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Plotting a function graph/ 2*sin((x-pi)/4)+1

Graphing y = 2*sin((x-pi)/4)+1

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
            /x - pi\    
f(x) = 2*sin|------| + 1
            \  4   /
f(x)=2sin⁡(x−π4)+1f{\left(x \right)} = 2 \sin{\left(\frac{x - \pi}{4} \right)} + 1 
f = 2*sin((x - pi)/4) + 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−π4)+1=02 \sin{\left(\frac{x - \pi}{4} \right)} + 1 = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=π3x_{1} = \frac{\pi}{3}
x2=17π3x_{2} = \frac{17 \pi}{3}
Numerical solution
x1=202.109127380943x_{1} = 202.109127380943
x2=118.333323285216x_{2} = 118.333323285216
x3=−82.7286065445312x_{3} = -82.7286065445312
x4=805.294916870184x_{4} = 805.294916870184
x5=93.2005820564972x_{5} = 93.2005820564972
x6=−74.3510261349584x_{6} = -74.3510261349584
x7=−7.33038285837618x_{7} = -7.33038285837618
x8=76.4454212373516x_{8} = 76.4454212373516
x9=1.0471975511966x_{9} = 1.0471975511966
x10=101.57816246607x_{10} = 101.57816246607
x11=−99.4837673636768x_{11} = -99.4837673636768
x12=42.9350995990605x_{12} = 42.9350995990605
x13=−24.0855436775217x_{13} = -24.0855436775217
x14=26.1799387799149x_{14} = 26.1799387799149
x15=−426.209403337015x_{15} = -426.209403337015
x16=17.8023583703422x_{16} = 17.8023583703422
x17=−32.4631240870945x_{17} = -32.4631240870945
x18=−57.5958653158129x_{18} = -57.5958653158129
x19=68.0678408277789x_{19} = 68.0678408277789
x20=−49.2182849062401x_{20} = -49.2182849062401
x21=51.3126800086333x_{21} = 51.3126800086333
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 2*sin((x - pi)/4) + 1.
2sin⁡((−1)π4)+12 \sin{\left(\frac{\left(-1\right) \pi}{4} \right)} + 1
The result:
f(0)=1−2f{\left(0 \right)} = 1 - \sqrt{2}
The point:
(0, 1 - sqrt(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
cos⁡(x−π4)2=0\frac{\cos{\left(\frac{x - \pi}{4} \right)}}{2} = 0
Solve this equation
The roots of this equation
x1=−πx_{1} = - \pi
x2=3πx_{2} = 3 \pi
The values of the extrema at the points:
(-pi, -1)

(3*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=3πx_{1} = 3 \pi
Decreasing at intervals
[−π,3π]\left[- \pi, 3 \pi\right]
Increasing at intervals
(−∞,−π]∪[3π,∞)\left(-\infty, - \pi\right] \cup \left[3 \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
−sin⁡(x−π4)8=0- \frac{\sin{\left(\frac{x - \pi}{4} \right)}}{8} = 0
Solve this equation
The roots of this equation
x1=πx_{1} = \pi
x2=5πx_{2} = 5 \pi

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
(−∞,π]∪[5π,∞)\left(-\infty, \pi\right] \cup \left[5 \pi, \infty\right)
Convex at the intervals
[π,5π]\left[\pi, 5 \pi\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−π4)+1)=⟨−1,3⟩\lim_{x \to -\infty}\left(2 \sin{\left(\frac{x - \pi}{4} \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−π4)+1)=⟨−1,3⟩\lim_{x \to \infty}\left(2 \sin{\left(\frac{x - \pi}{4} \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*sin((x - pi)/4) + 1, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(2sin⁡(x−π4)+1x)=0\lim_{x \to -\infty}\left(\frac{2 \sin{\left(\frac{x - \pi}{4} \right)} + 1}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(2sin⁡(x−π4)+1x)=0\lim_{x \to \infty}\left(\frac{2 \sin{\left(\frac{x - \pi}{4} \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−π4)+1=1−2sin⁡(x4+π4)2 \sin{\left(\frac{x - \pi}{4} \right)} + 1 = 1 - 2 \sin{\left(\frac{x}{4} + \frac{\pi}{4} \right)}
- No
2sin⁡(x−π4)+1=2sin⁡(x4+π4)−12 \sin{\left(\frac{x - \pi}{4} \right)} + 1 = 2 \sin{\left(\frac{x}{4} + \frac{\pi}{4} \right)} - 1
- No
so, the function
not is
neither even, nor odd
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