Quadratic function calculator
Calculator helpful during common operations related to quadratic function such as calculating value at given point, calculating discriminant or finding out function roots.

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This online calculator is currently under heavy development. It may or it may NOT work correctly.
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Calculations data - enter values, that you know here

Function value (yy)
(the function value at single point x, often marked as f(x))
=>
Dicriminant of the quadratic function (Δ\Delta)
=>
The first root of the function (x1x_1)
=>
The second root of the function (x2x_2)
=>
X coordinate of the parabola vertex (hh)
(for this argument the function reaches its local extremum)
=>
Y coordinate of the parabola vertex (kk)
=>
Function argument (xx)
<=
Coefficient before power of two (aa)
<=
Coefficient before power of one (bb)
<=
Free parameter (cc)
<=

Result: function value (yy)

Summary
Used formulaShow sourcey=a x2+b x+c y= a~{ x}^{2}+ b~ x+ c
ResultShow source33
Numerical resultShow source33
Result step by step
1Show source112+11+11\cdot{1}^{2}+1\cdot1+1Multiply by one
2Show source12+11+1{1}^{2}+1\cdot1+1Power of one number
3Show source1+11+11+1\cdot1+1Multiply by one
4Show source1+1+11+1+1Simplify arithmetic
5Show source33Result
Numerical result step by step
1Show source33Result

Some facts

  • The quadratic function is a function that can be prepresented in the form:
    y=a x2+b x+c y= a~{ x}^{2}+ b~ x+ c
    where:
    • yy - function value (the function value at single point x, often marked as f(x)),
    • xx - function argument (called also independent value),
    • aa, bb, cc - quadratic function coefficients (numbers just before x2, x and free parameter).
  • The graph of the quadratic function is parabola. Depending on the coefficient value at the second power (a), the following scenarios are possible:
    • when the coefficient on the second power is positive (a> 0) - the parabola's arms are directed upwards,
    • when the coefficient on the second power is negative (a < 0) - the parabola arms are directed downwards,
    • in the case when the coefficient on the second power is equal to zero (a = 0) - the quadratic function reduces to linear function.
  • A square function can have one, two, or have no zero points. To check the number of zero places (sometimes also called roots), we can calculate the discriminant of a quadratic function (colloquially called delta):
    Δ=b24 a c \Delta={ b}^{2}-4~ a~ c
    where:
    • Δ\Delta - dicriminant of the quadratic function,
    • aa, bb, cc - quadratic function coefficients (numbers just before x2, x and free parameter).
    then the following scenarios are possible:
    • discriminant is negative (Δ <0) - the function has no roots, the graph of the function is a parabola, which is located entirety above the OX axis or under the OX axis,
    • discriminant is equal to zero (Δ = 0) - the function has exactly one root, the graph of the function is a parabola whose vertex lies on the OX axis:
      h=b2 a h=\frac{- b}{2~ a}
    • discriminant is positive (Δ> 0) - the function has two different roots, the function graph is a parabola, whose arms cross the OX axis:
      x1=bΔ2 a x_1=\frac{- b-\sqrt{ \Delta}}{2~ a}
      x2=b+Δ2 a x_2=\frac{- b+\sqrt{ \Delta}}{2~ a}
  • A quadratic function is a special case of polynomial function in which the order is 2.

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