Quadratic Functions, Standard Form, Vertex Form, and Transformations

A comprehensive vocabulary set covering formulas, forms, characteristics, and transformations of quadratic functions from lecture notes.

Standard Form of a Quadratic Function

A quadratic function written in the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are real numbers with $a \neq 0$.

Vertex Form of a Quadratic Function

A quadratic function written in the form $f(x) = a(x - \alpha)^2 + \beta$, where $(\alpha, \beta)$ represents the coordinates of the vertex.

Formula for Vertex x-coordinate ($\alpha$)

The formula to calculate the x-coordinate of the parabola's vertex from standard form coefficients: $\alpha = -\frac{b}{2a}$.

Formula for Vertex y-coordinate ($\beta$)

The formula to calculate the y-coordinate of the parabola's vertex: $\beta = f(\alpha)$ or $\beta = \frac{4ac - b^2}{4a}$.

Linear Coefficient Relation ($b$)

The equation expressing the linear coefficient $b$ in terms of vertex parameters: $b = -2a\alpha$.

Constant Term Relation ($c$)

The equation expressing the constant term $c$ in terms of vertex parameters: $c = a\alpha^2 + \beta$.

Axis of Symmetry

The vertical line passing through the vertex of a parabola, given by the equation $s \equiv x = \alpha$ or $x = -\frac{b}{2a}$.

Vertex (Top)

The turning point of a parabola, represented by coordinates $T(\alpha, \beta)$ or $T\left(-\frac{b}{2a}, \frac{4ac - b^2}{4a}\right)$.

Dalparabool (Upward Parabola)

A parabola that opens upwards when $a > 0$, having a minimum value at its vertex.

Sign Table and Variation Table

Tables (Tekenverloop and Verloopschema) used to display the sign changes across roots $x_1$ and $x_2$ and the intervals where the function $f(x)$ decreases or increases around its vertex.

Algebraic Conversions between Standard and Vertex Forms

Worked conversions demonstrating transformations between standard form $f(x) = ax^2 + bx + c$ and vertex form $f(x) = a(x - \alpha)^2 + \beta$.

'Trace' Key on GRM

A feature on a graphing calculator (GRM) used to directly determine the y-value of the vertex by evaluating $f(\alpha)$.

Graph Transformations

Geometric alterations applied to a basic parabola $g(x) = x^2$, including horizontal shifts, vertical shifts, vertical stretching or compression, and reflections across the x-axis.