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Graphing y = x^2+|4x+5|

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f(x) = x  + |4*x + 5|

f{\left(x \right)} = x^{2} + \left|{4 x + 5}\right|

f = x^2 + |4*x + 5|

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:

x^{2} + \left|{4 x + 5}\right| = 0

Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X

The points of intersection with the Y axis coordinate

The graph crosses Y axis when x equals 0:
substitute x = 0 to x^2 + |4*x + 5|.

0^{2} + \left|{0 \cdot 4 + 5}\right|

The result:

f{\left(0 \right)} = 5

The point:

(0, 5)

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)} =

2 x + 4 \operatorname{sign}{\left(4 x + 5 \right)} = 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)} =

2 \left(16 \delta\left(4 x + 5\right) + 1\right) = 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(x^{2} + \left|{4 x + 5}\right|\right) = \infty

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

\lim_{x \to \infty}\left(x^{2} + \left|{4 x + 5}\right|\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^2 + |4*x + 5|, divided by x at x->+oo and x ->-oo

\lim_{x \to -\infty}\left(\frac{x^{2} + \left|{4 x + 5}\right|}{x}\right) = -\infty

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

\lim_{x \to \infty}\left(\frac{x^{2} + \left|{4 x + 5}\right|}{x}\right) = \infty

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

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:

x^{2} + \left|{4 x + 5}\right| = x^{2} + \left|{4 x - 5}\right|

- No

x^{2} + \left|{4 x + 5}\right| = - x^{2} - \left|{4 x - 5}\right|

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