[\frac{3 \pi}{4}, \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→−∞(−2cos⁡(x−π4))=⟨−2,2⟩\lim_{x \to -\infty}\left(- 2 \cos{\left(x - \frac{\pi}{4} \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→∞(−2cos⁡(x−π4))=⟨−2,2⟩\lim_{x \to \infty}\left(- 2 \cos{\left(x - \frac{\pi}{4} \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*cos(x - pi/4), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(−2cos⁡(x−π4)x)=0\lim_{x \to -\infty}\left(- \frac{2 \cos{\left(x - \frac{\pi}{4} \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(−2cos⁡(x−π4)x)=0\lim_{x \to \infty}\left(- \frac{2 \cos{\left(x - \frac{\pi}{4} \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:
−2cos⁡(x−π4)=−2cos⁡(x+π4)- 2 \cos{\left(x - \frac{\pi}{4} \right)} = - 2 \cos{\left(x + \frac{\pi}{4} \right)}
- No
−2cos⁡(x−π4)=2cos⁡(x+π4)- 2 \cos{\left(x - \frac{\pi}{4} \right)} = 2 \cos{\left(x + \frac{\pi}{4} \right)}
- No
so, the function
not is
neither even, nor odd
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