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Graphing y = x^1-4*x+1

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        1          
f(x) = x  - 4*x + 1

f{\left(x \right)} = \left(- 4 x + x^{1}\right) + 1

f = -4*x + x^1 + 1

The graph of the function

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:

\left(- 4 x + x^{1}\right) + 1 = 0

Solve this equation
The points of intersection with the axis X:

Analytical solution

x_{1} = \frac{1}{3}

Numerical solution

x_{1} = 0.333333333333333

The points of intersection with the Y axis coordinate

The graph crosses Y axis when x equals 0:
substitute x = 0 to x^1 - 4*x + 1.

\left(0^{1} - 0\right) + 1

The result:

f{\left(0 \right)} = 1

The point:

(0, 1)

Extrema of the function

In order to find the extrema, we need to solve the equation

\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:

\frac{d}{d x} f{\left(x \right)} =

-3 = 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

\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:

\frac{d^{2}}{d x^{2}} f{\left(x \right)} =

0 = 0

Solve this equation
Solutions are not found,
maybe, the function has no inflections

Horizontal asymptotes

Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo

\lim_{x \to -\infty}\left(\left(- 4 x + x^{1}\right) + 1\right) = \infty

Let's take the limit
so,
horizontal asymptote on the left doesn’t exist

\lim_{x \to \infty}\left(\left(- 4 x + x^{1}\right) + 1\right) = -\infty

Let's take the limit
so,
horizontal asymptote on the right doesn’t exist

Inclined asymptotes

Inclined asymptote can be found by calculating the limit of x^1 - 4*x + 1, divided by x at x->+oo and x ->-oo

\lim_{x \to -\infty}\left(\frac{\left(- 4 x + x^{1}\right) + 1}{x}\right) = -3

Let's take the limit
so,
inclined asymptote equation on the left:

y = - 3 x

\lim_{x \to \infty}\left(\frac{\left(- 4 x + x^{1}\right) + 1}{x}\right) = -3

Let's take the limit
so,
inclined asymptote equation on the right:

y = - 3 x

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:

\left(- 4 x + x^{1}\right) + 1 = 3 x + 1

- No

\left(- 4 x + x^{1}\right) + 1 = - 3 x - 1

- No
so, the function
not is
neither even, nor odd