math vocab chapter 5

best-fitting quadratic model

An equation that employs the variable x having the general form ax2 + bx + c = 0, where a, b, and c are constants and a does not equal zero; that is, the variable is squared but raised to no higher power.

complex number

An illustration of the complex plane. The real part of a complex number z = x + iy is x, and its imaginary part is y. A complex number is a number of the form a + bi, where a and b are real numbers and i is the imaginary unit, satisfying i2 = −1.

imaginary number

An imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit i, [note 1] which is defined by its property i2 = −1. The square of an imaginary number bi is −b2.

completing the square

Completing the Square is a technique used to solve quadratic equations, graph quadratic functions, and evaluate integrals. This technique can be used when factoring a quadratic equation does not work or to find irrational and complex roots.

zero product property

The Zero Product Property simply states that if ab = 0, then either a = 0 or b = 0 (or both). A product of factors is zero if and only if one or more of the factors is zero. This is particularly useful when solving quadratic equations.

complex conjugates

The complex conjugate of a + bi is a - bi, and similarly the complex conjugate of a - bi is a + bi. This consists of changing the sign of the imaginary part of a complex number. The real part is left unchanged.

monomial

The degree of the monomial is the sum of the exponents of all included variables. Constants have the monomial degree of 0.

intercept form of a quadratic function

The intercept form of a quadratic equation is y=a(x−p)(x−q), where a is the same value as in standard form, and p and q are the x−intercepts.

parabola

The parabola is the curve formed from all the points (x, y) that are equidistant from the directrix and the focus. The line perpendicular to the directrix and passing through the focus (that is, the line that splits the parabola up the middle) is called the "axis of symmetry".

rationalizing the denominator

The process by which a fraction is rewritten so that the denominator contains only rational numbers.

quadratic formula

The quadratic formula, , is used in algebra to solve quadratic equations (polynomial equations of the second degree). The general form of a quadratic equation is , where x represents a variable, and a, b, and c are constants, with . A quadratic equation has two solutions, called roots.

radicand

The radicand is the number found inside a radical symbol, and it is the number you want to find the root of. It could be a square root, cube root or other root, which will be defined by the subscript found outside and just before the radical symbol.

square root

The square root of a number is a value that, when multiplied by itself, gives the number.

axis of symmetry

The two sides of a graph on either side of the axis of symmetry look like mirror images of each other. Example: This is a graph of the parabola y = x2 - 4x + 2 together with its axis of symmetry x = 2. The axis of symmetry is the red vertical line.

vertex of a parabola

The vertex of a parabola is the point where the parabola crosses its axis of symmetry. If the coefficient of the x2 term is positive, the vertex will be the lowest point on the graph, the point at the bottom of the "U"-shape.

factoring

To factor a number means to break it up into numbers that can be multiplied together to get the original number.

pure imaginary number

a complex number of the form iy where y is a real number and i = rad -1

quadratic equation

a quadratic equation as an equation of degree 2, meaning that the highest exponent of this function is 2. The standard form of a quadratic is y = ax^2 + bx + c, where a, b, and c are numbers and a cannot be 0.

radical

a radical expression is defineimd as any expression containing a radical (√) symbol. Many people mistakenly call this a 'square root' symbol, and many times it is used to determine the square root of a number. However, it can also be used to describe a cube root, a fourth root or higher.

trinomial

an algebraic expression consisting of three terms.

binomial

an algebraic expression of the sum or the difference of two terms.

discriminant

an algebraic expression related to the coefficients of a polynomial equation whose value gives information about the roots of the polynomial: b² - 4ac is the discriminant of ax² + bx + c = 0.

complex plane

an infinite two-dimensional space representing the set of complex numbers, especially one in which Cartesian coordinates represent the real and imaginary parts of the complex numbers.

vertex form of a quadratic function

f (x) = a(x - h)2 + k, where (h, k) is the vertex of the parabola

imaginary unit, i

Although there are two possible square roots of any number, the square roots of a negative number cannot be distinguished until one of the two is defined as the imaginary unit, at which point and can then be distinguished.

standard form of a complex number

A complex number is any number that can be written in the standard form a + bi, where a and b are real numbers and i is the imaginary unit.

quadratic function

A quadratic function is one of the form f(x) = ax2 + bx + c, where a, b, and c are numbers with a not equal to zero. The graph of a quadratic function is a curve called a parabola. Parabolas may open upward or downward and vary in "width" or "steepness", but they all have the same basic "U" shape.

quadratic inequality

A quadratic inequality is a function whose degree is 2 and where the y is not always exactly equal to the function. These types of functions use symbols called inequality symbols that include the symbols we know as less than, greater than, less than or equal to, and greater than or equal to.

radical sign

A radical expression involving roots. The radical symbol (√), used to indicate the square root and other roots.

zero of a function

In other words, a "zero" of a function is an input value that produces an output of zero (0). If the function maps real numbers to real numbers, its zeroes are the x-coordinates of the points where its graph meets the x-axis. An alternative name for such a point (x,0) in this context is an x-intercept.

standard form of a quadratic function

Is a function that can be written in the form f(x) = ax^2 + bx + c where a, b, and c are real numbers and a = 0