Learning Objectives
In this section, you will:
5.7.1 – Finding the Domain and Range of a Quadratic Function
5.7.2 – Determining the Maximum and Minimum Values of Quadratic Functions
5.7.1 – 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 point, the range is restricted. Since the vertex of a parabola will be either a maximum or a minimum, the range will consist of all y-values greater than or equal to the y-coordinate at the turning point or less than or equal to the y-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 unless the context of the function presents some restrictions.
The range of a quadratic function written in general form [latex]\,f\left(x\right)=a{x}^{2}+bx+c\,[/latex] 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]
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] 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]
How To
Given a quadratic function, find the domain and range.
- Identify the domain of any quadratic function as all real numbers.
- 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.
- 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]
Example 1 – Finding the Domain and Range of a Quadratic Function
Find the domain and range of [latex]\,f\left(x\right)=-5{x}^{2}+9x-1.[/latex]
As with any quadratic function, the domain is all real numbers.
Because [latex]\,a\,[/latex] is negative, the parabola opens downward and has a maximum value. We need to determine the maximum value. We can begin by finding the [latex]\,x\text{-}[/latex] value of the vertex.
The maximum value is given by [latex]$$\,f\left(h\right).$$[/latex]
The range is [latex]\,f\left(x\right)\le \fra