Mister Exam
Graphing y = x^2+|4x+5|
The solution
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2 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
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
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:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$2 x + 4 \operatorname{sign}{\left(4 x + 5 \right)} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$2 x + 4 \operatorname{sign}{\left(4 x + 5 \right)} = 0$$
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:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$2 \left(16 \delta\left(4 x + 5\right) + 1\right) = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$2 \left(16 \delta\left(4 x + 5\right) + 1\right) = 0$$
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
$$\lim_{x \to -\infty}\left(x^{2} + \left|{4 x + 5}\right|\right) = \infty$$
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
$$\lim_{x \to -\infty}\left(\frac{x^{2} + \left|{4 x + 5}\right|}{x}\right) = -\infty$$
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
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