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tan(x-pi/4)
Plotting a function graph/ tan(x-pi/4)

You entered:

tan(x-pi/4)

What you mean?

tan((x - pi)/4)
 
tan(x - pi/4)
 
tan(x - pi/4)

Graphing y = tan(x-pi/4)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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          /    pi\
f(x) = tan|x - --|
          \    4 /
f(x)=tan⁡(x−π4)f{\left(x \right)} = \tan{\left(x - \frac{\pi}{4} \right)} 
f = tan(x - pi/4)
The graph of the function
0-80-70-60-50-40-30-20-101020-250250
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:
tan⁡(x−π4)=0\tan{\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}
Numerical solution
x1=−36.9137136796801x_{1} = -36.9137136796801
x2=25.9181393921158x_{2} = 25.9181393921158
x3=−80.8960108299372x_{3} = -80.8960108299372
x4=10.2101761241668x_{4} = 10.2101761241668
x5=−14.9225651045515x_{5} = -14.9225651045515
x6=95.0331777710912x_{6} = 95.0331777710912
x7=41.6261026600648x_{7} = 41.6261026600648
x8=85.6083998103219x_{8} = 85.6083998103219
x9=−46.3384916404494x_{9} = -46.3384916404494
x10=−93.4623814442964x_{10} = -93.4623814442964
x11=−11.7809724509617x_{11} = -11.7809724509617
x12=16.4933614313464x_{12} = 16.4933614313464
x13=−77.7544181763474x_{13} = -77.7544181763474
x14=−18.0641577581413x_{14} = -18.0641577581413
x15=−74.6128255227576x_{15} = -74.6128255227576
x16=35.3429173528852x_{16} = 35.3429173528852
x17=0.785398163397448x_{17} = 0.785398163397448
x18=98.174770424681x_{18} = 98.174770424681
x19=88.7499924639117x_{19} = 88.7499924639117
x20=66.7588438887831x_{20} = 66.7588438887831
x21=13.3517687777566x_{21} = 13.3517687777566
x22=79.3252145031423x_{22} = 79.3252145031423
x23=63.6172512351933x_{23} = 63.6172512351933
x24=82.4668071567321x_{24} = 82.4668071567321
x25=−21.2057504117311x_{25} = -21.2057504117311
x26=19.6349540849362x_{26} = 19.6349540849362
x27=−49.4800842940392x_{27} = -49.4800842940392
x28=−99.7455667514759x_{28} = -99.7455667514759
x29=44.7676953136546x_{29} = 44.7676953136546
x30=51.0508806208341x_{30} = 51.0508806208341
x31=−52.621676947629x_{31} = -52.621676947629
x32=76.1836218495525x_{32} = 76.1836218495525
x33=−68.329640215578x_{33} = -68.329640215578
x34=−33.7721210260903x_{34} = -33.7721210260903
x35=60.4756585816035x_{35} = 60.4756585816035
x36=−71.4712328691678x_{36} = -71.4712328691678
x37=38.484510006475x_{37} = 38.484510006475
x38=101.316363078271x_{38} = 101.316363078271
x39=−2.35619449019234x_{39} = -2.35619449019234
x40=−30.6305283725005x_{40} = -30.6305283725005
x41=−40.0553063332699x_{41} = -40.0553063332699
x42=22.776546738526x_{42} = 22.776546738526
x43=47.9092879672443x_{43} = 47.9092879672443
x44=7.06858347057703x_{44} = 7.06858347057703
x45=−90.3207887907066x_{45} = -90.3207887907066
x46=54.1924732744239x_{46} = 54.1924732744239
x47=57.3340659280137x_{47} = 57.3340659280137
x48=−43.1968989868597x_{48} = -43.1968989868597
x49=3.92699081698724x_{49} = 3.92699081698724
x50=−24.3473430653209x_{50} = -24.3473430653209
x51=−65.1880475619882x_{51} = -65.1880475619882
x52=−62.0464549083984x_{52} = -62.0464549083984
x53=−87.1791961371168x_{53} = -87.1791961371168
x54=−55.7632696012188x_{54} = -55.7632696012188
x55=−5.49778714378214x_{55} = -5.49778714378214
x56=−96.6039740978861x_{56} = -96.6039740978861
x57=−27.4889357189107x_{57} = -27.4889357189107
x58=73.0420291959627x_{58} = 73.0420291959627
x59=91.8915851175014x_{59} = 91.8915851175014
x60=29.0597320457056x_{60} = 29.0597320457056
x61=−8.63937979737193x_{61} = -8.63937979737193
x62=32.2013246992954x_{62} = 32.2013246992954
x63=69.9004365423729x_{63} = 69.9004365423729
x64=−58.9048622548086x_{64} = -58.9048622548086
x65=−84.037603483527x_{65} = -84.037603483527
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to tan(x - pi/4).
tan⁡(−π4+0)\tan{\left(- \frac{\pi}{4} + 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
tan⁡2(x−π4)+1=0\tan^{2}{\left(x - \frac{\pi}{4} \right)} + 1 = 0
Solve this equation
Solutions are not found,
function may have no extrema
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(cot⁡2(x+π4)+1)cot⁡(x+π4)=0- 2 \left(\cot^{2}{\left(x + \frac{\pi}{4} \right)} + 1\right) \cot{\left(x + \frac{\pi}{4} \right)} = 0
Solve this equation
The roots of this equation
x1=π4x_{1} = \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,∞)\left[\frac{\pi}{4}, \infty\right)
Convex at the intervals
(−∞,π4]\left(-\infty, \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→−∞tan⁡(x−π4)=⟨−∞,∞⟩\lim_{x \to -\infty} \tan{\left(x - \frac{\pi}{4} \right)} = \left\langle -\infty, \infty\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
lim⁡x→∞tan⁡(x−π4)=⟨−∞,∞⟩\lim_{x \to \infty} \tan{\left(x - \frac{\pi}{4} \right)} = \left\langle -\infty, \infty\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of tan(x - pi/4), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(tan⁡(x−π4)x)=lim⁡x→−∞(tan⁡(x−π4)x)\lim_{x \to -\infty}\left(\frac{\tan{\left(x - \frac{\pi}{4} \right)}}{x}\right) = \lim_{x \to -\infty}\left(\frac{\tan{\left(x - \frac{\pi}{4} \right)}}{x}\right)
Let's take the limit
so,
inclined asymptote equation on the left:
y=xlim⁡x→−∞(tan⁡(x−π4)x)y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(x - \frac{\pi}{4} \right)}}{x}\right)
lim⁡x→∞(tan⁡(x−π4)x)=lim⁡x→∞(tan⁡(x−π4)x)\lim_{x \to \infty}\left(\frac{\tan{\left(x - \frac{\pi}{4} \right)}}{x}\right) = \lim_{x \to \infty}\left(\frac{\tan{\left(x - \frac{\pi}{4} \right)}}{x}\right)
Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(tan⁡(x−π4)x)y = x \lim_{x \to \infty}\left(\frac{\tan{\left(x - \frac{\pi}{4} \right)}}{x}\right)
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:
tan⁡(x−π4)=−tan⁡(x+π4)\tan{\left(x - \frac{\pi}{4} \right)} = - \tan{\left(x + \frac{\pi}{4} \right)}
- No
tan⁡(x−π4)=tan⁡(x+π4)\tan{\left(x - \frac{\pi}{4} \right)} = \tan{\left(x + \frac{\pi}{4} \right)}
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = tan(x-pi/4)
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