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  • Graphing y =:
  • 3x^4-4x^3
  • 2x^4-4x^2+1
  • 2x^2-3x+4
  • (1/x)+4x^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|
Plotting a function graph/ |x^2-3*x+5|

Graphing y = |x^2-3*x+5|

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       | 2          |
f(x) = |x  - 3*x + 5|
f(x)=∣(x2−3x)+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
02468-8-6-4-2-10100200
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:
∣(x2−3x)+5∣=0\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
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to |x^2 - 3*x + 5|.
∣(02−0)+5∣\left|{\left(0^{2} - 0\right) + 5}\right|
The result:
f(0)=5f{\left(0 \right)} = 5
The point:
(0, 5)
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
d2dx2f(x)=0\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:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
2((2x−3)2δ(x2−3x+5)+sign⁡(x2−3x+5))=02 \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→−∞∣(x2−3x)+5∣=∞\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→∞∣(x2−3x)+5∣=∞\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→−∞(∣(x2−3x)+5∣x)=−∞\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→∞(∣(x2−3x)+5∣x)=∞\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:
∣(x2−3x)+5∣=∣x2+3x+5∣\left|{\left(x^{2} - 3 x\right) + 5}\right| = \left|{x^{2} + 3 x + 5}\right|
- No
∣(x2−3x)+5∣=−∣x2+3x+5∣\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
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