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cos((1/3)x-5)-1
Plotting a function graph/ cos((1/3)x-5)-1

Graphing y = cos((1/3)x-5)-1

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          /x    \    
f(x) = cos|- - 5| - 1
          \3    /
f(x)=cos⁡(x3−5)−1f{\left(x \right)} = \cos{\left(\frac{x}{3} - 5 \right)} - 1 
f = cos(x/3 - 1*5) - 1*1
The graph of the function
05-60-55-50-45-40-35-30-25-20-15-10-5102-4
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⁡(x3−5)−1=0\cos{\left(\frac{x}{3} - 5 \right)} - 1 = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=15x_{1} = 15
x2=15+6πx_{2} = 15 + 6 \pi
Numerical solution
x1=−60.398225167194x_{1} = -60.398225167194
x2=14.9999988261632x_{2} = 14.9999988261632
x3=−41.5486692725727x_{3} = -41.5486692725727
x4=90.3982227224037x_{4} = 90.3982227224037
x5=−79.2477781029098x_{5} = -79.2477781029098
x6=14.999998458996x_{6} = 14.999998458996
x7=−98.097334562244x_{7} = -98.097334562244
x8=14.9999997146293x_{8} = 14.9999997146293
x9=52.6991116051083x_{9} = 52.6991116051083
x10=−79.2477803321481x_{10} = -79.2477803321481
x11=71.5486662729243x_{11} = 71.5486662729243
x12=−98.0973340232512x_{12} = -98.0973340232512
x13=90.398223596696x_{13} = 90.398223596696
x14=−3.84955738562815x_{14} = -3.84955738562815
x15=33.8495569083246x_{15} = 33.8495569083246
x16=−79.2477782708537x_{16} = -79.2477782708537
x17=71.5486693049611x_{17} = 71.5486693049611
x18=52.6991107702266x_{18} = 52.6991107702266
x19=71.54866643115x_{19} = 71.54866643115
x20=−60.3982228243784x_{20} = -60.3982228243784
x21=90.3982251064929x_{21} = 90.3982251064929
x22=−79.2477789841353x_{22} = -79.2477789841353
x23=−41.5486663322582x_{23} = -41.5486663322582
x24=−41.5486688314017x_{24} = -41.5486688314017
x25=−22.6991114520374x_{25} = -22.6991114520374
x26=71.548668029236x_{26} = 71.548668029236
x27=−60.3982221493397x_{27} = -60.3982221493397
x28=90.3982251009654x_{28} = 90.3982251009654
x29=33.8495574342307x_{29} = 33.8495574342307
x30=71.54866948053x_{30} = 71.54866948053
x31=−60.3982226100855x_{31} = -60.3982226100855
x32=52.6991131935032x_{32} = 52.6991131935032
x33=−3.84955504975715x_{33} = -3.84955504975715
x34=−173.49556300409x_{34} = -173.49556300409
x35=−3.8495567900673x_{35} = -3.8495567900673
x36=33.8495551743668x_{36} = 33.8495551743668
x37=−60.398224329258x_{37} = -60.398224329258
x38=15.0000015363515x_{38} = 15.0000015363515
x39=90.3982200759343x_{39} = 90.3982200759343
x40=−41.5486711953331x_{40} = -41.5486711953331
x41=52.6991133380664x_{41} = 52.6991133380664
x42=33.849559522518x_{42} = 33.849559522518
x43=−98.0973362982451x_{43} = -98.0973362982451
x44=−98.0973354487839x_{44} = -98.0973354487839
x45=−79.2477810482348x_{45} = -79.2477810482348
x46=−79.2477798706684x_{46} = -79.2477798706684
x47=−41.5486710728419x_{47} = -41.5486710728419
x48=52.6991124893956x_{48} = 52.6991124893956
x49=71.5486671442574x_{49} = 71.5486671442574
x50=52.699111045689x_{50} = 52.699111045689
x51=−22.6991133695471x_{51} = -22.6991133695471
x52=−22.6991106660189x_{52} = -22.6991106660189
x53=52.6991103118927x_{53} = 52.6991103118927
x54=52.6991101149785x_{54} = 52.6991101149785
x55=−41.5486678817393x_{55} = -41.5486678817393
x56=71.5486685591574x_{56} = 71.5486685591574
x57=−22.6991131087023x_{57} = -22.6991131087023
x58=−3.84955445341867x_{58} = -3.84955445341867
x59=−41.5486687481687x_{59} = -41.5486687481687
x60=14.9999994759325x_{60} = 14.9999994759325
x61=33.849554516242x_{61} = 33.849554516242
x62=14.9999996226036x_{62} = 14.9999996226036
x63=−98.0973343764819x_{63} = -98.0973343764819
x64=−60.3982250331757x_{64} = -60.3982250331757
x65=−79.2477811421677x_{65} = -79.2477811421677
x66=−60.3982234469601x_{66} = -60.3982234469601
x67=−79.2477806989887x_{67} = -79.2477806989887
x68=33.8495560338965x_{68} = 33.8495560338965
x69=14.9999996057432x_{69} = 14.9999996057432
x70=90.3982225740357x_{70} = 90.3982225740357
x71=−41.5486663557103x_{71} = -41.5486663557103
x72=33.8495570322901x_{72} = 33.8495570322901
x73=90.3982244584232x_{73} = 90.3982244584232
x74=90.3982221847585x_{74} = 90.3982221847585
x75=−41.5486670141914x_{75} = -41.5486670141914
x76=−3.84955720208902x_{76} = -3.84955720208902
x77=71.5486688591282x_{77} = 71.5486688591282
x78=−3.84955461250715x_{78} = -3.84955461250715
x79=33.8495545081941x_{79} = 33.8495545081941
x80=15.0000012688973x_{80} = 15.0000012688973
x81=−22.699111269591x_{81} = -22.699111269591
x82=−60.3982216595791x_{82} = -60.3982216595791
x83=15.0000015442757x_{83} = 15.0000015442757
x84=−22.6991102947873x_{84} = -22.6991102947873
x85=15.0000005154463x_{85} = 15.0000005154463
x86=−3.8495557964631x_{86} = -3.8495557964631
x87=−22.6991123554547x_{87} = -22.6991123554547
x88=−22.6991114630313x_{88} = -22.6991114630313
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cos(x/3 - 1*5) - 1*1.
(−1)1+cos⁡((−1)5+13⋅0)\left(-1\right) 1 + \cos{\left(\left(-1\right) 5 + \frac{1}{3} \cdot 0 \right)}
The result:
f(0)=−1+cos⁡(5)f{\left(0 \right)} = -1 + \cos{\left(5 \right)}
The point:
(0, -1 + cos(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
−sin⁡(x3−5)3=0- \frac{\sin{\left(\frac{x}{3} - 5 \right)}}{3} = 0
Solve this equation
The roots of this equation
x1=15x_{1} = 15
x2=3π+15x_{2} = 3 \pi + 15
The values of the extrema at the points:
(15, 1 - 1)

