Key Concepts & Glossary

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