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3sinx-4cosx
Plotting a function graph/ 3sinx-4cosx

Graphing y = 3sinx-4cosx

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = 3*sin(x) - 4*cos(x)
f(x)=3sin⁡(x)−4cos⁡(x)f{\left(x \right)} = 3 \sin{\left(x \right)} - 4 \cos{\left(x \right)} 
f = 3*sin(x) - 4*cos(x)
The graph of the function
0-80-60-40-2020406080-1010
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:
3sin⁡(x)−4cos⁡(x)=03 \sin{\left(x \right)} - 4 \cos{\left(x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=atan⁡(43)x_{1} = \operatorname{atan}{\left(\frac{4}{3} \right)}
Numerical solution
x1=−87.0372990825126x_{1} = -87.0372990825126
x2=92.0334821721056x_{2} = 92.0334821721056
x3=−2.21429743558818x_{3} = -2.21429743558818
x4=−21.0638533571269x_{4} = -21.0638533571269
x5=−80.754113775333x_{5} = -80.754113775333
x6=104.599852786465x_{6} = 104.599852786465
x7=−58357.297837866x_{7} = -58357.297837866
x8=−96.462077043282x_{8} = -96.462077043282
x9=−68.1877431609738x_{9} = -68.1877431609738
x10=−30.4886313178963x_{10} = -30.4886313178963
x11=−14.7806680499474x_{11} = -14.7806680499474
x12=41.7679997146689x_{12} = 41.7679997146689
x13=−71.3293358145636x_{13} = -71.3293358145636
x14=−39.9134092786657x_{14} = -39.9134092786657
x15=38.6264070610791x_{15} = 38.6264070610791
x16=66.9007409433873x_{16} = 66.9007409433873
x17=−24.2054460107167x_{17} = -24.2054460107167
x18=95.1750748256954x_{18} = 95.1750748256954
x19=−27.3470386643065x_{19} = -27.3470386643065
x20=−5.35589008917797x_{20} = -5.35589008917797
x21=−77.6125211217432x_{21} = -77.6125211217432
x22=−33.6302239714861x_{22} = -33.6302239714861
x23=−17.9222607035371x_{23} = -17.9222607035371
x24=35.4848144074893x_{24} = 35.4848144074893
x25=48.0511850218485x_{25} = 48.0511850218485
x26=13.4936658323608x_{26} = 13.4936658323608
x27=−58.7629652002045x_{27} = -58.7629652002045
x28=10.352073178771x_{28} = 10.352073178771
x29=60.6175556362077x_{29} = 60.6175556362077
x30=−99.6036696968718x_{30} = -99.6036696968718
x31=70.0423335969771x_{31} = 70.0423335969771
x32=76.3255189041567x_{32} = 76.3255189041567
x33=22.9184437931302x_{33} = 22.9184437931302
x34=16.6352584859506x_{34} = 16.6352584859506
x35=19.7768511395404x_{35} = 19.7768511395404
x36=−74.4709284681534x_{36} = -74.4709284681534
x37=57.4759629826179x_{37} = 57.4759629826179
x38=−206.417819918925x_{38} = -206.417819918925
x39=−36.7718166250759x_{39} = -36.7718166250759
x40=44.9095923682587x_{40} = 44.9095923682587
x41=7.2104805251812x_{41} = 7.2104805251812
x42=29.2016291003098x_{42} = 29.2016291003098
x43=88.8918895185158x_{43} = 88.8918895185158
x44=−49.3381872394351x_{44} = -49.3381872394351
x45=−93.3204843896922x_{45} = -93.3204843896922
x46=−65.046150507384x_{46} = -65.046150507384
x47=−11.6390753963576x_{47} = -11.6390753963576
x48=−43.0550019322555x_{48} = -43.0550019322555
x49=−52.4797798930249x_{49} = -52.4797798930249
x50=32.3432217538995x_{50} = 32.3432217538995
x51=63.7591482897975x_{51} = 63.7591482897975
x52=54.3343703290281x_{52} = 54.3343703290281
x53=82.6087042113362x_{53} = 82.6087042113362
x54=−83.8957064289228x_{54} = -83.8957064289228
x55=98.3166674792852x_{55} = 98.3166674792852
x56=51.1927776754383x_{56} = 51.1927776754383
x57=−8.49748274276777x_{57} = -8.49748274276777
x58=0.927295218001612x_{58} = 0.927295218001612
x59=73.1839262505669x_{59} = 73.1839262505669
x60=4.06888787159141x_{60} = 4.06888787159141
x61=85.750296864926x_{61} = 85.750296864926
x62=−90.1788917361024x_{62} = -90.1788917361024
x63=−61.9045578537943x_{63} = -61.9045578537943
x64=26.06003644672x_{64} = 26.06003644672
x65=79.4671115577464x_{65} = 79.4671115577464
x66=−46.1965945858453x_{66} = -46.1965945858453
x67=−55.6213725466147x_{67} = -55.6213725466147
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 3*sin(x) - 4*cos(x).
−4cos⁡(0)+3sin⁡(0)- 4 \cos{\left(0 \right)} + 3 \sin{\left(0 \right)}
The result:
f(0)=−4f{\left(0 \right)} = -4
The point:
(0, -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
4sin⁡(x)+3cos⁡(x)=04 \sin{\left(x \right)} + 3 \cos{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=−atan⁡(34)x_{1} = - \operatorname{atan}{\left(\frac{3}{4} \right)}
The values of the extrema at the points:
(-atan(3/4), -5)


