Mister Exam

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Graphing y = |x^2+8x+12|

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f(x) = |x  + 8*x + 12|

f{\left(x \right)} = \left|{\left(x^{2} + 8 x\right) + 12}\right|

f = |x^2 + 8*x + 12|

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|{\left(x^{2} + 8 x\right) + 12}\right| = 0

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

Analytical solution

x_{1} = -6

x_{2} = -2

Numerical solution

x_{1} = -2

x_{2} = -6

The points of intersection with the Y axis coordinate

The graph crosses Y axis when x equals 0:
substitute x = 0 to |x^2 + 8*x + 12|.

\left|{\left(0^{2} + 0 \cdot 8\right) + 12}\right|

The result:

f{\left(0 \right)} = 12

The point:

(0, 12)

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(4 \left(x + 4\right)^{2} \delta\left(x^{2} + 8 x + 12\right) + \operatorname{sign}{\left(x^{2} + 8 x + 12 \right)}\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|{\left(x^{2} + 8 x\right) + 12}\right| = \infty

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

\lim_{x \to \infty} \left|{\left(x^{2} + 8 x\right) + 12}\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 + 8*x + 12|, divided by x at x->+oo and x ->-oo

\lim_{x \to -\infty}\left(\frac{\left|{\left(x^{2} + 8 x\right) + 12}\right|}{x}\right) = -\infty

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

\lim_{x \to \infty}\left(\frac{\left|{\left(x^{2} + 8 x\right) + 12}\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:

\left|{\left(x^{2} + 8 x\right) + 12}\right| = \left|{x^{2} - 8 x + 12}\right|

- No

\left|{\left(x^{2} + 8 x\right) + 12}\right| = - \left|{x^{2} - 8 x + 12}\right|

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