- Identify quadratic functions in both general and standard form
- Determine the domain and range of a quadratic function by recognizing whether the vertex represents a maximum or minimum point
- Recognize key features of a parabola’s graph: vertex, axis of symmetry, y-intercept, and minimum or maximum value
- Create graphs of quadratic functions using tables and transformations
Equations of Quadratic Functions
general form of a quadratic function
The general form of a quadratic function presents the function in the form
[latex]f\left(x\right)=a{x}^{2}+bx+c[/latex]
where [latex]a[/latex], [latex]b[/latex], and [latex]c[/latex] are real numbers and [latex]a\ne 0[/latex].
If [latex]a>0[/latex], the parabola opens upward. If [latex]a<0[/latex], the parabola opens downward.
We can use the general form of a parabola to find the equation for the axis of symmetry.
axis of symmetry
The axis of symmetry is defined by [latex]x=-\dfrac{b}{2a}[/latex]. If we use the quadratic formula, [latex]x=\dfrac{-b\pm \sqrt{{b}^{2}-4ac}}{2a}[/latex], to solve [latex]a{x}^{2}+bx+c=0[/latex] for the [latex]x[/latex]-intercepts, or zeros, we find the value of [latex]x[/latex] halfway between them is always [latex]x=-\dfrac{b}{2a}[/latex], the equation for the axis of symmetry.
The figure below shows the graph of the quadratic function written in general form as [latex]y={x}^{2}+4x+3[/latex]. In this form, [latex]a=1,\text{ }b=4[/latex], and [latex]c=3[/latex]. Because [latex]a>0[/latex], the parabola opens upward. The axis of symmetry is [latex]x=-\dfrac{4}{2\left(1\right)}=-2[/latex]. This also makes sense because we can see from the graph that the vertical line [latex]x=-2[/latex] divides the graph in half. The vertex always occurs along the axis of symmetry. For a parabola that opens upward, the vertex occurs at the lowest point on the graph, in this instance, [latex]\left(-2,-1\right)[/latex]. The [latex]x[/latex]-intercepts, those points where the parabola crosses the [latex]x[/latex]-axis, occur at [latex]\left(-3,0\right)[/latex] and [latex]\left(-1,0\right)[/latex].
standard form of a quadratic function
The standard form of a quadratic function presents the function in the form
[latex]f\left(x\right)=a{\left(x-h\right)}^{2}+k[/latex]
where [latex]\left(h,\text{ }k\right)[/latex] is the vertex.
Because the vertex appears in the standard form of the quadratic function, this form is also known as the vertex form of a quadratic function.
Given a quadratic function in general form, find the vertex of the parabola.
One reason we may want to identify the vertex of the parabola is that this point will inform us where the maximum or minimum value of the output occurs, [latex]k[/latex], and where it occurs, [latex]h[/latex].
vertex of the parabola
If we are given the general form of a quadratic function:
[latex]f(x)=ax^2+bx+c[/latex]
We can define the vertex, [latex](h,k)[/latex], by doing the following:
- Identify [latex]a[/latex], [latex]b[/latex], and [latex]c[/latex].
- Find [latex]h[/latex], the [latex]x[/latex]-coordinate of the vertex, by substituting [latex]a[/latex] and [latex]b[/latex] into [latex]h=-\dfrac{b}{2a}[/latex].
- Find [latex]k[/latex], the [latex]y[/latex]-coordinate of the vertex, by evaluating [latex]k=f\left(h\right)=f\left(-\dfrac{b}{2a}\right)[/latex]
Find the vertex of the quadratic function [latex]f\left(x\right)=2{x}^{2}-6x+7[/latex]. Rewrite the quadratic in standard form (vertex form).