Mister Exam
Graphing y = |x^2-2x-3|
The solution
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| 2 | f(x) = |x - 2*x - 3|
$$f{\left(x \right)} = \left|{\left(x^{2} - 2 x\right) - 3}\right|$$
f = |x^2 - 2*x - 3|
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:
$$\left|{\left(x^{2} - 2 x\right) - 3}\right| = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = -1$$
$$x_{2} = 3$$
Numerical solution
$$x_{1} = 3$$
$$x_{2} = -1$$
so we need to solve the equation:
$$\left|{\left(x^{2} - 2 x\right) - 3}\right| = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = -1$$
$$x_{2} = 3$$
Numerical solution
$$x_{1} = 3$$
$$x_{2} = -1$$
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
$$2 \left(4 \left(x - 1\right)^{2} \delta\left(- x^{2} + 2 x + 3\right) - \operatorname{sign}{\left(- x^{2} + 2 x + 3 \right)}\right) = 0$$
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
$$2 \left(4 \left(x - 1\right)^{2} \delta\left(- x^{2} + 2 x + 3\right) - \operatorname{sign}{\left(- x^{2} + 2 x + 3 \right)}\right) = 0$$
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
$$\lim_{x \to -\infty} \left|{\left(x^{2} - 2 x\right) - 3}\right| = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \left|{\left(x^{2} - 2 x\right) - 3}\right| = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty} \left|{\left(x^{2} - 2 x\right) - 3}\right| = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \left|{\left(x^{2} - 2 x\right) - 3}\right| = \infty$$
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 - 2*x - 3|, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\left|{\left(x^{2} - 2 x\right) - 3}\right|}{x}\right) = -\infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{\left|{\left(x^{2} - 2 x\right) - 3}\right|}{x}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\frac{\left|{\left(x^{2} - 2 x\right) - 3}\right|}{x}\right) = -\infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{\left|{\left(x^{2} - 2 x\right) - 3}\right|}{x}\right) = \infty$$
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:
$$\left|{\left(x^{2} - 2 x\right) - 3}\right| = \left|{x^{2} + 2 x - 3}\right|$$
- No
$$\left|{\left(x^{2} - 2 x\right) - 3}\right| = - \left|{x^{2} + 2 x - 3}\right|$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\left|{\left(x^{2} - 2 x\right) - 3}\right| = \left|{x^{2} + 2 x - 3}\right|$$
- No
$$\left|{\left(x^{2} - 2 x\right) - 3}\right| = - \left|{x^{2} + 2 x - 3}\right|$$
- No
so, the function
not is
neither even, nor odd