Mister Exam

Other calculators

Plotting a function graph/ sin(5x+4)

Graphing y = sin(5x+4)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
f(x) = sin(5*x + 4)
f(x)=sin⁡(5x+4)f{\left(x \right)} = \sin{\left(5 x + 4 \right)} 
f = sin(5*x + 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⁡(5x+4)=0\sin{\left(5 x + 4 \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−45x_{1} = - \frac{4}{5}
x2=−45+π5x_{2} = - \frac{4}{5} + \frac{\pi}{5}
Numerical solution
x1=36.2707933123596x_{1} = 36.2707933123596
x2=90.306186954104x_{2} = 90.306186954104
x3=9.8814150222053x_{3} = 9.8814150222053
x4=80.2530904626167x_{4} = 80.2530904626167
x5=63.9168086639497x_{5} = 63.9168086639497
x6=28.1026524130261x_{6} = 28.1026524130261
x7=100.359283445591x_{7} = 100.359283445591
x8=−61.746897479642x_{8} = -61.746897479642
x9=−27.8176968208722x_{9} = -27.8176968208722
x10=−83.7380460547705x_{10} = -83.7380460547705
x11=92.1911425462579x_{11} = 92.1911425462579
x12=−68.0300827868216x_{12} = -68.0300827868216
x13=−46.038934211693x_{13} = -46.038934211693
x14=34.3858377202057x_{14} = 34.3858377202057
x15=78.3681348704628x_{15} = 78.3681348704628
x16=53.8637121724624x_{16} = 53.8637121724624
x17=−78.0831792783089x_{17} = -78.0831792783089
x18=−2.05663706143592x_{18} = -2.05663706143592
x19=−69.9150383789755x_{19} = -69.9150383789755
x20=7.99645943005142x_{20} = 7.99645943005142
x21=58.2619418874881x_{21} = 58.2619418874881
x22=14.279644737231x_{22} = 14.279644737231
x23=−0.8x_{23} = -0.8
x24=−19.6495559215388x_{24} = -19.6495559215388
x25=−91.906186954104x_{25} = -91.906186954104
x26=−25.9327412287183x_{26} = -25.9327412287183
x27=2.34159265358979x_{27} = 2.34159265358979
x28=62.0318530717959x_{28} = 62.0318530717959
x29=31.8725635973339x_{29} = 31.8725635973339
x30=70.1999939711293x_{30} = 70.1999939711293
x31=−17.7646003293849x_{31} = -17.7646003293849
x32=19.9345115136926x_{32} = 19.9345115136926
x33=−35.9858377202057x_{33} = -35.9858377202057
x34=−12.1097335529233x_{34} = -12.1097335529233
x35=89.0495498926681x_{35} = 89.0495498926681
x36=38.1557489045134x_{36} = 38.1557489045134
x37=4.22654824574367x_{37} = 4.22654824574367
x38=−73.6849495632832x_{38} = -73.6849495632832
x39=−13.9946891450771x_{39} = -13.9946891450771
x40=739.359229185755x_{40} = 739.359229185755
x41=60.146897479642x_{41} = 60.146897479642
x42=−7.71150383789755x_{42} = -7.71150383789755
x43=72.0849495632832x_{43} = 72.0849495632832
x44=56.3769862953342x_{44} = 56.3769862953342
x45=−95.6760981384118x_{45} = -95.6760981384118
x46=96.5893722612836x_{46} = 96.5893722612836
x47=−37.8707933123596x_{47} = -37.8707933123596
x48=16.1646003293849x_{48} = 16.1646003293849
x49=94.0760981384118x_{49} = 94.0760981384118
x50=111.040698467797x_{50} = 111.040698467797
x51=−81.8530904626167x_{51} = -81.8530904626167
x52=−100.074327853437x_{52} = -100.074327853437
x53=−90.0212313619501x_{53} = -90.0212313619501
x54=18.0495559215388x_{54} = 18.0495559215388
x55=−59.8619418874881x_{55} = -59.8619418874881
x56=−15.879644737231x_{56} = -15.879644737231
x57=−63.6318530717959x_{57} = -63.6318530717959
x58=14836.3137843739x_{58} = 14836.3137843739
x59=−34.1008821280518x_{59} = -34.1008821280518
x60=17.4212373908208x_{60} = 17.4212373908208
x61=−47.9238898038469x_{61} = -47.9238898038469
x62=−35.3575191894877x_{62} = -35.3575191894877
x63=−56.0920307031804x_{63} = -56.0920307031804
x64=82.1380460547705x_{64} = 82.1380460547705
x65=−49.8088453960008x_{65} = -49.8088453960008
x66=−57.9769862953342x_{66} = -57.9769862953342
x67=40.0407044966673x_{67} = 40.0407044966673
x68=−29.7026524130261x_{68} = -29.7026524130261
x69=112.925654059951x_{69} = 112.925654059951
x70=−52.9504380495906x_{70} = -52.9504380495906
x71=24.3327412287183x_{71} = 24.3327412287183
x72=26.2176968208722x_{72} = 26.2176968208722
x73=−45.4106156809751x_{73} = -45.4106156809751
x74=−39.7557489045134x_{74} = -39.7557489045134
x75=46.3238898038469x_{75} = 46.3238898038469
x76=85.9079572390783x_{76} = 85.9079572390783
x77=−41.6407044966673x_{77} = -41.6407044966673
x78=6.11150383789755x_{78} = 6.11150383789755
x79=51.9787565803085x_{79} = 51.9787565803085
x80=29.98760800518x_{80} = 29.98760800518
x81=−51.6938009881546x_{81} = -51.6938009881546
x82=−8.3398223686155x_{82} = -8.3398223686155
x83=−3.94159265358979x_{83} = -3.94159265358979
x84=73.9699051554371x_{84} = 73.9699051554371
x85=−88.1362757697963x_{85} = -88.1362757697963
x86=−5.82654824574367x_{86} = -5.82654824574367
x87=95.9610537305656x_{87} = 95.9610537305656
x88=50.0938009881546x_{88} = 50.0938009881546
x89=−24.0477856365645x_{89} = -24.0477856365645
x90=68.3150383789754x_{90} = 68.3150383789754
x91=−93.7911425462579x_{91} = -93.7911425462579
x92=48.2088453960008x_{92} = 48.2088453960008
x93=84.0230016469244x_{93} = 84.0230016469244
x94=−71.7999939711293x_{94} = -71.7999939711293
x95=−85.6230016469244x_{95} = -85.6230016469244
x96=41.9256600888212x_{96} = 41.9256600888212
x97=−79.9681348704628x_{97} = -79.9681348704628
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(5*x + 4).
sin⁡(0⋅5+4)\sin{\left(0 \cdot 5 + 4 \right)}
The result:
f(0)=sin⁡(4)f{\left(0 \right)} = \sin{\left(4 \right)}
The point:
(0, sin(4))
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
5cos⁡(5x+4)=05 \cos{\left(5 x + 4 \right)} = 0
Solve this equation
The roots of this equation
x1=−45+π10x_{1} = - \frac{4}{5} + \frac{\pi}{10}
x2=−45+3π10x_{2} = - \frac{4}{5} + \frac{3 \pi}{10}
The values of the extrema at the points:
   4   pi    
(- - + --, 1)
   5   10    

