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Plotting a function graph/ 2cosx-1,5

Graphing y = 2cosx-1,5

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = 2*cos(x) - 3/2
f(x)=2cos⁡(x)−32f{\left(x \right)} = 2 \cos{\left(x \right)} - \frac{3}{2} 
f = 2*cos(x) - 3/2
The graph of the function
02468-8-6-4-2-10105-5
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:
2cos⁡(x)−32=02 \cos{\left(x \right)} - \frac{3}{2} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−acos⁡(34)+2πx_{1} = - \operatorname{acos}{\left(\frac{3}{4} \right)} + 2 \pi
x2=acos⁡(34)x_{2} = \operatorname{acos}{\left(\frac{3}{4} \right)}
Numerical solution
x1=−38.4218460908909x_{1} = -38.4218460908909
x2=82.404143241148x_{2} = 82.404143241148
x3=−57.2714020124297x_{3} = -57.2714020124297
x4=−50.9882167052501x_{4} = -50.9882167052501
x5=68.392304131162x_{5} = 68.392304131162
x6=−63.5545873196093x_{6} = -63.5545873196093
x7=−24.4100069809049x_{7} = -24.4100069809049
x8=−32.1386607837113x_{8} = -32.1386607837113
x9=−30.6931922880845x_{9} = -30.6931922880845
x10=43.2595629024437x_{10} = 43.2595629024437
x11=63.5545873196093x_{11} = 63.5545873196093
x12=−55.8259335168029x_{12} = -55.8259335168029
x13=−7.005919554993x_{13} = -7.005919554993
x14=13.2891048621726x_{14} = 13.2891048621726
x15=−43.2595629024437x_{15} = -43.2595629024437
x16=38.4218460908909x_{16} = 38.4218460908909
x17=−68.392304131162x_{17} = -68.392304131162
x18=−62.1091188239824x_{18} = -62.1091188239824
x19=−11.8436363665458x_{19} = -11.8436363665458
x20=−36.9763775952641x_{20} = -36.9763775952641
x21=−0.722734247813416x_{21} = -0.722734247813416
x22=62.1091188239824x_{22} = 62.1091188239824
x23=−13.2891048621726x_{23} = -13.2891048621726
x24=87.2418600527008x_{24} = 87.2418600527008
x25=36.9763775952641x_{25} = 36.9763775952641
x26=11.8436363665458x_{26} = 11.8436363665458
x27=−99.80823066706x_{27} = -99.80823066706
x28=49.5427482096233x_{28} = 49.5427482096233
x29=−44.7050313980705x_{29} = -44.7050313980705
x30=16770.5443191101x_{30} = 16770.5443191101
x31=69.8377726267889x_{31} = 69.8377726267889
x32=−93.5250453598804x_{32} = -93.5250453598804
x33=−76.1209579339685x_{33} = -76.1209579339685
x34=420.250681333219x_{34} = 420.250681333219
x35=−82.404143241148x_{35} = -82.404143241148
x36=30.6931922880845x_{36} = 30.6931922880845
x37=−18.1268216737253x_{37} = -18.1268216737253
x38=−69.8377726267889x_{38} = -69.8377726267889
x39=157.802366927303x_{39} = 157.802366927303
x40=−19.5722901693522x_{40} = -19.5722901693522
x41=94.9705138555072x_{41} = 94.9705138555072
x42=18.1268216737253x_{42} = 18.1268216737253
x43=76.1209579339685x_{43} = 76.1209579339685
x44=80.9586747455212x_{44} = 80.9586747455212
x45=57.2714020124297x_{45} = 57.2714020124297
x46=55.8259335168029x_{46} = 55.8259335168029
x47=50.9882167052501x_{47} = 50.9882167052501
x48=88.6873285483276x_{48} = 88.6873285483276
x49=7.005919554993x_{49} = 7.005919554993
x50=−94.9705138555072x_{50} = -94.9705138555072
x51=19.5722901693522x_{51} = 19.5722901693522
x52=−101.253699162687x_{52} = -101.253699162687
x53=25.8554754765318x_{53} = 25.8554754765318
x54=0.722734247813416x_{54} = 0.722734247813416
x55=99.80823066706x_{55} = 99.80823066706
x56=−5.56045105936617x_{56} = -5.56045105936617
x57=−25.8554754765318x_{57} = -25.8554754765318
x58=5.56045105936617x_{58} = 5.56045105936617
x59=−74.6754894383416x_{59} = -74.6754894383416
x60=32.1386607837113x_{60} = 32.1386607837113
x61=93.5250453598804x_{61} = 93.5250453598804
x62=−88.6873285483276x_{62} = -88.6873285483276
x63=−80.9586747455212x_{63} = -80.9586747455212
x64=−49.5427482096233x_{64} = -49.5427482096233
x65=44.7050313980705x_{65} = 44.7050313980705
x66=74.6754894383416x_{66} = 74.6754894383416
x67=−87.2418600527008x_{67} = -87.2418600527008
x68=24.4100069809049x_{68} = 24.4100069809049
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 2*cos(x) - 3/2.
−32+2cos⁡(0)- \frac{3}{2} + 2 \cos{\left(0 \right)}
The result:
f(0)=12f{\left(0 \right)} = \frac{1}{2}
The point:
(0, 1/2)
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
−2sin⁡(x)=0- 2 \sin{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=πx_{2} = \pi
The values of the extrema at the points:
(0, 1/2)

(pi, -7/2)


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=πx_{1} = \pi
Maxima of the function at points:
x1=0x_{1} = 0
Decreasing at intervals
(−∞,0]∪[π,∞)\left(-\infty, 0\right] \cup \left[\pi, \infty\right)
Increasing at intervals
[0,π]\left[0, \pi\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
−2cos⁡(x)=0- 2 \cos{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=π2x_{1} = \frac{\pi}{2}
x2=3π2x_{2} = \frac{3 \pi}{2}

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
[π2,3π2]\left[\frac{\pi}{2}, \frac{3 \pi}{2}\right]
Convex at the intervals
(−∞,π2]∪[3π2,∞)\left(-\infty, \frac{\pi}{2}\right] \cup \left[\frac{3 \pi}{2}, \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→−∞(2cos⁡(x)−32)=⟨−72,12⟩\lim_{x \to -\infty}\left(2 \cos{\left(x \right)} - \frac{3}{2}\right) = \left\langle - \frac{7}{2}, \frac{1}{2}\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−72,12⟩y = \left\langle - \frac{7}{2}, \frac{1}{2}\right\rangle
lim⁡x→∞(2cos⁡(x)−32)=⟨−72,12⟩\lim_{x \to \infty}\left(2 \cos{\left(x \right)} - \frac{3}{2}\right) = \left\langle - \frac{7}{2}, \frac{1}{2}\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−72,12⟩y = \left\langle - \frac{7}{2}, \frac{1}{2}\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 2*cos(x) - 3/2, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(2cos⁡(x)−32x)=0\lim_{x \to -\infty}\left(\frac{2 \cos{\left(x \right)} - \frac{3}{2}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(2cos⁡(x)−32x)=0\lim_{x \to \infty}\left(\frac{2 \cos{\left(x \right)} - \frac{3}{2}}{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:
2cos⁡(x)−32=2cos⁡(x)−322 \cos{\left(x \right)} - \frac{3}{2} = 2 \cos{\left(x \right)} - \frac{3}{2}
- Yes
2cos⁡(x)−32=32−2cos⁡(x)2 \cos{\left(x \right)} - \frac{3}{2} = \frac{3}{2} - 2 \cos{\left(x \right)}
- No
so, the function
is
even
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