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sin^4x-cos^4x
Plotting a function graph/ sin^4x-cos^4x

Graphing y = sin^4x-cos^4x

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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          4         4   
f(x) = sin (x) - cos (x)
f(x)=sin⁡4(x)−cos⁡4(x)f{\left(x \right)} = \sin^{4}{\left(x \right)} - \cos^{4}{\left(x \right)} 
f = sin(x)^4 - cos(x)^4
The graph of the function
0-80-70-60-50-40-30-20-10102-2
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:
sin⁡4(x)−cos⁡4(x)=0\sin^{4}{\left(x \right)} - \cos^{4}{\left(x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−π4x_{1} = - \frac{\pi}{4}
x2=π4x_{2} = \frac{\pi}{4}
Numerical solution
x1=−85.6083998103219x_{1} = -85.6083998103219
x2=−68.329640215578x_{2} = -68.329640215578
x3=−62.0464549083984x_{3} = -62.0464549083984
x4=−84.037603483527x_{4} = -84.037603483527
x5=5.49778714378214x_{5} = 5.49778714378214
x6=162.577419823272x_{6} = 162.577419823272
x7=−13.3517687777566x_{7} = -13.3517687777566
x8=16.4933614313464x_{8} = 16.4933614313464
x9=90.3207887907066x_{9} = 90.3207887907066
x10=−11.7809724509617x_{10} = -11.7809724509617
x11=27.4889357189107x_{11} = 27.4889357189107
x12=2.35619449019234x_{12} = 2.35619449019234
x13=−82.4668071567321x_{13} = -82.4668071567321
x14=24.3473430653209x_{14} = 24.3473430653209
x15=96.6039740978861x_{15} = 96.6039740978861
x16=55.7632696012188x_{16} = 55.7632696012188
x17=−24.3473430653209x_{17} = -24.3473430653209
x18=99.7455667514759x_{18} = 99.7455667514759
x19=−60.4756585816035x_{19} = -60.4756585816035
x20=−32.2013246992954x_{20} = -32.2013246992954
x21=−69.9004365423729x_{21} = -69.9004365423729
x22=44.7676953136546x_{22} = 44.7676953136546
x23=−99.7455667514759x_{23} = -99.7455667514759
x24=63.6172512351933x_{24} = 63.6172512351933
x25=88.7499924639117x_{25} = 88.7499924639117
x26=66.7588438887831x_{26} = 66.7588438887831
x27=−57.3340659280137x_{27} = -57.3340659280137
x28=46.3384916404494x_{28} = 46.3384916404494
x29=32.2013246992954x_{29} = 32.2013246992954
x30=69.9004365423729x_{30} = 69.9004365423729
x31=74.6128255227576x_{31} = 74.6128255227576
x32=−55.7632696012188x_{32} = -55.7632696012188
x33=49.4800842940392x_{33} = 49.4800842940392
x34=−47.9092879672443x_{34} = -47.9092879672443
x35=−5.49778714378214x_{35} = -5.49778714378214
x36=−63.6172512351933x_{36} = -63.6172512351933
x37=−41.6261026600648x_{37} = -41.6261026600648
x38=−79.3252145031423x_{38} = -79.3252145031423
x39=68.329640215578x_{39} = 68.329640215578
x40=8.63937979737193x_{40} = 8.63937979737193
x41=40.0553063332699x_{41} = 40.0553063332699
x42=60.4756585816035x_{42} = 60.4756585816035
x43=10.2101761241668x_{43} = 10.2101761241668
x44=19.6349540849362x_{44} = 19.6349540849362
x45=−76.1836218495525x_{45} = -76.1836218495525
x46=−27.4889357189107x_{46} = -27.4889357189107
x47=3.92699081698724x_{47} = 3.92699081698724
x48=−98.174770424681x_{48} = -98.174770424681
x49=−40.0553063332699x_{49} = -40.0553063332699
x50=−77.7544181763474x_{50} = -77.7544181763474
x51=−54.1924732744239x_{51} = -54.1924732744239
x52=−1144.32512407008x_{52} = -1144.32512407008
x53=−3.92699081698724x_{53} = -3.92699081698724
x54=98.174770424681x_{54} = 98.174770424681
x55=22.776546738526x_{55} = 22.776546738526
x56=84.037603483527x_{56} = 84.037603483527
x57=−33.7721210260903x_{57} = -33.7721210260903
x58=33.7721210260903x_{58} = 33.7721210260903
x59=−91.8915851175014x_{59} = -91.8915851175014
x60=−90.3207887907066x_{60} = -90.3207887907066
x61=91.8915851175014x_{61} = 91.8915851175014
x62=38.484510006475x_{62} = 38.484510006475
x63=−71.4712328691678x_{63} = -71.4712328691678
x64=−93.4623814442964x_{64} = -93.4623814442964
x65=52.621676947629x_{65} = 52.621676947629
x66=384.059701901352x_{66} = 384.059701901352
x67=62.0464549083984x_{67} = 62.0464549083984
x68=−2.35619449019234x_{68} = -2.35619449019234
x69=−25.9181393921158x_{69} = -25.9181393921158
x70=85.6083998103219x_{70} = 85.6083998103219
x71=41.6261026600648x_{71} = 41.6261026600648
x72=30.6305283725005x_{72} = 30.6305283725005
x73=−38.484510006475x_{73} = -38.484510006475
x74=−16.4933614313464x_{74} = -16.4933614313464
x75=18.0641577581413x_{75} = 18.0641577581413
x76=−49.4800842940392x_{76} = -49.4800842940392
x77=54.1924732744239x_{77} = 54.1924732744239
x78=−19.6349540849362x_{78} = -19.6349540849362
x79=−18.0641577581413x_{79} = -18.0641577581413
x80=11.7809724509617x_{80} = 11.7809724509617
x81=76.1836218495525x_{81} = 76.1836218495525
x82=−35.3429173528852x_{82} = -35.3429173528852
x83=87.1791961371168x_{83} = 87.1791961371168
x84=−10.2101761241668x_{84} = -10.2101761241668
x85=25.9181393921158x_{85} = 25.9181393921158
x86=−46.3384916404494x_{86} = -46.3384916404494
x87=82.4668071567321x_{87} = 82.4668071567321
x88=77.7544181763474x_{88} = 77.7544181763474
x89=47.9092879672443x_{89} = 47.9092879672443
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(x)^4 - cos(x)^4.
−cos⁡4(0)+sin⁡4(0)- \cos^{4}{\left(0 \right)} + \sin^{4}{\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
4sin⁡3(x)cos⁡(x)+4sin⁡(x)cos⁡3(x)=04 \sin^{3}{\left(x \right)} \cos{\left(x \right)} + 4 \sin{\left(x \right)} \cos^{3}{\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     
(----, 1)
  2      

