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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 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=−38.6264070610791x_{1} = -38.6264070610791
x2=83.8957064289228x_{2} = 83.8957064289228
x3=−29.2016291003098x_{3} = -29.2016291003098
x4=55.6213725466147x_{4} = 55.6213725466147
x5=−35.4848144074893x_{5} = -35.4848144074893
x6=24.2054460107167x_{6} = 24.2054460107167
x7=77.6125211217432x_{7} = 77.6125211217432
x8=−88.8918895185158x_{8} = -88.8918895185158
x9=−26.06003644672x_{9} = -26.06003644672
x10=−19.7768511395404x_{10} = -19.7768511395404
x11=−76.3255189041567x_{11} = -76.3255189041567
x12=27.3470386643065x_{12} = 27.3470386643065
x13=−7.2104805251812x_{13} = -7.2104805251812
x14=−73.1839262505669x_{14} = -73.1839262505669
x15=−32.3432217538995x_{15} = -32.3432217538995
x16=−4.06888787159141x_{16} = -4.06888787159141
x17=17.9222607035371x_{17} = 17.9222607035371
x18=90.1788917361024x_{18} = 90.1788917361024
x19=74.4709284681534x_{19} = 74.4709284681534
x20=96.462077043282x_{20} = 96.462077043282
x21=−0.927295218001612x_{21} = -0.927295218001612
x22=11.6390753963576x_{22} = 11.6390753963576
x23=93.3204843896922x_{23} = 93.3204843896922
x24=−51.1927776754383x_{24} = -51.1927776754383
x25=87.0372990825126x_{25} = 87.0372990825126
x26=109.028447657641x_{26} = 109.028447657641
x27=−44.9095923682587x_{27} = -44.9095923682587
x28=−92.0334821721056x_{28} = -92.0334821721056
x29=68.1877431609738x_{29} = 68.1877431609738
x30=8.49748274276777x_{30} = 8.49748274276777
x31=−79.4671115577464x_{31} = -79.4671115577464
x32=52.4797798930249x_{32} = 52.4797798930249
x33=21.0638533571269x_{33} = 21.0638533571269
x34=5.35589008917797x_{34} = 5.35589008917797
x35=−48.0511850218485x_{35} = -48.0511850218485
x36=2.21429743558818x_{36} = 2.21429743558818
x37=65.046150507384x_{37} = 65.046150507384
x38=168.718708075847x_{38} = 168.718708075847
x39=−66.9007409433873x_{39} = -66.9007409433873
x40=−710.927234929295x_{40} = -710.927234929295
x41=61.9045578537943x_{41} = 61.9045578537943
x42=43.0550019322555x_{42} = 43.0550019322555
x43=49.3381872394351x_{43} = 49.3381872394351
x44=−70.0423335969771x_{44} = -70.0423335969771
x45=58.7629652002045x_{45} = 58.7629652002045
x46=−41.7679997146689x_{46} = -41.7679997146689
x47=71.3293358145636x_{47} = 71.3293358145636
x48=−85.750296864926x_{48} = -85.750296864926
x49=−10.352073178771x_{49} = -10.352073178771
x50=33.6302239714861x_{50} = 33.6302239714861
x51=36.7718166250759x_{51} = 36.7718166250759
x52=99.6036696968718x_{52} = 99.6036696968718
x53=−82.6087042113362x_{53} = -82.6087042113362
x54=−54.3343703290281x_{54} = -54.3343703290281
x55=−63.7591482897975x_{55} = -63.7591482897975
x56=−57.4759629826179x_{56} = -57.4759629826179
x57=39.9134092786657x_{57} = 39.9134092786657
x58=−98.3166674792852x_{58} = -98.3166674792852
x59=−22.9184437931302x_{59} = -22.9184437931302
x60=14.7806680499474x_{60} = 14.7806680499474
x61=30.4886313178963x_{61} = 30.4886313178963
x62=46.1965945858453x_{62} = 46.1965945858453
x63=−95.1750748256954x_{63} = -95.1750748256954
x64=−60.6175556362077x_{64} = -60.6175556362077
x65=80.754113775333x_{65} = 80.754113775333
x66=−13.4936658323608x_{66} = -13.4936658323608
x67=−16.6352584859506x_{67} = -16.6352584859506
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).
3sin⁡(0)+4cos⁡(0)3 \sin{\left(0 \right)} + 4 \cos{\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)=0- 4 \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:
The function has no minima
Maxima of the function at points:
x1=atan⁡(34)x_{1} = \operatorname{atan}{\left(\frac{3}{4} \right)}
Decreasing at intervals
(−∞,atan⁡(34)]\left(-\infty, \operatorname{atan}{\left(\frac{3}{4} \right)}\right]
Increasing at intervals
[atan⁡(34),∞)\left[\operatorname{atan}{\left(\frac{3}{4} \right)}, \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
−(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
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