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Plotting a function graph/ cos^2x

Graphing y = cos^2x

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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          2   
f(x) = cos (x)
f(x)=cos⁡2(x)f{\left(x \right)} = \cos^{2}{\left(x \right)} 
f = cos(x)^2
The graph of the function
02468-8-6-4-2-101002
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⁡2(x)=0\cos^{2}{\left(x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=π2x_{1} = \frac{\pi}{2}
x2=3π2x_{2} = \frac{3 \pi}{2}
Numerical solution
x1=98.9601683381274x_{1} = 98.9601683381274
x2=86.393797888273x_{2} = 86.393797888273
x3=80.1106126771746x_{3} = 80.1106126771746
x4=−17.2787590276524x_{4} = -17.2787590276524
x5=−73.8274272800405x_{5} = -73.8274272800405
x6=39.2699081179815x_{6} = 39.2699081179815
x7=76.9690207492347x_{7} = 76.9690207492347
x8=−4.71238872430683x_{8} = -4.71238872430683
x9=−98.9601684414698x_{9} = -98.9601684414698
x10=−67.5442421675773x_{10} = -67.5442421675773
x11=4.71238876848081x_{11} = 4.71238876848081
x12=70.6858345016621x_{12} = 70.6858345016621
x13=48.6946859238715x_{13} = 48.6946859238715
x14=−58.1194639993376x_{14} = -58.1194639993376
x15=98.9601685932308x_{15} = 98.9601685932308
x16=23.5619451230057x_{16} = 23.5619451230057
x17=89.5353908552844x_{17} = 89.5353908552844
x18=58.1194644379895x_{18} = 58.1194644379895
x19=61.2610569989704x_{19} = 61.2610569989704
x20=−39.2699083866483x_{20} = -39.2699083866483
x21=−20.4203520321877x_{21} = -20.4203520321877
x22=32.9867226137576x_{22} = 32.9867226137576
x23=73.8274274795554x_{23} = 73.8274274795554
x24=−1.57079642969308x_{24} = -1.57079642969308
x25=80.1106131434937x_{25} = 80.1106131434937
x26=−29.8451300963672x_{26} = -29.8451300963672
x27=14.1371671048484x_{27} = 14.1371671048484
x28=−92.6769830239371x_{28} = -92.6769830239371
x29=−54.9778713137198x_{29} = -54.9778713137198
x30=−23.5619450090417x_{30} = -23.5619450090417
x31=7.85398174058521x_{31} = 7.85398174058521
x32=64.4026493086922x_{32} = 64.4026493086922
x33=−42.4115006098842x_{33} = -42.4115006098842
x34=54.9778711883962x_{34} = 54.9778711883962
x35=−70.685834448838x_{35} = -70.685834448838
x36=17.2787595624179x_{36} = 17.2787595624179
x37=29.845130320338x_{37} = 29.845130320338
x38=54.9778714849733x_{38} = 54.9778714849733
x39=67.5442422779275x_{39} = 67.5442422779275
x40=−32.9867231091652x_{40} = -32.9867231091652
x41=−92.6769831823972x_{41} = -92.6769831823972
x42=−95.8185758681287x_{42} = -95.8185758681287
x43=−64.4026491876462x_{43} = -64.4026491876462
x44=−4.7123889912442x_{44} = -4.7123889912442
x45=−86.393797765473x_{45} = -86.393797765473
x46=61.2610566752601x_{46} = 61.2610566752601
x47=−98.96016883042x_{47} = -98.96016883042
x48=−10.9955745350309x_{48} = -10.9955745350309
x49=42.4115007291722x_{49} = 42.4115007291722
x50=−26.7035372990183x_{50} = -26.7035372990183
x51=−61.2610569641117x_{51} = -61.2610569641117
x52=−76.9690202568697x_{52} = -76.9690202568697
x53=−54.9778716831146x_{53} = -54.9778716831146
x54=−48.6946858738636x_{54} = -48.6946858738636
x55=541.924732890135x_{55} = 541.924732890135
x56=−14.1371668392726x_{56} = -14.1371668392726
x57=−70.6858346386357x_{57} = -70.6858346386357
x58=45.553093700501x_{58} = 45.553093700501
x59=10.9955740392793x_{59} = 10.9955740392793
x60=36.1283156002139x_{60} = 36.1283156002139
x61=20.4203521497111x_{61} = 20.4203521497111
x62=10.9955743696636x_{62} = 10.9955743696636
x63=76.9690197631883x_{63} = 76.9690197631883
x64=51.8362788999928x_{64} = 51.8362788999928
x65=23.5619449395428x_{65} = 23.5619449395428
x66=−48.6946860920117x_{66} = -48.6946860920117
x67=95.8185760590309x_{67} = 95.8185760590309
x68=−51.8362786897497x_{68} = -51.8362786897497
x69=−61.2610562242523x_{69} = -61.2610562242523
x70=−98.960168684456x_{70} = -98.960168684456
x71=−7.85398149857354x_{71} = -7.85398149857354
x72=39.2699084246933x_{72} = 39.2699084246933
x73=92.6769830795146x_{73} = 92.6769830795146
x74=83.2522052340866x_{74} = 83.2522052340866
x75=1.5707965454425x_{75} = 1.5707965454425
x76=−83.2522055415057x_{76} = -83.2522055415057
x77=32.986722928111x_{77} = 32.986722928111
x78=−76.9690198771149x_{78} = -76.9690198771149
x79=−89.5353907467661x_{79} = -89.5353907467661
x80=−36.1283154192437x_{80} = -36.1283154192437
x81=−10.9955741902138x_{81} = -10.9955741902138
x82=83.2522055730903x_{82} = 83.2522055730903
x83=76.9690200400775x_{83} = 76.9690200400775
x84=−80.1106125795659x_{84} = -80.1106125795659
x85=26.7035373461441x_{85} = 26.7035373461441
x86=17.2787598502655x_{86} = 17.2787598502655
x87=−39.2699081528781x_{87} = -39.2699081528781
x88=−17.2787598091171x_{88} = -17.2787598091171
x89=−26.7035375427973x_{89} = -26.7035375427973
x90=−45.5530935883361x_{90} = -45.5530935883361
x91=−32.9867227513827x_{91} = -32.9867227513827
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.
cos⁡2(0)\cos^{2}{\left(0 \right)}
The result:
f(0)=1f{\left(0 \right)} = 1
The point:
(0, 1)
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)cos⁡(x)=0- 2 \sin{\left(x \right)} \cos{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=−π2x_{2} = - \frac{\pi}{2}
x3=π2x_{3} = \frac{\pi}{2}
The values of the extrema at the points:
(0, 1)

