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Plotting a function graph/ 2*cos(x-1)+1

Graphing y = 2*cos(x-1)+1

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = 2*cos(x - 1) + 1
f(x)=2cos⁡(x−1)+1f{\left(x \right)} = 2 \cos{\left(x - 1 \right)} + 1 
f = 2*cos(x - 1) + 1
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−1)+1=02 \cos{\left(x - 1 \right)} + 1 = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=1+2π3x_{1} = 1 + \frac{2 \pi}{3}
x2=1+4π3x_{2} = 1 + \frac{4 \pi}{3}
Numerical solution
x1=−59.7374579694027x_{1} = -59.7374579694027
x2=−45.0766922526503x_{2} = -45.0766922526503
x3=−63.9262481741891x_{3} = -63.9262481741891
x4=−19.943951023932x_{4} = -19.943951023932
x5=84.7758040957278x_{5} = 84.7758040957278
x6=−91.1533845053006x_{6} = -91.1533845053006
x7=97.342174710087x_{7} = 97.342174710087
x8=15.6607657167524x_{8} = 15.6607657167524
x9=−7.37758040957278x_{9} = -7.37758040957278
x10=93.1533845053006x_{10} = 93.1533845053006
x11=−32.5103216382911x_{11} = -32.5103216382911
x12=−40.8879020478639x_{12} = -40.8879020478639
x13=−53.4542726622231x_{13} = -53.4542726622231
x14=40.7935069454707x_{14} = 40.7935069454707
x15=−11537.0226190841x_{15} = -11537.0226190841
x16=99.4365698124802x_{16} = 99.4365698124802
x17=49.1710873550435x_{17} = 49.1710873550435
x18=9.37758040957278x_{18} = 9.37758040957278
x19=−26.2271363311115x_{19} = -26.2271363311115
x20=65.9262481741891x_{20} = 65.9262481741891
x21=24.0383461263252x_{21} = 24.0383461263252
x22=−70.2094334813686x_{22} = -70.2094334813686
x23=−13.6607657167524x_{23} = -13.6607657167524
x24=72.2094334813686x_{24} = 72.2094334813686
x25=−3.18879020478639x_{25} = -3.18879020478639
x26=53.3598775598299x_{26} = 53.3598775598299
x27=−72.3038285837618x_{27} = -72.3038285837618
x28=−15.7551608191456x_{28} = -15.7551608191456
x29=17.7551608191456x_{29} = 17.7551608191456
x30=−51.3598775598299x_{30} = -51.3598775598299
x31=−76.4926187885482x_{31} = -76.4926187885482
x32=78.4926187885482x_{32} = 78.4926187885482
x33=−78.5870138909414x_{33} = -78.5870138909414
x34=11.471975511966x_{34} = 11.471975511966
x35=−97.4365698124802x_{35} = -97.4365698124802
x36=91.0589894029074x_{36} = 91.0589894029074
x37=−82.7758040957278x_{37} = -82.7758040957278
x38=36.6047167406843x_{38} = 36.6047167406843
x39=−1.0943951023932x_{39} = -1.0943951023932
x40=5.18879020478639x_{40} = 5.18879020478639
x41=21.943951023932x_{41} = 21.943951023932
x42=34.5103216382911x_{42} = 34.5103216382911
x43=3.0943951023932x_{43} = 3.0943951023932
x44=80.5870138909414x_{44} = 80.5870138909414
x45=47.0766922526503x_{45} = 47.0766922526503
x46=−66.0206432765823x_{46} = -66.0206432765823
x47=−34.6047167406843x_{47} = -34.6047167406843
x48=−95.342174710087x_{48} = -95.342174710087
x49=−22.0383461263252x_{49} = -22.0383461263252
x50=68.0206432765823x_{50} = 68.0206432765823
x51=−38.7935069454707x_{51} = -38.7935069454707
x52=−47.1710873550435x_{52} = -47.1710873550435
x53=−89.0589894029074x_{53} = -89.0589894029074
x54=61.7374579694027x_{54} = 61.7374579694027
x55=28.2271363311115x_{55} = 28.2271363311115
x56=1330.94089001968x_{56} = 1330.94089001968
x57=−28.3215314335047x_{57} = -28.3215314335047
x58=55.4542726622231x_{58} = 55.4542726622231
x59=59.6430628670095x_{59} = 59.6430628670095
x60=−57.6430628670095x_{60} = -57.6430628670095
x61=−103.71975511966x_{61} = -103.71975511966
x62=30.3215314335047x_{62} = 30.3215314335047
x63=−84.870199198121x_{63} = -84.870199198121
x64=74.3038285837618x_{64} = 74.3038285837618
x65=42.8879020478639x_{65} = 42.8879020478639
x66=86.870199198121x_{66} = 86.870199198121
x67=−9.47197551196598x_{67} = -9.47197551196598
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 - 1) + 1.
1+2cos⁡(−1)1 + 2 \cos{\left(-1 \right)}
The result:
f(0)=1+2cos⁡(1)f{\left(0 \right)} = 1 + 2 \cos{\left(1 \right)}
The point:
(0, 1 + 2*cos(1))
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−1)=0- 2 \sin{\left(x - 1 \right)} = 0
Solve this equation
The roots of this equation
x1=1x_{1} = 1
x2=1+πx_{2} = 1 + \pi
The values of the extrema at the points:
(1, 3)

(1 + pi, -1)


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=1+πx_{1} = 1 + \pi
Maxima of the function at points:
x1=1x_{1} = 1
Decreasing at intervals
(−∞,1]∪[1+π,∞)\left(-\infty, 1\right] \cup \left[1 + \pi, \infty\right)
Increasing at intervals
[1,1+π]\left[1, 1 + \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−1)=0- 2 \cos{\left(x - 1 \right)} = 0
Solve this equation
The roots of this equation
x1=1+π2x_{1} = 1 + \frac{\pi}{2}
x2=1+3π2x_{2} = 1 + \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
[1+π2,1+3π2]\left[1 + \frac{\pi}{2}, 1 + \frac{3 \pi}{2}\right]
Convex at the intervals
(−∞,1+π2]∪[1+3π2,∞)\left(-\infty, 1 + \frac{\pi}{2}\right] \cup \left[1 + \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−1)+1)=⟨−1,3⟩\lim_{x \to -\infty}\left(2 \cos{\left(x - 1 \right)} + 1\right) = \left\langle -1, 3\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−1,3⟩y = \left\langle -1, 3\right\rangle
lim⁡x→∞(2cos⁡(x−1)+1)=⟨−1,3⟩\lim_{x \to \infty}\left(2 \cos{\left(x - 1 \right)} + 1\right) = \left\langle -1, 3\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−1,3⟩y = \left\langle -1, 3\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 2*cos(x - 1) + 1, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(2cos⁡(x−1)+1x)=0\lim_{x \to -\infty}\left(\frac{2 \cos{\left(x - 1 \right)} + 1}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(2cos⁡(x−1)+1x)=0\lim_{x \to \infty}\left(\frac{2 \cos{\left(x - 1 \right)} + 1}{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−1)+1=2cos⁡(x+1)+12 \cos{\left(x - 1 \right)} + 1 = 2 \cos{\left(x + 1 \right)} + 1
- No
2cos⁡(x−1)+1=−2cos⁡(x+1)−12 \cos{\left(x - 1 \right)} + 1 = - 2 \cos{\left(x + 1 \right)} - 1
- No
so, the function
not is
neither even, nor odd
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