## Key Equations

general form of a quadratic function | [latex]f\left(x\right)=a{x}^{2}+bx+c[/latex] |

the quadratic formula | [latex]x=\frac{-b\pm \sqrt{{b}^{2}-4ac}}{2a}[/latex] |

standard form of a quadratic function | [latex]f\left(x\right)=a{\left(x-h\right)}^{2}+k[/latex] |

## Key Concepts

- A polynomial function of degree two is called a quadratic function.
- The graph of a quadratic function is a parabola. A parabola is a U-shaped curve that can open either up or down.
- The axis of symmetry is the vertical line passing through the vertex. The zeros, or
*x*-intercepts, are the points at which the parabola crosses the*x*-axis. The*y*-intercept is the point at which the parabola crosses the*y-*axis. - Quadratic functions are often written in general form. Standard or vertex form is useful to easily identify the vertex of a parabola. Either form can be written from a graph.
- The vertex can be found from an equation representing a quadratic function.
- The domain of a quadratic function is all real numbers. The range varies with the function.
- A quadratic function’s minimum or maximum value is given by the
*y*-value of the vertex. - The minimum or maximum value of a quadratic function can be used to determine the range of the function and to solve many kinds of real-world problems, including problems involving area and revenue.
- Some quadratic equations must be solved by using the quadratic formula.
- The vertex and the intercepts can be identified and interpreted to solve real-world problems.

## Glossary

**axis of symmetry**- a vertical line drawn through the vertex of a parabola around which the parabola is symmetric; it is defined by [latex]x=-\frac{b}{2a}.[/latex]

**general form of a quadratic function**- the function that describes a parabola, written in the form [latex]f\left(x\right)=a{x}^{2}+bx+c,[/latex] where
*a*,*b*, and*c*are real numbers and [latex]a\ne 0.[/latex]

**standard form of a quadratic function**- the function that describes a parabola, written 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.

**vertex**- the point at which a parabola changes direction, corresponding to the minimum or maximum value of the quadratic function

**vertex form of a quadratic function**- another name for the standard form of a quadratic function

**zeros**- in a given function, the values of
*x*at which*y*= 0, also called roots