Mister Exam
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- Graph of implicit function
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- Graph in Polar Coordinates
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- The intersection points of functions
-
How to use it?
- Graphing y =:
- x^2+3x-10
-
(x^2+3)/(x-1)
- x^2-10x+27
- ((x+1)^2)/(x-2)
-
Identical expressions
- |x^ two - three *x+ five |
- module of x squared minus 3 multiply by x plus 5|
- module of x to the power of two minus three multiply by x plus five |
- |x2-3*x+5|
- |x²-3*x+5|
- |x to the power of 2-3*x+5|
- |x^2-3x+5|
- |x2-3x+5|
-
Similar expressions
- |x^2+3*x+5|
- |x^2-3*x-5|
Graphing y = |x^2-3*x+5|
The solution
You have entered
[src]
| 2 | f(x) = |x - 3*x + 5|
$$f{\left(x \right)} = \left|{\left(x^{2} - 3 x\right) + 5}\right|$$
f = |x^2 - 3*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:
$$\left|{\left(x^{2} - 3 x\right) + 5}\right| = 0$$
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:
$$\left|{\left(x^{2} - 3 x\right) + 5}\right| = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
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
$$2 \left(\left(2 x - 3\right)^{2} \delta\left(x^{2} - 3 x + 5\right) + \operatorname{sign}{\left(x^{2} - 3 x + 5 \right)}\right) = 0$$
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
$$2 \left(\left(2 x - 3\right)^{2} \delta\left(x^{2} - 3 x + 5\right) + \operatorname{sign}{\left(x^{2} - 3 x + 5 \right)}\right) = 0$$
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
$$\lim_{x \to -\infty} \left|{\left(x^{2} - 3 x\right) + 5}\right| = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \left|{\left(x^{2} - 3 x\right) + 5}\right| = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty} \left|{\left(x^{2} - 3 x\right) + 5}\right| = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \left|{\left(x^{2} - 3 x\right) + 5}\right| = \infty$$
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 - 3*x + 5|, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\left|{\left(x^{2} - 3 x\right) + 5}\right|}{x}\right) = -\infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{\left|{\left(x^{2} - 3 x\right) + 5}\right|}{x}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\frac{\left|{\left(x^{2} - 3 x\right) + 5}\right|}{x}\right) = -\infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{\left|{\left(x^{2} - 3 x\right) + 5}\right|}{x}\right) = \infty$$
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:
$$\left|{\left(x^{2} - 3 x\right) + 5}\right| = \left|{x^{2} + 3 x + 5}\right|$$
- No
$$\left|{\left(x^{2} - 3 x\right) + 5}\right| = - \left|{x^{2} + 3 x + 5}\right|$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\left|{\left(x^{2} - 3 x\right) + 5}\right| = \left|{x^{2} + 3 x + 5}\right|$$
- No
$$\left|{\left(x^{2} - 3 x\right) + 5}\right| = - \left|{x^{2} + 3 x + 5}\right|$$
- No
so, the function
not is
neither even, nor odd