Mister Exam

Other calculators

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

Graphing y = sin(x-pi/4)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          /    pi\
f(x) = sin|x - --|
          \    4 /
f(x)=sin⁡(x−π4)f{\left(x \right)} = \sin{\left(x - \frac{\pi}{4} \right)} 
f = sin(x - pi/4)
The graph of the function
02468-8-6-4-2-10102-2
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⁡(x−π4)=0\sin{\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=5π4x_{2} = \frac{5 \pi}{4}
Numerical solution
x1=0.785398163397448x_{1} = 0.785398163397448
x2=−21.2057504117311x_{2} = -21.2057504117311
x3=−55.7632696012188x_{3} = -55.7632696012188
x4=76.1836218495525x_{4} = 76.1836218495525
x5=41.6261026600648x_{5} = 41.6261026600648
x6=−43.1968989868597x_{6} = -43.1968989868597
x7=−62.0464549083984x_{7} = -62.0464549083984
x8=−49.4800842940392x_{8} = -49.4800842940392
x9=51.0508806208341x_{9} = 51.0508806208341
x10=101.316363078271x_{10} = 101.316363078271
x11=79.3252145031423x_{11} = 79.3252145031423
x12=−52.621676947629x_{12} = -52.621676947629
x13=19.6349540849362x_{13} = 19.6349540849362
x14=−36.9137136796801x_{14} = -36.9137136796801
x15=3.92699081698724x_{15} = 3.92699081698724
x16=−24.3473430653209x_{16} = -24.3473430653209
x17=35.3429173528852x_{17} = 35.3429173528852
x18=82.4668071567321x_{18} = 82.4668071567321
x19=−46.3384916404494x_{19} = -46.3384916404494
x20=60.4756585816035x_{20} = 60.4756585816035
x21=−27.4889357189107x_{21} = -27.4889357189107
x22=69.9004365423729x_{22} = 69.9004365423729
x23=−74.6128255227576x_{23} = -74.6128255227576
x24=−1535.45340944201x_{24} = -1535.45340944201
x25=32.2013246992954x_{25} = 32.2013246992954
x26=29.0597320457056x_{26} = 29.0597320457056
x27=−2.35619449019234x_{27} = -2.35619449019234
x28=88.7499924639117x_{28} = 88.7499924639117
x29=−93.4623814442964x_{29} = -93.4623814442964
x30=−87.1791961371168x_{30} = -87.1791961371168
x31=−90.3207887907066x_{31} = -90.3207887907066
x32=47.9092879672443x_{32} = 47.9092879672443
x33=−77.7544181763474x_{33} = -77.7544181763474
x34=−99.7455667514759x_{34} = -99.7455667514759
x35=73.0420291959627x_{35} = 73.0420291959627
x36=95.0331777710912x_{36} = 95.0331777710912
x37=54.1924732744239x_{37} = 54.1924732744239
x38=38.484510006475x_{38} = 38.484510006475
x39=−65.1880475619882x_{39} = -65.1880475619882
x40=−96.6039740978861x_{40} = -96.6039740978861
x41=66.7588438887831x_{41} = 66.7588438887831
x42=−5.49778714378214x_{42} = -5.49778714378214
x43=−80.8960108299372x_{43} = -80.8960108299372
x44=−68.329640215578x_{44} = -68.329640215578
x45=1672.11268987317x_{45} = 1672.11268987317
x46=57.3340659280137x_{46} = 57.3340659280137
x47=7.06858347057703x_{47} = 7.06858347057703
x48=85.6083998103219x_{48} = 85.6083998103219
x49=−11.7809724509617x_{49} = -11.7809724509617
x50=−71.4712328691678x_{50} = -71.4712328691678
x51=−8.63937979737193x_{51} = -8.63937979737193
x52=16.4933614313464x_{52} = 16.4933614313464
x53=98.174770424681x_{53} = 98.174770424681
x54=44.7676953136546x_{54} = 44.7676953136546
x55=−40.0553063332699x_{55} = -40.0553063332699
x56=22.776546738526x_{56} = 22.776546738526
x57=−14.9225651045515x_{57} = -14.9225651045515
x58=−30.6305283725005x_{58} = -30.6305283725005
x59=63.6172512351933x_{59} = 63.6172512351933
x60=−228.550865548657x_{60} = -228.550865548657
x61=−58.9048622548086x_{61} = -58.9048622548086
x62=91.8915851175014x_{62} = 91.8915851175014
x63=25.9181393921158x_{63} = 25.9181393921158
x64=−84.037603483527x_{64} = -84.037603483527
x65=−18.0641577581413x_{65} = -18.0641577581413
x66=−33.7721210260903x_{66} = -33.7721210260903
x67=10.2101761241668x_{67} = 10.2101761241668
x68=13.3517687777566x_{68} = 13.3517687777566
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(x - pi/4).
sin⁡(−π4)\sin{\left(- \frac{\pi}{4} \right)}
The result:
f(0)=−22f{\left(0 \right)} = - \frac{\sqrt{2}}{2}
The point:
(0, -sqrt(2)/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
cos⁡(x−π4)=0\cos{\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}
The values of the extrema at the points:
 -pi       /pi   pi\ 
(----, -sin|-- + --|)
  4        \4    4 / 

 3*pi     /pi   pi\ 
(----, 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=3π4x_{1} = \frac{3 \pi}{4}
Decreasing at intervals
[−π4,3π4]\left[- \frac{\pi}{4}, \frac{3 \pi}{4}\right]
Increasing at intervals
(−∞,−π4]∪[3π4,∞)\left(-\infty, - \frac{\pi}{4}\right] \cup \left[\frac{3 \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
cos⁡(x+π4)=0\cos{\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}

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]∪[5π4,∞)\left(-\infty, \frac{\pi}{4}\right] \cup \left[\frac{5 \pi}{4}, \infty\right)
Convex at the intervals
[π4,5π4]\left[\frac{\pi}{4}, \frac{5 \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→−∞sin⁡(x−π4)=⟨−1,1⟩\lim_{x \to -\infty} \sin{\left(x - \frac{\pi}{4} \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⁡(x−π4)=⟨−1,1⟩\lim_{x \to \infty} \sin{\left(x - \frac{\pi}{4} \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 - pi/4), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡(x−π4)x)=0\lim_{x \to -\infty}\left(\frac{\sin{\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→∞(sin⁡(x−π4)x)=0\lim_{x \to \infty}\left(\frac{\sin{\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:
sin⁡(x−π4)=−sin⁡(x+π4)\sin{\left(x - \frac{\pi}{4} \right)} = - \sin{\left(x + \frac{\pi}{4} \right)}
- No
sin⁡(x−π4)=sin⁡(x+π4)\sin{\left(x - \frac{\pi}{4} \right)} = \sin{\left(x + \frac{\pi}{4} \right)}
- No
so, the function
not is
neither even, nor odd
    © Mister Exam – Calculator