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

Graphing y = 2*cos(x)-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
f(x)=2cos⁡(x)−1f{\left(x \right)} = 2 \cos{\left(x \right)} - 1 
f = 2*cos(x) - 1
The graph of the function
5.00005.01005.00105.00205.00305.00405.00505.00605.00705.00805.0090-0.44-0.42
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=02 \cos{\left(x \right)} - 1 = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=π3x_{1} = \frac{\pi}{3}
x2=5π3x_{2} = \frac{5 \pi}{3}
Numerical solution
x1=38.7463093942741x_{1} = 38.7463093942741
x2=13.6135681655558x_{2} = 13.6135681655558
x3=11.5191730631626x_{3} = 11.5191730631626
x4=−55.5014702134197x_{4} = -55.5014702134197
x5=−30.3687289847013x_{5} = -30.3687289847013
x6=−61.7846555205993x_{6} = -61.7846555205993
x7=−86.9173967493176x_{7} = -86.9173967493176
x8=89.0117918517108x_{8} = 89.0117918517108
x9=−51.3126800086333x_{9} = -51.3126800086333
x10=1.0471975511966x_{10} = 1.0471975511966
x11=−68.0678408277789x_{11} = -68.0678408277789
x12=36.6519142918809x_{12} = 36.6519142918809
x13=−70.162235930172x_{13} = -70.162235930172
x14=93.2005820564972x_{14} = 93.2005820564972
x15=−95.2949771588904x_{15} = -95.2949771588904
x16=49.2182849062401x_{16} = 49.2182849062401
x17=30.3687289847013x_{17} = 30.3687289847013
x18=17.8023583703422x_{18} = 17.8023583703422
x19=−32.4631240870945x_{19} = -32.4631240870945
x20=−99.4837673636768x_{20} = -99.4837673636768
x21=63.8790506229925x_{21} = 63.8790506229925
x22=42.9350995990605x_{22} = 42.9350995990605
x23=−225.147473507269x_{23} = -225.147473507269
x24=−93.2005820564972x_{24} = -93.2005820564972
x25=−17.8023583703422x_{25} = -17.8023583703422
x26=7.33038285837618x_{26} = 7.33038285837618
x27=−11.5191730631626x_{27} = -11.5191730631626
x28=57.5958653158129x_{28} = 57.5958653158129
x29=−42.9350995990605x_{29} = -42.9350995990605
x30=−24.0855436775217x_{30} = -24.0855436775217
x31=70.162235930172x_{31} = 70.162235930172
x32=−36.6519142918809x_{32} = -36.6519142918809
x33=−26.1799387799149x_{33} = -26.1799387799149
x34=19.8967534727354x_{34} = 19.8967534727354
x35=−76.4454212373516x_{35} = -76.4454212373516
x36=76.4454212373516x_{36} = 76.4454212373516
x37=−1.0471975511966x_{37} = -1.0471975511966
x38=1651.43053823704x_{38} = 1651.43053823704
x39=55.5014702134197x_{39} = 55.5014702134197
x40=−5.23598775598299x_{40} = -5.23598775598299
x41=51.3126800086333x_{41} = 51.3126800086333
x42=45.0294947014537x_{42} = 45.0294947014537
x43=68.0678408277789x_{43} = 68.0678408277789
x44=−45.0294947014537x_{44} = -45.0294947014537
x45=−82.7286065445312x_{45} = -82.7286065445312
x46=−19.8967534727354x_{46} = -19.8967534727354
x47=−57.5958653158129x_{47} = -57.5958653158129
x48=74.3510261349584x_{48} = 74.3510261349584
x49=−38.7463093942741x_{49} = -38.7463093942741
x50=80.634211442138x_{50} = 80.634211442138
x51=99.4837673636768x_{51} = 99.4837673636768
x52=−359.188760060433x_{52} = -359.188760060433
x53=26.1799387799149x_{53} = 26.1799387799149
x54=−63.8790506229925x_{54} = -63.8790506229925
x55=5.23598775598299x_{55} = 5.23598775598299
x56=−89.0117918517108x_{56} = -89.0117918517108
x57=82.7286065445312x_{57} = 82.7286065445312
x58=−13.6135681655558x_{58} = -13.6135681655558
x59=−80.634211442138x_{59} = -80.634211442138
x60=−74.3510261349584x_{60} = -74.3510261349584
x61=86.9173967493176x_{61} = 86.9173967493176
x62=24.0855436775217x_{62} = 24.0855436775217
x63=−49.2182849062401x_{63} = -49.2182849062401
x64=61.7846555205993x_{64} = 61.7846555205993
x65=95.2949771588904x_{65} = 95.2949771588904
x66=32.4631240870945x_{66} = 32.4631240870945
x67=−7.33038285837618x_{67} = -7.33038285837618
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+2cos⁡(0)-1 + 2 \cos{\left(0 \right)}
The result:
f(0)=1f{\left(0 \right)} = 1
The point:
(0, 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)=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)

(pi, -3)


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