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Plotting a function graph/ arctg(4*x-1)

Graphing y = arctg(4*x-1)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = atan(4*x - 1)
f(x)=atan⁡(4x−1)f{\left(x \right)} = \operatorname{atan}{\left(4 x - 1 \right)} 
f = atan(4*x - 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:
atan⁡(4x−1)=0\operatorname{atan}{\left(4 x - 1 \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=14x_{1} = \frac{1}{4}
Numerical solution
x1=0.25x_{1} = 0.25
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to atan(4*x - 1).
atan⁡(−1+0⋅4)\operatorname{atan}{\left(-1 + 0 \cdot 4 \right)}
The result:
f(0)=−π4f{\left(0 \right)} = - \frac{\pi}{4}
The point:
(0, -pi/4)
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
4(4x−1)2+1=0\frac{4}{\left(4 x - 1\right)^{2} + 1} = 0
Solve this equation
Solutions are not found,
function may have no extrema
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
−32(4x−1)((4x−1)2+1)2=0- \frac{32 \left(4 x - 1\right)}{\left(\left(4 x - 1\right)^{2} + 1\right)^{2}} = 0
Solve this equation
The roots of this equation
x1=14x_{1} = \frac{1}{4}

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
(−∞,14]\left(-\infty, \frac{1}{4}\right]
Convex at the intervals
[14,∞)\left[\frac{1}{4}, \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→−∞atan⁡(4x−1)=−π2\lim_{x \to -\infty} \operatorname{atan}{\left(4 x - 1 \right)} = - \frac{\pi}{2}
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=−π2y = - \frac{\pi}{2}
lim⁡x→∞atan⁡(4x−1)=π2\lim_{x \to \infty} \operatorname{atan}{\left(4 x - 1 \right)} = \frac{\pi}{2}
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=π2y = \frac{\pi}{2}
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of atan(4*x - 1), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(atan⁡(4x−1)x)=0\lim_{x \to -\infty}\left(\frac{\operatorname{atan}{\left(4 x - 1 \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(atan⁡(4x−1)x)=0\lim_{x \to \infty}\left(\frac{\operatorname{atan}{\left(4 x - 1 \right)}}{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:
atan⁡(4x−1)=−atan⁡(4x+1)\operatorname{atan}{\left(4 x - 1 \right)} = - \operatorname{atan}{\left(4 x + 1 \right)}
- No
atan⁡(4x−1)=atan⁡(4x+1)\operatorname{atan}{\left(4 x - 1 \right)} = \operatorname{atan}{\left(4 x + 1 \right)}
- No
so, the function
not is
neither even, nor odd
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