Quadratic Functions and Complex Numbers Review

Comprehensive vocabulary flashcards covering solving methods, graphing attributes, transformations, complex numbers, and discriminants as found in the Unit 3 lecture notes.

Standard Form (Quadratic)

$$ax^2 + bx + c = 0$$

Quadratic Formula

A formula used to solve any quadratic equation, expressed as $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Factoring Method

A technique to solve quadratics by setting the equation to 0 and identifying factors such as GCF, Difference of Squares, or Trinomials.

Square Root Method

A solving method used when there is only an $x^2$ term (no $x$), involving isolating the square and taking the square root of both sides.

Completing the Square

A method to solve quadratics by isolating $c$, ensuring $a=1$, and adding the square of half the $b$ term to both sides.

Parent Function

The most basic quadratic function, $y = x^2$, which creates a parabola with a vertex at $(0,0)$.

Many-to-one

The classification of the quadratic parent function's mapping, where multiple inputs can result in the same output.

Vertex

The highest or lowest point on a parabola, denoted as $(h, k)$; it is a minimum if $a > 0$ and a maximum if $a < 0$.

Axis of Symmetry

The vertical line that divides a parabola into two symmetrical halves, defined by the equation $x = -\frac{b}{2a}$.

Vertex Form

$$y = a(x - h)^2 + k$$

Factored Form

$y = a(x - r_1)(x - r_2)$ where $r_1$ and $r_2$ represent the $x$-intercepts.

Domain of Quadratic Functions

$$(-\infty, \infty)$$

Range

The set of possible $y$-values for a function, determined by the vertex and the direction the parabola opens.

Vertical Translation

A transformation that shifts the graph up ($+k$) or down ($-k$) based on values outside the parentheses.

Horizontal Translation

A transformation that shifts the graph left ($+h$) or right ($-h$) based on values inside the parentheses.

Vertical Stretch

Occurs when the coefficient $|a| > 1$, making the parabola narrower.

Vertical Compression

Occurs when the coefficient $|a| < 1$, making the parabola wider.

Vertical Reflection

Occurs when $a < 0$, causing the parabola to flip over the $x$-axis.

Discriminant

The value $b^2 - 4ac$ from the quadratic formula, used to determine the number and type of solutions.

Positive Discriminant ($b^2 - 4ac > 0$)

Indicates the equation has two real solutions (roots/zeros).

Zero Discriminant ($b^2 - 4ac = 0$)

Indicates the equation has exactly one real solution.

Negative Discriminant ($b^2 - 4ac < 0$)

Indicates the equation has two imaginary solutions.

Imaginary Unit ($i$)

Defined as $i = \sqrt{-1}$, where $i^2 = -1$.

Complex Number

A number in the form $a + bi$, where $a$ is the real part and $bi$ is the imaginary part.

Complex Conjugates

Numbers in the form $(a + bi)$ and $(a - bi)$; their product is always a real number.

Quadratic Inequality

An inequality involving a quadratic expression, graphed with shaded regions and either solid (for $\leq, \geq$) or dashed (for $<, >$) lines.