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
Licenses & Attributions
CC licensed content, Shared previously
- Precalculus. Provided by: OpenStax Authored by: Jay Abramson, et al.. Located at: https://openstax.org/books/precalculus/pages/1-introduction-to-functions. License: CC BY: Attribution. License terms: Download For Free at : http://cnx.org/contents/[email protected]..