Mister Exam
Other calculators
- Graph of implicit function
- Derivative Step by Step
- Integral Step by Step
- Graph of parametric function
- Graph in Polar Coordinates
- Surface defined parametrically
- Surface defined by equation
- Curve in 3D space defined parametrically
- Area between lines
- The intersection points of functions
-
How to use it?
- Graphing y =:
- y=(6x^2-x^4)/9
- y=x^6
- y=x^3+3x^2-9x+5
- y=x^3+1.5x^2-6x
-
Identical expressions
- abs(x^ two -2abs(x)- fifteen)
- abs(x squared minus 2abs(x) minus 15)
- abs(x to the power of two minus 2abs(x) minus fifteen)
- abs(x2-2abs(x)-15)
- absx2-2absx-15
- abs(x²-2abs(x)-15)
- abs(x to the power of 2-2abs(x)-15)
- absx^2-2absx-15
-
Similar expressions
- abs(x^2+2abs(x)-15)
- abs(x^2-2abs(x)+15)
Graphing y = abs(x^2-2abs(x)-15)
The solution
You have entered
[src]
| 2 | f(x) = |x - 2*|x| - 15|
$$f{\left(x \right)} = \left|{x^{2} - 2 \left|{x}\right| - 15}\right|$$
f = Abs(x^2 - 2*|x| - 1*15)
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:
$$\left|{x^{2} - 2 \left|{x}\right| - 15}\right| = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = -5$$
$$x_{2} = 5$$
Numerical solution
$$x_{1} = 5.00000000000004$$
$$x_{2} = 5$$
$$x_{3} = -5$$
so we need to solve the equation:
$$\left|{x^{2} - 2 \left|{x}\right| - 15}\right| = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = -5$$
$$x_{2} = 5$$
Numerical solution
$$x_{1} = 5.00000000000004$$
$$x_{2} = 5$$
$$x_{3} = -5$$
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
$$2 \cdot \left(4 \left(x - \operatorname{sign}{\left(x \right)}\right)^{2} \delta\left(- x^{2} + 2 \left|{x}\right| + 15\right) + \left(2 \delta\left(x\right) - 1\right) \operatorname{sign}{\left(- x^{2} + 2 \left|{x}\right| + 15 \right)}\right) = 0$$
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
$$2 \cdot \left(4 \left(x - \operatorname{sign}{\left(x \right)}\right)^{2} \delta\left(- x^{2} + 2 \left|{x}\right| + 15\right) + \left(2 \delta\left(x\right) - 1\right) \operatorname{sign}{\left(- x^{2} + 2 \left|{x}\right| + 15 \right)}\right) = 0$$
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
$$\lim_{x \to -\infty} \left|{x^{2} - 2 \left|{x}\right| - 15}\right| = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \left|{x^{2} - 2 \left|{x}\right| - 15}\right| = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty} \left|{x^{2} - 2 \left|{x}\right| - 15}\right| = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \left|{x^{2} - 2 \left|{x}\right| - 15}\right| = \infty$$
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 Abs(x^2 - 2*|x| - 1*15), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\left|{x^{2} - 2 \left|{x}\right| - 15}\right|}{x}\right) = -\infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{\left|{x^{2} - 2 \left|{x}\right| - 15}\right|}{x}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\frac{\left|{x^{2} - 2 \left|{x}\right| - 15}\right|}{x}\right) = -\infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{\left|{x^{2} - 2 \left|{x}\right| - 15}\right|}{x}\right) = \infty$$
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:
$$\left|{x^{2} - 2 \left|{x}\right| - 15}\right| = \left|{x^{2} - 2 \left|{x}\right| - 15}\right|$$
- Yes
$$\left|{x^{2} - 2 \left|{x}\right| - 15}\right| = - \left|{x^{2} - 2 \left|{x}\right| - 15}\right|$$
- No
so, the function
is
even
So, check:
$$\left|{x^{2} - 2 \left|{x}\right| - 15}\right| = \left|{x^{2} - 2 \left|{x}\right| - 15}\right|$$
- Yes
$$\left|{x^{2} - 2 \left|{x}\right| - 15}\right| = - \left|{x^{2} - 2 \left|{x}\right| - 15}\right|$$
- No
so, the function
is
even
The graph