 pi    
(--, 1)
 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=0x_{1} = 0
Maxima of the function at points:
x1=−π2x_{1} = - \frac{\pi}{2}
x1=π2x_{1} = \frac{\pi}{2}
Decreasing at intervals
(−∞,−π2]∪[0,∞)\left(-\infty, - \frac{\pi}{2}\right] \cup \left[0, \infty\right)
Increasing at intervals
(−∞,0]∪[π2,∞)\left(-\infty, 0\right] \cup \left[\frac{\pi}{2}, \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
4(−sin⁡4(x)+cos⁡4(x))=04 \left(- \sin^{4}{\left(x \right)} + \cos^{4}{\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[- \frac{\pi}{4}, \frac{\pi}{4}\right]
Convex at the intervals
(−∞,−π4]∪[π4,∞)\left(-\infty, - \frac{\pi}{4}\right] \cup \left[\frac{\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→−∞(sin⁡4(x)−cos⁡4(x))=⟨−1,1⟩\lim_{x \to -\infty}\left(\sin^{4}{\left(x \right)} - \cos^{4}{\left(x \right)}\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→∞(sin⁡4(x)−cos⁡4(x))=⟨−1,1⟩\lim_{x \to \infty}\left(\sin^{4}{\left(x \right)} - \cos^{4}{\left(x \right)}\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 sin(x)^4 - cos(x)^4, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡4(x)−cos⁡4(x)x)=0\lim_{x \to -\infty}\left(\frac{\sin^{4}{\left(x \right)} - \cos^{4}{\left(x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(sin⁡4(x)−cos⁡4(x)x)=0\lim_{x \to \infty}\left(\frac{\sin^{4}{\left(x \right)} - \cos^{4}{\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:
sin⁡4(x)−cos⁡4(x)=sin⁡4(x)−cos⁡4(x)\sin^{4}{\left(x \right)} - \cos^{4}{\left(x \right)} = \sin^{4}{\left(x \right)} - \cos^{4}{\left(x \right)}
- Yes
sin⁡4(x)−cos⁡4(x)=−sin⁡4(x)+cos⁡4(x)\sin^{4}{\left(x \right)} - \cos^{4}{\left(x \right)} = - \sin^{4}{\left(x \right)} + \cos^{4}{\left(x \right)}
- No
so, the function
is
even
The graph
Graphing y = sin^4x-cos^4x
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