Finding the Equation of a Quadratic Function

Vocabulary flashcards covering key terms, parts of a parabola, forms of quadratic equations, and methods for determining the equation from tables, graphs, and zeros.

Parabola

The characteristic curve formed by the graph of a quadratic function.

Vertex

The minimum or maximum point of a quadratic function, depending on whether the parabola opens upward or downward.

Axis of Symmetry

A vertical line that divides a parabola into two mirror halves.

$y$-intercept

The point where the graph of a quadratic function crosses the $y$-axis.

$x$-intercepts

The points where the graph crosses the $x$-axis, corresponding to the roots of the equation and the zeros of the function.

Discriminant

Calculated using $d = b^2 - 4ac$, this value indicates the nature of the roots and how many times a parabola crosses the $x$-axis.

Standard Form

The quadratic equation written as $y = ax^2 + bx + c$ where $a \neq 0$, commonly used when given a table of values or three ordinary points.

Vertex Form

The quadratic equation written as $y = a(x - h)^2 + k$, where $(h, k)$ represents the vertex of the parabola.

Factored Form

The quadratic equation written as $y = a(x - r_1)(x - r_2)$, where $r_1$ and $r_2$ represent the zeros or $x$-intercepts of the function.

Second Difference

The difference between consecutive first differences of the $y$-values in a table of values; when constant, it confirms a quadratic relationship.

$2a$ Rule

A relationship stating that for $x$-values increasing by $1$, the constant second difference of the $y$-values is equal to $2a$.

Three-point Method

A procedure for finding the equation $y = ax^2 + bx + c$ by selecting three ordered pairs and solving the resulting system of three linear equations for $a$, $b$, and $c$.