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=−atan⁡(34)x_{1} = - \operatorname{atan}{\left(\frac{3}{4} \right)}
The function has no maxima
Decreasing at intervals
[−atan⁡(34),∞)\left[- \operatorname{atan}{\left(\frac{3}{4} \right)}, \infty\right)
Increasing at intervals
(−∞,−atan⁡(34)]\left(-\infty, - \operatorname{atan}{\left(\frac{3}{4} \right)}\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
−3sin⁡(x)+4cos⁡(x)=0- 3 \sin{\left(x \right)} + 4 \cos{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=atan⁡(43)x_{1} = \operatorname{atan}{\left(\frac{4}{3} \right)}

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
(−∞,atan⁡(43)]\left(-\infty, \operatorname{atan}{\left(\frac{4}{3} \right)}\right]
Convex at the intervals
[atan⁡(43),∞)\left[\operatorname{atan}{\left(\frac{4}{3} \right)}, \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→−∞(3sin⁡(x)−4cos⁡(x))=⟨−7,7⟩\lim_{x \to -\infty}\left(3 \sin{\left(x \right)} - 4 \cos{\left(x \right)}\right) = \left\langle -7, 7\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−7,7⟩y = \left\langle -7, 7\right\rangle
lim⁡x→∞(3sin⁡(x)−4cos⁡(x))=⟨−7,7⟩\lim_{x \to \infty}\left(3 \sin{\left(x \right)} - 4 \cos{\left(x \right)}\right) = \left\langle -7, 7\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−7,7⟩y = \left\langle -7, 7\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 3*sin(x) - 4*cos(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(3sin⁡(x)−4cos⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{3 \sin{\left(x \right)} - 4 \cos{\left(x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(3sin⁡(x)−4cos⁡(x)x)=0\lim_{x \to \infty}\left(\frac{3 \sin{\left(x \right)} - 4 \cos{\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:
3sin⁡(x)−4cos⁡(x)=−3sin⁡(x)−4cos⁡(x)3 \sin{\left(x \right)} - 4 \cos{\left(x \right)} = - 3 \sin{\left(x \right)} - 4 \cos{\left(x \right)}
- No
3sin⁡(x)−4cos⁡(x)=3sin⁡(x)+4cos⁡(x)3 \sin{\left(x \right)} - 4 \cos{\left(x \right)} = 3 \sin{\left(x \right)} + 4 \cos{\left(x \right)}
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = 3sinx-4cosx
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