   4   3*pi     
(- - + ----, -1)
   5    10


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=−45+3π10x_{1} = - \frac{4}{5} + \frac{3 \pi}{10}
Maxima of the function at points:
x1=−45+π10x_{1} = - \frac{4}{5} + \frac{\pi}{10}
Decreasing at intervals
(−∞,−45+π10]∪[−45+3π10,∞)\left(-\infty, - \frac{4}{5} + \frac{\pi}{10}\right] \cup \left[- \frac{4}{5} + \frac{3 \pi}{10}, \infty\right)
Increasing at intervals
[−45+π10,−45+3π10]\left[- \frac{4}{5} + \frac{\pi}{10}, - \frac{4}{5} + \frac{3 \pi}{10}\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
−25sin⁡(5x+4)=0- 25 \sin{\left(5 x + 4 \right)} = 0
Solve this equation
The roots of this equation
x1=−45x_{1} = - \frac{4}{5}
x2=−45+π5x_{2} = - \frac{4}{5} + \frac{\pi}{5}

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
(−∞,−45]∪[−45+π5,∞)\left(-\infty, - \frac{4}{5}\right] \cup \left[- \frac{4}{5} + \frac{\pi}{5}, \infty\right)
Convex at the intervals
[−45,−45+π5]\left[- \frac{4}{5}, - \frac{4}{5} + \frac{\pi}{5}\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⁡(5x+4)=⟨−1,1⟩\lim_{x \to -\infty} \sin{\left(5 x + 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⁡(5x+4)=⟨−1,1⟩\lim_{x \to \infty} \sin{\left(5 x + 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(5*x + 4), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡(5x+4)x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(5 x + 4 \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(sin⁡(5x+4)x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(5 x + 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⁡(5x+4)=−sin⁡(5x−4)\sin{\left(5 x + 4 \right)} = - \sin{\left(5 x - 4 \right)}
- No
sin⁡(5x+4)=sin⁡(5x−4)\sin{\left(5 x + 4 \right)} = \sin{\left(5 x - 4 \right)}
- No
so, the function
not is
neither even, nor odd
    © Mister Exam – Calculator