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Plotting a function graph/ 1+2sin(x-п/2)

Graphing y = 1+2sin(x-п/2)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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                /    pi\
f(x) = 1 + 2*sin|x - --|
                \    2 /
f(x)=2sin⁡(x−π2)+1f{\left(x \right)} = 2 \sin{\left(x - \frac{\pi}{2} \right)} + 1 
f = 2*sin(x - pi/2) + 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:
2sin⁡(x−π2)+1=02 \sin{\left(x - \frac{\pi}{2} \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=−13.6135681655558x_{1} = -13.6135681655558
x2=89.0117918517108x_{2} = 89.0117918517108
x3=−93.2005820564972x_{3} = -93.2005820564972
x4=−55.5014702134197x_{4} = -55.5014702134197
x5=−36.6519142918809x_{5} = -36.6519142918809
x6=−30.3687289847013x_{6} = -30.3687289847013
x7=−68.0678408277789x_{7} = -68.0678408277789
x8=51.3126800086333x_{8} = 51.3126800086333
x9=32.4631240870945x_{9} = 32.4631240870945
x10=11.5191730631626x_{10} = 11.5191730631626
x11=−82.7286065445312x_{11} = -82.7286065445312
x12=−89.0117918517108x_{12} = -89.0117918517108
x13=63.8790506229925x_{13} = 63.8790506229925
x14=−26.1799387799149x_{14} = -26.1799387799149
x15=−42.9350995990605x_{15} = -42.9350995990605
x16=7.33038285837618x_{16} = 7.33038285837618
x17=82.7286065445312x_{17} = 82.7286065445312
x18=1651.43053823704x_{18} = 1651.43053823704
x19=95.2949771588904x_{19} = 95.2949771588904
x20=−19.8967534727354x_{20} = -19.8967534727354
x21=−45.0294947014537x_{21} = -45.0294947014537
x22=−24.0855436775217x_{22} = -24.0855436775217
x23=−95.2949771588904x_{23} = -95.2949771588904
x24=13.6135681655558x_{24} = 13.6135681655558
x25=5.23598775598299x_{25} = 5.23598775598299
x26=99.4837673636768x_{26} = 99.4837673636768
x27=−99.4837673636768x_{27} = -99.4837673636768
x28=45.0294947014537x_{28} = 45.0294947014537
x29=74.3510261349584x_{29} = 74.3510261349584
x30=42.9350995990605x_{30} = 42.9350995990605
x31=61.7846555205993x_{31} = 61.7846555205993
x32=36.6519142918809x_{32} = 36.6519142918809
x33=24.0855436775217x_{33} = 24.0855436775217
x34=−5.23598775598299x_{34} = -5.23598775598299
x35=93.2005820564972x_{35} = 93.2005820564972
x36=−86.9173967493176x_{36} = -86.9173967493176
x37=−1.0471975511966x_{37} = -1.0471975511966
x38=−11.5191730631626x_{38} = -11.5191730631626
x39=−74.3510261349584x_{39} = -74.3510261349584
x40=49.2182849062401x_{40} = 49.2182849062401
x41=26.1799387799149x_{41} = 26.1799387799149
x42=80.634211442138x_{42} = 80.634211442138
x43=55.5014702134197x_{43} = 55.5014702134197
x44=−76.4454212373516x_{44} = -76.4454212373516
x45=−32.4631240870945x_{45} = -32.4631240870945
x46=30.3687289847013x_{46} = 30.3687289847013
x47=−63.8790506229925x_{47} = -63.8790506229925
x48=86.9173967493176x_{48} = 86.9173967493176
x49=−70.162235930172x_{49} = -70.162235930172
x50=17.8023583703422x_{50} = 17.8023583703422
x51=−17.8023583703422x_{51} = -17.8023583703422
x52=−225.147473507269x_{52} = -225.147473507269
x53=−57.5958653158129x_{53} = -57.5958653158129
x54=1.0471975511966x_{54} = 1.0471975511966
x55=−359.188760060433x_{55} = -359.188760060433
x56=−51.3126800086333x_{56} = -51.3126800086333
x57=70.162235930172x_{57} = 70.162235930172
x58=−61.7846555205993x_{58} = -61.7846555205993
x59=38.7463093942741x_{59} = 38.7463093942741
x60=68.0678408277789x_{60} = 68.0678408277789
x61=76.4454212373516x_{61} = 76.4454212373516
x62=−7.33038285837618x_{62} = -7.33038285837618
x63=19.8967534727354x_{63} = 19.8967534727354
x64=−80.634211442138x_{64} = -80.634211442138
x65=57.5958653158129x_{65} = 57.5958653158129
x66=−38.7463093942741x_{66} = -38.7463093942741
x67=−49.2182849062401x_{67} = -49.2182849062401
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 1 + 2*sin(x - pi/2).
2sin⁡(−π2)+12 \sin{\left(- \frac{\pi}{2} \right)} + 1
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
2cos⁡(x−π2)=02 \cos{\left(x - \frac{\pi}{2} \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=0x_{1} = 0
Maxima of the function at points:
x1=πx_{1} = \pi
Decreasing at intervals
[0,π]\left[0, \pi\right]
Increasing at intervals
(−∞,0]∪[π,∞)\left(-\infty, 0\right] \cup \left[\pi, \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
2cos⁡(x)=02 \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(-\infty, \frac{\pi}{2}\right] \cup \left[\frac{3 \pi}{2}, \infty\right)
Convex at the intervals
[π2,3π2]\left[\frac{\pi}{2}, \frac{3 \pi}{2}\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(2sin⁡(x−π2)+1)=⟨−1,3⟩\lim_{x \to -\infty}\left(2 \sin{\left(x - \frac{\pi}{2} \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→∞(2sin⁡(x−π2)+1)=⟨−1,3⟩\lim_{x \to \infty}\left(2 \sin{\left(x - \frac{\pi}{2} \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 1 + 2*sin(x - pi/2), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(2sin⁡(x−π2)+1x)=0\lim_{x \to -\infty}\left(\frac{2 \sin{\left(x - \frac{\pi}{2} \right)} + 1}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(2sin⁡(x−π2)+1x)=0\lim_{x \to \infty}\left(\frac{2 \sin{\left(x - \frac{\pi}{2} \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:
2sin⁡(x−π2)+1=1−2sin⁡(x+π2)2 \sin{\left(x - \frac{\pi}{2} \right)} + 1 = 1 - 2 \sin{\left(x + \frac{\pi}{2} \right)}
- No
2sin⁡(x−π2)+1=2sin⁡(x+π2)−12 \sin{\left(x - \frac{\pi}{2} \right)} + 1 = 2 \sin{\left(x + \frac{\pi}{2} \right)} - 1
- No
so, the function
not is
neither even, nor odd
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