 -pi     
(----, 0)
  2      

 pi    
(--, 0)
 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=−π2x_{1} = - \frac{\pi}{2}
x2=π2x_{2} = \frac{\pi}{2}
Maxima of the function at points:
x2=0x_{2} = 0
Decreasing at intervals
[−π2,0]∪[π2,∞)\left[- \frac{\pi}{2}, 0\right] \cup \left[\frac{\pi}{2}, \infty\right)
Increasing at intervals
(−∞,−π2]∪[0,π2]\left(-\infty, - \frac{\pi}{2}\right] \cup \left[0, \frac{\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
2(sin⁡2(x)−cos⁡2(x))=02 \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) = 0
Solve this equation
The roots of this equation
x1=−π4x_{1} = - \frac{\pi}{4}
x2=π4x_{2} = \frac{\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]∪[π4,∞)\left(-\infty, - \frac{\pi}{4}\right] \cup \left[\frac{\pi}{4}, \infty\right)
Convex at the intervals
[−π4,π4]\left[- \frac{\pi}{4}, \frac{\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→−∞cos⁡2(x)=⟨0,1⟩\lim_{x \to -\infty} \cos^{2}{\left(x \right)} = \left\langle 0, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨0,1⟩y = \left\langle 0, 1\right\rangle
lim⁡x→∞cos⁡2(x)=⟨0,1⟩\lim_{x \to \infty} \cos^{2}{\left(x \right)} = \left\langle 0, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨0,1⟩y = \left\langle 0, 1\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cos(x)^2, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(cos⁡2(x)x)=0\lim_{x \to -\infty}\left(\frac{\cos^{2}{\left(x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(cos⁡2(x)x)=0\lim_{x \to \infty}\left(\frac{\cos^{2}{\left(x \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⁡2(x)=cos⁡2(x)\cos^{2}{\left(x \right)} = \cos^{2}{\left(x \right)}
- Yes
cos⁡2(x)=−cos⁡2(x)\cos^{2}{\left(x \right)} = - \cos^{2}{\left(x \right)}
- No
so, the function
is
even
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