Polynomial relationship with highest power of variable as two

Quadratic functions describe a special type of polynomial relationship with the highest power of the variable being two

Graph of a Quadratic Function

Parabola

The graph of a quadratic function is a curve called a parabola

Direction of Opening

The direction of opening of a parabola is determined by the coefficient of the quadratic term

Forms of Quadratic Functions

Quadratic functions can be written in three primary forms: standard form, factored form, and vertex form

Definition of Standard Form

The standard form of a quadratic function is \(y=ax^2+bx+c\), where \(a\), \(b\), and \(c\) are constants and \(a \neq 0\)

Attributes of Standard Form

Direction of Opening

The coefficient \(a\) in the standard form determines the direction of the parabola's opening

Y-Intercept

The y-intercept of a parabola can be found by evaluating \(f(0)\), which equals \(c\) in the standard form

Axis of Symmetry

The axis of symmetry of a parabola is given by the equation \(x=-\frac{b}{2a}\) in the standard form

Quadratic Formula

The quadratic formula, \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\), derived from the standard form, allows for the determination of the parabola's roots

Definition of Factored Form

The factored form of a quadratic function is expressed as \(f(x)=a(x-p)(x-q)\), where \(p\) and \(q\) are the roots of the quadratic equation

Usefulness of Factored Form

X-Intercepts

The factored form is useful for quickly identifying the x-intercepts of a parabola, which occur at \(x=p\) and \(x=q\)

Zero Product Property

The zero product property facilitates the determination of x-intercepts by stating that if a product of factors equals zero, then at least one of the factors must be zero

Limitations of Factored Form

Not all quadratic equations can be factored over the real numbers, as some may have complex roots

Definition of Vertex Form

The vertex form of a quadratic function is given by \(f(x)=a(x-h)^2+k\), where \((h, k)\) are the coordinates of the vertex

Importance of Vertex Form

Vertex Properties

The vertex form is particularly valuable for identifying the location and properties of the vertex, such as maximum or minimum values

Parameter \(a\)

The parameter \(a\) in the vertex form affects the parabola's opening direction and vertical stretch or compression

Derivation of Vertex Form

The vertex form can be derived from completing the square on a quadratic equation in standard form