Quadratic Functions: Learn It 2

Finding the Domain and Range of a Quadratic Function

Any number can be the input value of a quadratic function. Therefore the domain of any quadratic function is all real numbers. Because parabolas have a maximum or a minimum at the vertex, the range is restricted. Since the vertex of a parabola will be either a maximum or a minimum, the range will consist of all [latex]y[/latex]-values greater than or equal to the [latex]y[/latex]-coordinate of the vertex or less than or equal to the [latex]y[/latex]-coordinate at the turning point, depending on whether the parabola opens up or down.

domain and range of a quadratic function

The domain of any quadratic function is all real numbers.

 

Determining the range of a quadratic formula is different depending on which form the quadratic function is in:

 

General Form

  • The range of a quadratic function written in general form with a positive [latex]a[/latex] value is [latex]f\left(x\right)\ge f\left(-\frac{b}{2a}\right)[/latex], or [latex]\left[f\left(-\frac{b}{2a}\right),\infty \right)[/latex]
  • The range of a quadratic function written in general form with a negative [latex]a[/latex] value is [latex]f\left(x\right)\le f\left(-\frac{b}{2a}\right)[/latex], or [latex]\left(-\infty ,f\left(-\frac{b}{2a}\right)\right][/latex].

Standard Form

  • The range of a quadratic function written in standard form [latex]f\left(x\right)=a{\left(x-h\right)}^{2}+k[/latex] with a positive [latex]a[/latex] value is [latex]f\left(x\right)\ge k[/latex] or [latex][k,\infty)[/latex].
  • The range of a quadratic function written in standard form with a negative [latex]a[/latex] value is [latex]f\left(x\right)\le k[/latex] or or [latex](-\infty,k][/latex].
How to: Determine the Domain and Range from the Vertex

  1. The domain of any quadratic function is all real numbers.
  2. Determine whether [latex]a[/latex] is positive or negative.
             If [latex]a[/latex] is positive, the parabola has a minimum.
             If [latex]a[/latex] is negative, the parabola has a maximum.
  3. Determine the maximum or minimum value of the parabola, [latex]k[/latex].
             If the parabola has a minimum, the range is given by [latex]f\left(x\right)\ge k[/latex], or [latex]\left[k,\infty \right)[/latex].
             If the parabola has a maximum, the range is given by [latex]f\left(x\right)\le k[/latex], or [latex]\left(-\infty ,k\right][/latex].

Find the domain and range of [latex]f\left(x\right)=-5{x}^{2}+9x - 1[/latex].