5.1 Quadratic functions  (Page 7/15)

Key equations

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 $x\text{-}$ intercepts, are the points at which the parabola crosses the $x\text{-}$ axis. The $y\text{-}$ intercept is the point at which the parabola crosses the $y\text{-}$ axis. See [link] , [link] , and [link] .
• 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. See [link] .
• The vertex can be found from an equation representing a quadratic function. See [link] .
• The domain of a quadratic function is all real numbers. The range varies with the function. See [link] .
• A quadratic function’s minimum or maximum value is given by the $y\text{-}$ 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. See [link] and [link] .
• The vertex and the intercepts can be identified and interpreted to solve real-world problems. See [link] .

Section exercises

Verbal

Algebraic

For the following exercises, rewrite the quadratic functions in standard form and give the vertex.

For the following exercises, determine whether there is a minimum or maximum value to each quadratic function. Find the value and the axis of symmetry.

For the following exercises, determine the domain and range of the quadratic function.