Summary: Analysis of Quadratic Functions

Key Equations

general form of a quadratic function [latex]f\left(x\right)=a{x}^{2}+bx+c[/latex]
standard form of a quadratic function [latex]f\left(x\right)=a{\left(x-h\right)}^{2}+k[/latex]
Vertex of a parabola (h,k) [latex]h=-b/2a[/latex]         [latex]k=f\left(-b/2a\right)[/latex]
The Quadratic Formula [latex]x=\frac{-b\pm \sqrt{{b}^{2}-4ac}}{2a}[/latex]

Key Concepts

  • The zeros, or [latex]x[/latex]-intercepts, are the points at which the parabola crosses the [latex]x[/latex]-axis. The [latex]y[/latex]-intercept is the point at which the parabola crosses the [latex]y[/latex]–axis.
  • The vertex can be found from an equation representing a quadratic function.
  • A quadratic function’s minimum or maximum value is given by the [latex]y[/latex]-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

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 [latex]x[/latex] at which [latex]y=0[/latex], also called roots