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How to use it?
- Graphing y =:
- x+4x
- x^4-4x^2
-
-x^3+x^2-x+6
- sin(x)
-
Identical expressions
- (|x+ two |*(x- six)^ two)/((x- two)*(x- six))
- ( module of x plus 2| multiply by (x minus 6) squared ) divide by ((x minus 2) multiply by (x minus 6))
- ( module of x plus two | multiply by (x minus six) to the power of two) divide by ((x minus two) multiply by (x minus six))
- (|x+2|*(x-6)2)/((x-2)*(x-6))
- |x+2|*x-62/x-2*x-6
- (|x+2|*(x-6)²)/((x-2)*(x-6))
- (|x+2|*(x-6) to the power of 2)/((x-2)*(x-6))
- (|x+2|(x-6)^2)/((x-2)(x-6))
- (|x+2|(x-6)2)/((x-2)(x-6))
- |x+2|x-62/x-2x-6
- |x+2|x-6^2/x-2x-6
- (|x+2|*(x-6)^2) divide by ((x-2)*(x-6))
-
Similar expressions
- (|x-2|*(x-6)^2)/((x-2)*(x-6))
- (|x+2|*(x+6)^2)/((x-2)*(x-6))
- (|x+2|*(x-6)^2)/((x-2)*(x+6))
- (|x+2|*(x-6)^2)/((x+2)*(x-6))
Graphing y = (|x+2|*(x-6)^2)/((x-2)*(x-6))
The solution
You have entered
[src]
2
|x + 2|*(x - 6)
f(x) = ----------------
(x - 2)*(x - 6)
$$f{\left(x \right)} = \frac{\left(x - 6\right)^{2} \left|{x + 2}\right|}{\left(x - 6\right) \left(x - 2\right)}$$
f = ((x - 6)^2*|x + 2|)/(((x - 6)*(x - 2)))
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 2$$
$$x_{2} = 6$$
$$x_{1} = 2$$
$$x_{2} = 6$$
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:
$$\frac{\left(x - 6\right)^{2} \left|{x + 2}\right|}{\left(x - 6\right) \left(x - 2\right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = -2$$
$$x_{2} = 6$$
Numerical solution
$$x_{1} = 6.00000000000016$$
$$x_{2} = 6$$
$$x_{3} = -2$$
$$x_{4} = 6$$
so we need to solve the equation:
$$\frac{\left(x - 6\right)^{2} \left|{x + 2}\right|}{\left(x - 6\right) \left(x - 2\right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = -2$$
$$x_{2} = 6$$
Numerical solution
$$x_{1} = 6.00000000000016$$
$$x_{2} = 6$$
$$x_{3} = -2$$
$$x_{4} = 6$$
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
$$\frac{\left(8 - 2 x\right) \left|{x + 2}\right|}{\left(x - 2\right)^{2}} + \frac{1}{\left(x - 6\right) \left(x - 2\right)} \left(\left(x - 6\right)^{2} \operatorname{sign}{\left(x + 2 \right)} + \left(2 x - 12\right) \left|{x + 2}\right|\right) = 0$$
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
$$\frac{\left(8 - 2 x\right) \left|{x + 2}\right|}{\left(x - 2\right)^{2}} + \frac{1}{\left(x - 6\right) \left(x - 2\right)} \left(\left(x - 6\right)^{2} \operatorname{sign}{\left(x + 2 \right)} + \left(2 x - 12\right) \left|{x + 2}\right|\right) = 0$$
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
$$\frac{2 \left(\frac{\left(\left(x - 4\right) \left(\frac{1}{x - 2} + \frac{1}{x - 6}\right) + \frac{x - 4}{x - 2} - 1 + \frac{x - 4}{x - 6}\right) \left|{x + 2}\right|}{x - 2} - \frac{2 \left(x - 4\right) \left(\left(x - 6\right) \operatorname{sign}{\left(x + 2 \right)} + 2 \left|{x + 2}\right|\right)}{\left(x - 6\right) \left(x - 2\right)} + \frac{\left(x - 6\right)^{2} \delta\left(x + 2\right) + 2 \left(x - 6\right) \operatorname{sign}{\left(x + 2 \right)} + \left|{x + 2}\right|}{x - 6}\right)}{x - 2} = 0$$
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
$$\frac{2 \left(\frac{\left(\left(x - 4\right) \left(\frac{1}{x - 2} + \frac{1}{x - 6}\right) + \frac{x - 4}{x - 2} - 1 + \frac{x - 4}{x - 6}\right) \left|{x + 2}\right|}{x - 2} - \frac{2 \left(x - 4\right) \left(\left(x - 6\right) \operatorname{sign}{\left(x + 2 \right)} + 2 \left|{x + 2}\right|\right)}{\left(x - 6\right) \left(x - 2\right)} + \frac{\left(x - 6\right)^{2} \delta\left(x + 2\right) + 2 \left(x - 6\right) \operatorname{sign}{\left(x + 2 \right)} + \left|{x + 2}\right|}{x - 6}\right)}{x - 2} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
$$x_{1} = 2$$
$$x_{2} = 6$$
$$x_{1} = 2$$
$$x_{2} = 6$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{\left(x - 6\right)^{2} \left|{x + 2}\right|}{\left(x - 6\right) \left(x - 2\right)}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{\left(x - 6\right)^{2} \left|{x + 2}\right|}{\left(x - 6\right) \left(x - 2\right)}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\frac{\left(x - 6\right)^{2} \left|{x + 2}\right|}{\left(x - 6\right) \left(x - 2\right)}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{\left(x - 6\right)^{2} \left|{x + 2}\right|}{\left(x - 6\right) \left(x - 2\right)}\right) = \infty$$
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|*(x - 6)^2)/(((x - 2)*(x - 6))), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\frac{1}{\left(x - 6\right) \left(x - 2\right)} \left(x - 6\right)^{2} \left|{x + 2}\right|}{x}\right) = -1$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = - x$$
$$\lim_{x \to \infty}\left(\frac{\frac{1}{\left(x - 6\right) \left(x - 2\right)} \left(x - 6\right)^{2} \left|{x + 2}\right|}{x}\right) = 1$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x$$
$$\lim_{x \to -\infty}\left(\frac{\frac{1}{\left(x - 6\right) \left(x - 2\right)} \left(x - 6\right)^{2} \left|{x + 2}\right|}{x}\right) = -1$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = - x$$
$$\lim_{x \to \infty}\left(\frac{\frac{1}{\left(x - 6\right) \left(x - 2\right)} \left(x - 6\right)^{2} \left|{x + 2}\right|}{x}\right) = 1$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x$$
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:
$$\frac{\left(x - 6\right)^{2} \left|{x + 2}\right|}{\left(x - 6\right) \left(x - 2\right)} = \frac{\left(- x - 6\right) \left|{x - 2}\right|}{- x - 2}$$
- No
$$\frac{\left(x - 6\right)^{2} \left|{x + 2}\right|}{\left(x - 6\right) \left(x - 2\right)} = - \frac{\left(- x - 6\right) \left|{x - 2}\right|}{- x - 2}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{\left(x - 6\right)^{2} \left|{x + 2}\right|}{\left(x - 6\right) \left(x - 2\right)} = \frac{\left(- x - 6\right) \left|{x - 2}\right|}{- x - 2}$$
- No
$$\frac{\left(x - 6\right)^{2} \left|{x + 2}\right|}{\left(x - 6\right) \left(x - 2\right)} = - \frac{\left(- x - 6\right) \left|{x - 2}\right|}{- x - 2}$$
- No
so, the function
not is
neither even, nor odd