(15 + 3*pi, -1 - 1)


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=3π+15x_{1} = 3 \pi + 15
Maxima of the function at points:
x1=15x_{1} = 15
Decreasing at intervals
(−∞,15]∪[3π+15,∞)\left(-\infty, 15\right] \cup \left[3 \pi + 15, \infty\right)
Increasing at intervals
[15,3π+15]\left[15, 3 \pi + 15\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⁡(x3−5)9=0- \frac{\cos{\left(\frac{x}{3} - 5 \right)}}{9} = 0
Solve this equation
The roots of this equation
x1=3π2+15x_{1} = \frac{3 \pi}{2} + 15
x2=9π2+15x_{2} = \frac{9 \pi}{2} + 15

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
[3π2+15,9π2+15]\left[\frac{3 \pi}{2} + 15, \frac{9 \pi}{2} + 15\right]
Convex at the intervals
(−∞,3π2+15]∪[9π2+15,∞)\left(-\infty, \frac{3 \pi}{2} + 15\right] \cup \left[\frac{9 \pi}{2} + 15, \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⁡(x3−5)−1)=⟨−2,0⟩\lim_{x \to -\infty}\left(\cos{\left(\frac{x}{3} - 5 \right)} - 1\right) = \left\langle -2, 0\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−2,0⟩y = \left\langle -2, 0\right\rangle
lim⁡x→∞(cos⁡(x3−5)−1)=⟨−2,0⟩\lim_{x \to \infty}\left(\cos{\left(\frac{x}{3} - 5 \right)} - 1\right) = \left\langle -2, 0\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−2,0⟩y = \left\langle -2, 0\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cos(x/3 - 1*5) - 1*1, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(cos⁡(x3−5)−1x)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(\frac{x}{3} - 5 \right)} - 1}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(cos⁡(x3−5)−1x)=0\lim_{x \to \infty}\left(\frac{\cos{\left(\frac{x}{3} - 5 \right)} - 1}{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⁡(x3−5)−1=cos⁡(x3+5)−1\cos{\left(\frac{x}{3} - 5 \right)} - 1 = \cos{\left(\frac{x}{3} + 5 \right)} - 1
- No
cos⁡(x3−5)−1=1−cos⁡(x3+5)\cos{\left(\frac{x}{3} - 5 \right)} - 1 = 1 - \cos{\left(\frac{x}{3} + 5 \right)}
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = cos((1/3)x-5)-1
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