Illustrating Quadratic Equations and Functions

Vocabulary terms covering definitions, properties, and methods for solving quadratic equations and graphing quadratic functions based on the lecture material.

Quadratic Equation

An equation of degree $2$ or a second degree equation that can be written in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real numbers and $a eq 0$.

Linear Equation

An equation that does not have a squared exponent; if the coefficient $a$ in $ax^2 + bx + c = 0$ equals $0$, the equation becomes linear.

Extracting Square Roots

A method of solving quadratic equations by taking the square root of both sides, used when the equation is in the form $x^2 = k$.

Perfect Square Trinomial

An expression that can be expressed as a square of a binomial, such as $x^2 + 10x + 25 = (x+5)^2$.

Zero Product Property

A property stating that if $ab = 0$, then either $a = 0$, $b = 0$, or both are zero.

Quadratic Formula

The formula used to solve quadratic equations: x = rac{-b ext{ extpm} ext{ extsqrt}(b^2 - 4ac)}{2a}.

Discriminant

The radicand $b^2 - 4ac$ in the quadratic formula used to determine the number and types of solutions.

Nature of Roots: Zero

When the discriminant is zero, the roots are real numbers and equal.

Nature of Roots: Positive Perfect Square

When the discriminant is a positive perfect square, the roots are rational numbers and not equal.

Nature of Roots: Positive Non-Perfect Square

When the discriminant is positive but not a perfect square, the roots are irrational numbers and not equal.

Nature of Roots: Negative

When the discriminant is less than zero, the equation has no real roots or solutions.

Sum of the Roots

The sum of the solutions of a quadratic equation, calculated as ext{-rac{b}{a}}.

Product of the Roots

The product of the solutions of a quadratic equation, calculated as ext{rac{c}{a}}.

Completing the Square

A process where a term is added to $x^2 + bx$ to create a perfect square trinomial, specifically by adding the square of half the coefficient of $x$: (rac{b}{2})^2.

Quadratic Inequality

An inequality that contains a polynomial of degree $2$ and can be written in forms such as ax^2 + bx + c > 0, ax^2 + bx + c < 0, $ax^2 + bx + c ext{ extgeq} 0$, or $ax^2 + bx + c ext{ extleq} 0$.

Quadratic Function

A second degree function that can be written in the form $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are real numbers and $a eq 0$.

Parabola

The smooth curve that represents the graph of a quadratic function.

Vertex

The minimum or maximum point on the graph of a quadratic function, denoted as $(h, k)$.

Axis of Symmetry

The vertical line $x = h$ that divides a parabola into two equal halves such that one half reflects the other.

General Form of Quadratic Function

The expression of a quadratic function as $y = ax^2 + bx + c$.

Standard/Vertex Form of Quadratic Function

The expression of a quadratic function as $y = a(x - h)^2 + k$.

Domain of a Quadratic Function

The set of all possible input values for $x$, which for any quadratic function is the set of all real numbers.

Range of a Quadratic Function

The set of all output values $y$; if a > 0, the range is $ext{ extbraceleft} y ext{|} y ext{ extgeq} k ext{ extbraceright}$; if a < 0, the range is $ext{ extbraceleft} y ext{|} y ext{ extleq} k ext{ extbraceright}$, where $k$ is the $y$-coordinate of the vertex.

Zeros of the Function

The values of $x$ for which the function result is zero, also known as the roots or $x$-intercepts.

Parent Function

The most basic form of a function group from which others are derived; for quadratics, it is $y = x^2$.

Horizontal Translation

The effect of changing the value of $h$ in $y = a(x - h)^2 + k$; if h > 0, the parabola translates right; if h < 0, it translates left.

Vertical Translation

The effect of changing the value of $k$ in $y = a(x - h)^2 + k$; if k > 0, the parabola translates upward; if k < 0, it translates downward.